tag:blogger.com,1999:blog-16732447063249410012024-03-14T06:00:01.257-04:00Autotelic ComputingChristian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.comBlogger34125tag:blogger.com,1999:blog-1673244706324941001.post-44232808695186755932015-03-02T23:17:00.000-05:002015-03-07T12:16:53.019-05:00The Knight of Coursera
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<p>I just began the new Coursera MOOC on <a href="https://www.coursera.org/course/probability">probability</a>, given by UPenn Dr. Santosh S. Venkatesh, and at the very first lecture, my interest was piqued by this intriguing idea, called the Chevalier de Méré's Paradox.. If I ask you which is more likely:</p>
<ol>
<li>Getting a "6" four times in four throws of a die</li>
<li>Getting a "double 6" twenty-four times in twenty-four throws of a pair of dice</li>
</ol>
<p>You probably won't come up with actual numbers, but your intuition will align nicely with reality in telling you that while the probability associated with (1) is pretty small.. the probability associated with (2) is astronomically so:</p>
<ol>
<li>$P(all\ 6s) = P(6) ^ 4 = \frac{1}{6} ^ 4 \approx ~0.00077$</li>
<li>$P(all\ \left\{6,6\right\}s) = P(\left\{6,6\right\})^{24} = \frac{1}{36} ^{24} \approx ~4.45 \times 10^{-38}$</li>
</ol>
<p>However if I change the wording of the question slightly:</p>
<ol>
<li>Getting a "6" <em>at least once</em> in four throws of a die</li>
<li>Getting a "double 6" <em>at least once</em> in twenty-four throws of a pair of dice</li>
</ol>
<p>You might be then tempted to answer that they are now both equally likely, by a reasoning which could be similar to this : there are a certain number of independent events and I'm asked about the probability of their <em>union</em>, therefore I must consider the sum of their probabilities, i.e.:</p>
<ol>
<li>$P(at\ least\ one\ 6) = P(6) \cdot 4 = \frac{1}{6} \cdot 4 = \frac{2}{3}$</li>
<li>$P(at\ least\ one\ \left\{6,6\right\}) = P(\left\{6,6\right\}) \cdot 24 = \frac{1}{36} \cdot {24} = \frac{2}{3}$</li>
</ol>
<p>But in that case of course your intuition would be completely wrong (just consider what would happen to the probability of getting a "6" in <em>six</em> throws of a die, under this reasoning). In the case of (1), the independent events that we should be counting are not the individual results, from 1 to 6, but rather, the different <em>configurations</em> in which the four throws could end up, which is quite different. Let's examine a simpler case with only two throws instead of four. The 36 different configurations are:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([(</span><span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">,</span> <span class="s">'Yes'</span> <span class="k">if</span> <span class="mi">6</span> <span class="ow">in</span> <span class="p">{</span><span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">}</span> <span class="k">else</span> <span class="s">''</span><span class="p">)</span>
<span class="k">for</span> <span class="n">t1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="k">for</span> <span class="n">t2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)],</span>
<span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">'Throw 1'</span><span class="p">,</span> <span class="s">'Throw 2'</span><span class="p">,</span> <span class="s">'Has 6'</span><span class="p">])</span>
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<th></th>
<th>Throw 1</th>
<th>Throw 2</th>
<th>Has 6</th>
</tr>
</thead>
<tbody>
<tr>
<th>0 </th>
<td> 1</td>
<td> 1</td>
<td> </td>
</tr>
<tr>
<th>1 </th>
<td> 1</td>
<td> 2</td>
<td> </td>
</tr>
<tr>
<th>2 </th>
<td> 1</td>
<td> 3</td>
<td> </td>
</tr>
<tr>
<th>3 </th>
<td> 1</td>
<td> 4</td>
<td> </td>
</tr>
<tr>
<th>4 </th>
<td> 1</td>
<td> 5</td>
<td> </td>
</tr>
<tr>
<th>5 </th>
<td> 1</td>
<td> 6</td>
<td> Yes</td>
</tr>
<tr>
<th>6 </th>
<td> 2</td>
<td> 1</td>
<td> </td>
</tr>
<tr>
<th>7 </th>
<td> 2</td>
<td> 2</td>
<td> </td>
</tr>
<tr>
<th>8 </th>
<td> 2</td>
<td> 3</td>
<td> </td>
</tr>
<tr>
<th>9 </th>
<td> 2</td>
<td> 4</td>
<td> </td>
</tr>
<tr>
<th>10</th>
<td> 2</td>
<td> 5</td>
<td> </td>
</tr>
<tr>
<th>11</th>
<td> 2</td>
<td> 6</td>
<td> Yes</td>
</tr>
<tr>
<th>12</th>
<td> 3</td>
<td> 1</td>
<td> </td>
</tr>
<tr>
<th>13</th>
<td> 3</td>
<td> 2</td>
<td> </td>
</tr>
<tr>
<th>14</th>
<td> 3</td>
<td> 3</td>
<td> </td>
</tr>
<tr>
<th>15</th>
<td> 3</td>
<td> 4</td>
<td> </td>
</tr>
<tr>
<th>16</th>
<td> 3</td>
<td> 5</td>
<td> </td>
</tr>
<tr>
<th>17</th>
<td> 3</td>
<td> 6</td>
<td> Yes</td>
</tr>
<tr>
<th>18</th>
<td> 4</td>
<td> 1</td>
<td> </td>
</tr>
<tr>
<th>19</th>
<td> 4</td>
<td> 2</td>
<td> </td>
</tr>
<tr>
<th>20</th>
<td> 4</td>
<td> 3</td>
<td> </td>
</tr>
<tr>
<th>21</th>
<td> 4</td>
<td> 4</td>
<td> </td>
</tr>
<tr>
<th>22</th>
<td> 4</td>
<td> 5</td>
<td> </td>
</tr>
<tr>
<th>23</th>
<td> 4</td>
<td> 6</td>
<td> Yes</td>
</tr>
<tr>
<th>24</th>
<td> 5</td>
<td> 1</td>
<td> </td>
</tr>
<tr>
<th>25</th>
<td> 5</td>
<td> 2</td>
<td> </td>
</tr>
<tr>
<th>26</th>
<td> 5</td>
<td> 3</td>
<td> </td>
</tr>
<tr>
<th>27</th>
<td> 5</td>
<td> 4</td>
<td> </td>
</tr>
<tr>
<th>28</th>
<td> 5</td>
<td> 5</td>
<td> </td>
</tr>
<tr>
<th>29</th>
<td> 5</td>
<td> 6</td>
<td> Yes</td>
</tr>
<tr>
<th>30</th>
<td> 6</td>
<td> 1</td>
<td> Yes</td>
</tr>
<tr>
<th>31</th>
<td> 6</td>
<td> 2</td>
<td> Yes</td>
</tr>
<tr>
<th>32</th>
<td> 6</td>
<td> 3</td>
<td> Yes</td>
</tr>
<tr>
<th>33</th>
<td> 6</td>
<td> 4</td>
<td> Yes</td>
</tr>
<tr>
<th>34</th>
<td> 6</td>
<td> 5</td>
<td> Yes</td>
</tr>
<tr>
<th>35</th>
<td> 6</td>
<td> 6</td>
<td> Yes</td>
</tr>
</tbody>
</table>
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<p>The third row highlights the configurations where a "6" occurs, and you'll notice that there are 11 of them (and not 12, as intuition would lead you to falsely believe), which entails that:</p>
<p>$ P(at\ least\ a\ 6\ in\ two\ throws) = \frac{11}{36} = 0.30\overline{5} $</p>
<p>We notice that this is a bit less than $\frac{1}{3}$ of course. By a similar reasoning, and by noticing that it's actually easier to count the configurations which we must reject (instead of those that are of interest), we arrive at the conclusion that in four throws of a die and twenty-four throws of a pair of dice:</p>
<ol>
<li>$P(at\ least\ one\ 6) = 1 - \frac{5^4}{6^4} \approx 0.512 $</li>
<li>$P(at\ least\ one\ \left\{6,6\right\}) = 1 - \frac{35^{24}}{36^{24}} \approx 0.491 $</li>
</ol>
<p>Finally, if we are still a bit skeptical, nothing beats a little simulation to bring intuition, math and reality together:</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">random</span> <span class="k">import</span> <span class="n">randint</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100000</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[{</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)}</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="k">if</span> <span class="mi">6</span> <span class="ow">in</span> <span class="n">s</span><span class="p">])</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[{(</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">24</span><span class="p">)}</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="k">if</span> <span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="ow">in</span> <span class="n">s</span><span class="p">])</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
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<pre>0.51723
0.49148
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</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-32880101928886884222014-09-05T12:20:00.000-04:002014-09-05T12:34:39.075-04:00An Unpredictable Bug<p>There's something deeply fascinating in the fact that complexity can sometimes emerge from simple rules.. Here's a <a href="http://en.wikipedia.org/wiki/Langton%27s_ant">Langton's ant</a>, which is always headed in one of four directions (up, down, left, right), which it changes before going a new step:</p>
<ul>
<li>By turning 90° right on a white pixel</li>
<li>Or by turning 90° left on a black pixel</li>
</ul>
<p>This simple pattern leads to a surprinsingly chaotic behavior..</p>
<div id="counter">[Press the Start button below..]</div>
<canvas id="grid"></canvas><br />
<button onclick="start()">Start</button>
<button onclick="stop()">Stop</button>
<script>
// http://stackoverflow.com/a/11798652/787842
var ctx = document.getElementById('grid').getContext('2d');
var div = document.getElementById('counter');
ctx.canvas.width = ctx.canvas.height = 512;
var speed = 100;
ctx.strokeStyle = "lightgray";
var size = 8;
var grid_shape = [ctx.canvas.width / size, ctx.canvas.height / size];
var grid = [];
for (var i = 0; i < grid_shape[0]; i++) {
grid[i] = [];
for (var j = 0; j < grid_shape[1]; j++) {
grid[i][j] = false;
}
}
// draw grid
var w = ctx.canvas.width,
h = ctx.canvas.height;
ctx.beginPath();
for (var x = 0; x <= w; x += size) {
ctx.moveTo(x - 0.5, 0);
ctx.lineTo(x - 0.5, h);
}
for (var y = 0; y <= h; y += size) {
ctx.moveTo(0, y - 0.5);
ctx.lineTo(w, y - 0.5);
}
ctx.stroke();
function gridAt(p) {
return grid[p[0]][p[1]];
}
// turn([0, 1], true) -> [1, 0]
function turn(d, right) {
return right ? [d[1], -d[0]] : [-d[1], d[0]];
}
// go([10, 10], [0, 1]) -> [10, 11]
function go(pos, dir) {
return [pos[0] + dir[0], pos[1] + dir[1]];
}
function flip(p) {
var i = p[0],
j = p[1],
j_inv = grid_shape[1] - j - 1;
ctx.fillStyle = 'black';
if (gridAt(p)) {
ctx.clearRect(i * size, j_inv * size, size, size);
grid[i][j] = false;
} else {
ctx.fillRect(i * size, j_inv * size, size, size);
grid[i][j] = true;
}
ctx.stroke();
}
var ant_dir = [-1, 0]; // left
var ant_pos = [grid_shape[0] / 2, grid_shape[1] / 2];
var n_iters = 0;
var interval = null;
ctx.fillStyle = 'red';
ctx.fillRect(ant_pos[0] * size, (grid_shape[1] - ant_pos[1] - 1) * size, size, size);
function start() {
if (interval !== null) {
return;
}
interval = setInterval(function() {
if (n_iters > 11000) {
clearInterval(interval);
return;
}
ant_dir = turn(ant_dir, !gridAt(ant_pos));
flip(ant_pos);
ant_pos = go(ant_pos, ant_dir);
ctx.fillStyle = 'red';
ctx.fillRect(ant_pos[0] * size, (grid_shape[1] - ant_pos[1] - 1) * size, size, size);
div.textContent = n_iters + ' iterations';
n_iters++;
}, speed);
}
function stop() {
if (interval !== null) {
clearInterval(interval);
interval = null;
}
}
</script>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-75022864088450346062014-06-17T23:21:00.000-04:002015-08-04T10:24:23.402-04:00DBSCAN Blues
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My previous machine learning posts:
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<ul>
<li><a href="http://cjauvin.blogspot.ca/2014/03/k-means-vs-louvain.html" target="_blank">K-means vs Louvain</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/12/robustified-linear-regression.html" target="_blank">Robustified Linear Regression</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/neural-network-101.html" target="_blank">Neural Network 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/softmax-regression-101.html" target="_blank">Softmax Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/logistic-regression-101.html" target="_blank">Logistic Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html" target="_blank">Linear Regression 101</a></li>
</ul>
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<p>Suppose I give you this dataset of 2D points (with the added twist that the generating code has been obfuscated, to avoid making the answer too obvious), and ask you to group them into \(k\) meaningful clusters. Is there a way to do it without knowing (or guessing) \(k\) in advance, as \(k\)-means would require? Yes, with the DBSCAN algorithm, but at the expense of introducing <em>two</em> additional parameters! In this post, I try to show why it's not as bad as it may sound by first describing the intuition behind DBSCAN, fiddling next with the problem of parameter estimation and finally showing an equivalent way of looking at it, in terms of graph operations.</p>
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">base64</span>
<span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c"># this obfuscated code just defines a 'points' dataset, as a numpy array</span>
<span class="n">obfuscated_code</span> <span class="o">=</span> <span class="p">(</span><span class="s">'cG9pbnRzMCA9IHJhbmRvbS5tdWx0aXZhcmlhdGVfbm9ybWFs'</span>
<span class="s">'KFswLCAwXSwgZXllKDIpLCAzMDApCnBvaW50czEgPSByYW5k'</span>
<span class="s">'b20ubXVsdGl2YXJpYXRlX25vcm1hbChbNSwgNV0sIGV5ZSgy'</span>
<span class="s">'KSArIDIsIDMwMCkKcG9pbnRzMiA9IHJhbmRvbS5tdWx0aXZh'</span>
<span class="s">'cmlhdGVfbm9ybWFsKFs4LCAyXSwgZXllKDIpLCAzMDApCnBv'</span>
<span class="s">'aW50cyA9IHZzdGFjaygocG9pbnRzMCwgcG9pbnRzMSwgcG9p'</span>
<span class="s">'bnRzMikp'</span><span class="p">)</span>
<span class="nb">eval</span><span class="p">(</span><span class="nb">compile</span><span class="p">(</span><span class="n">base64</span><span class="o">.</span><span class="n">b64decode</span><span class="p">(</span><span class="n">obfuscated_code</span><span class="p">),</span> <span class="s">'<string>'</span><span class="p">,</span><span class="s">'exec'</span><span class="p">))</span>
<span class="c"># 'points' is defined at this point</span>
<span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">5</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">);</span>
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</div>
</div>
</div>
</div>
</div>
<div class="text_cell_render border-box-sizing rendered_html">
<p>Like \(k\)-means, DBSCAN is quite a simple idea. But to be honest, I find the formal introduction (from the <a href="http://en.wikipedia.org/wiki/DBSCAN#Preliminaries">Wikipedia</a> article) slightly confusing, whereas the basic idea is really simple: a point is included in a cluster if either:</p>
<ol style="list-style-type: decimal">
<li>It's surrounded by at least a certain number of neighbors inside a certain radius.</li>
<li>It's inside a certain distance of a point as defined in 1.</li>
</ol>
<p>Let's call the first type <em>core</em> points, and the second <em>reachable</em> points. The remaining points (if any) are <em>noise</em>, and we don't care about them (contrary to \(k\)-means, DBSCAN can easily result in uncategorized points, which can be meaningful for certain applications).</p>
<p>So first things first, if we are to implement an algorithm based on the notions of <em>distance</em> and <em>neighborhoods</em>, we'd better have at our disposal an efficient way of retrieving the nearest neighbors of a point in terms of a certain distance metric. The naive way of doing it (computing every pairwise possibilities) would be \(O(n^2)\), but it turns out that for relatively low-dimensional (i.e. < 20-<em>ish</em>) Euclidean spaces, a <a href="http://en.wikipedia.org/wiki/K-d_tree">k-d tree</a> is a very reasonable and efficient way of doing it. So here's how to use a <code>scipy.cKDTree</code> to retrieve all the neighbors inside a radius of 1 of the first dataset point:</p>
</div>
<div class="cell border-box-sizing code_cell vbox">
<div class="input hbox">
<div class="prompt input_prompt">
In [2]:
</div>
<div class="input_area box-flex1">
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">scipy.spatial</span>
<span class="n">tree</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">spatial</span><span class="o">.</span><span class="n">cKDTree</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">query_ball_point</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">75</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">neighbors</span><span class="p">[</span><span class="mi">0</span><span class="p">]]),</span> <span class="n">color</span><span class="o">=</span><span class="s">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s">'b'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">Circle</span><span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">25</span><span class="p">));</span>
</pre></div>
</div>
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<p>We now have everything we need to define and implement the DBSCAN algorithm. Although Python is itself stylistiscally very close to pseudocode, the essence of the algorithm can be summarized in words as: for every unvisited point with enough neighbors, start a cluster by adding them all in, and then, for each, recursively expand the cluster if they also have enough neighbors, and stop expanding otherwise. Although I used the word "recursively", you'll notice that our implementation is actually not:</p>
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<div class="highlight"><pre><span class="k">def</span> <span class="nf">DBSCAN</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">min_pts</span><span class="p">):</span>
<span class="n">tree</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">spatial</span><span class="o">.</span><span class="n">cKDTree</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">query_ball_point</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="n">clusters</span> <span class="o">=</span> <span class="p">[]</span> <span class="c"># list of (set, set)'s, to distinguish </span>
<span class="c"># core/reachable points</span>
<span class="n">visited</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)):</span>
<span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">>=</span> <span class="n">min_pts</span><span class="p">:</span>
<span class="n">clusters</span><span class="o">.</span><span class="n">append</span><span class="p">(({</span><span class="n">i</span><span class="p">},</span> <span class="nb">set</span><span class="p">()))</span> <span class="c"># core</span>
<span class="n">to_merge_in_cluster</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="k">while</span> <span class="n">to_merge_in_cluster</span><span class="p">:</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">to_merge_in_cluster</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">j</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
<span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">>=</span> <span class="n">min_pts</span><span class="p">:</span>
<span class="n">to_merge_in_cluster</span> <span class="o">|=</span> <span class="nb">set</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">([</span><span class="n">j</span> <span class="ow">in</span> <span class="n">c[0] | c[1]</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">clusters</span><span class="p">]):</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">>=</span> <span class="n">min_pts</span><span class="p">:</span>
<span class="n">clusters</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="c"># core</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">clusters</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="c"># reachable</span>
<span class="k">return</span> <span class="n">clusters</span>
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<p>Now that we have an algorithm, you may ask a very legitimate question: how do we choose <code>eps</code> (the minimal distance/radius) and <code>min_pts</code> (the minimal number of points inside that radius to start/expand a cluster)? If we pick an arbitrary value for <code>min_pts</code> (say 10), we could try looking at the distribution of distances separating every point from their 10 nearest neighbors:</p>
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<div class="highlight"><pre><span class="n">min_pts</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">nn_dists</span> <span class="o">=</span> <span class="n">ravel</span><span class="p">(</span><span class="n">tree</span><span class="o">.</span><span class="n">query</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="n">min_pts</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">0</span><span class="p">][:,</span> <span class="mi">1</span><span class="p">:])</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">nn_dists</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">*</span> <span class="n">min_pts</span>
<span class="n">hist</span><span class="p">(</span><span class="n">nn_dists</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">);</span>
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<p>So it would seem that a value of <code>eps</code> somewhere between 0.25 and 1 would make sense.. let's try:</p>
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In [5]:
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<div class="highlight"><pre><span class="n">clusters</span> <span class="o">=</span> <span class="n">DBSCAN</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="n">min_pts</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">75</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="s">'rbgycm'</span> <span class="o">*</span> <span class="mi">10</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">core</span><span class="p">,</span> <span class="n">reachable</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clusters</span><span class="p">):</span>
<span class="n">core</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">core</span><span class="p">);</span> <span class="n">reachable</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">reachable</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">core</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">reachable</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">4</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
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<p>Clearly not working very well.. I'm a somewhat suspicious coder: could it be a bug with our implementation of DBSCAN? To verify, let's try <a href="http://scikit-learn.org/stable/modules/generated/sklearn.cluster.DBSCAN.html">scikit-learn</a>'s version, which is certainly right:</p>
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In [6]:
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">sklearn.cluster</span>
<span class="n">clusters</span> <span class="o">=</span> <span class="n">sklearn</span><span class="o">.</span><span class="n">cluster</span><span class="o">.</span><span class="n">DBSCAN</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">min_pts</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">75</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">clusters</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span> <span class="k">continue</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">clusters</span> <span class="o">==</span> <span class="n">i</span><span class="p">)[</span><span class="mi">0</span><span class="p">]]),</span>
<span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)],</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
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">
</div>
</div>
</div>
</div>
</div>
<div class="text_cell_render border-box-sizing rendered_html">
<p>Identical results (apart from the color code).. which spells good news for our implementation, but bad news for our choice of parameters.. let's see what happens with <code>eps = 0.5</code>:</p>
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In [7]:
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<div class="highlight"><pre><span class="n">clusters</span> <span class="o">=</span> <span class="n">DBSCAN</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">min_pts</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">75</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">core</span><span class="p">,</span> <span class="n">reachable</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clusters</span><span class="p">):</span>
<span class="n">core</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">core</span><span class="p">);</span> <span class="n">reachable</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">reachable</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">core</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">[</span><span class="n">reachable</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">4</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
</pre></div>
</div>
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<p>All right, much better (also note the graphical distinction between <em>core</em> and <em>reachable</em> points, the latter being a bit paler)! And it wasn't that hard to come up with reasonable parameter values.. but are we really convinced yet that DBSCAN is such an improvement over \(k\)-means? Picking a value for <code>min_pts</code> was still pretty much arbitrary.. To continue our exploration, let's create a dataset which would be very hard (in fact impossible) for \(k\)-means to make sense of:</p>
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In [8]:
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<div class="highlight"><pre><span class="n">inner</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">500</span><span class="p">)</span>
<span class="n">rr</span><span class="p">,</span> <span class="n">rn</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">,</span> <span class="n">random</span><span class="o">.</span><span class="n">normal</span>
<span class="n">outer</span> <span class="o">=</span> <span class="p">[(</span><span class="n">r</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">r</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
<span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">t</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">rn</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">rr</span><span class="p">())</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">500</span><span class="p">)]]</span>
<span class="n">points2</span> <span class="o">=</span> <span class="n">vstack</span><span class="p">((</span><span class="n">inner</span><span class="p">,</span> <span class="n">outer</span><span class="p">))</span>
<span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">points2</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points2</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">4</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">);</span>
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In [9]:
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<div class="highlight"><pre><span class="n">clusters</span> <span class="o">=</span> <span class="n">DBSCAN</span><span class="p">(</span><span class="n">points2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">min_pts</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points2</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">75</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">core</span><span class="p">,</span> <span class="n">reachable</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clusters</span><span class="p">):</span>
<span class="n">core</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">core</span><span class="p">);</span> <span class="n">reachable</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">reachable</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points2</span><span class="p">[</span><span class="n">core</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">points2</span><span class="p">[</span><span class="n">reachable</span><span class="p">]),</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">4</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
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<p>Very reasonable and interesting results! To conclude, let's examine an alternative way of looking at the DBSCAN algorithm, from the perspective of graph theory. It turns out that DBSCAN is equivalent to finding the <a href="http://en.wikipedia.org/wiki/Connected_component_(graph_theory">connected components</a> of a graph defined as: "for every node, draw an edge to every neighbors inside a certain radius (<code>eps</code>), if they are in sufficient number (<code>min_pts</code>)". This is very easy to implement using the <a href="https://networkx.github.io/">networkx</a> library, and yields the expected results:</p>
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">query_ball_point</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="n">g</span><span class="o">.</span><span class="n">add_nodes_from</span><span class="p">(</span><span class="nb">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)))</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">with_labels</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">node_size</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="n">node_color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">5</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">width</span><span class="o">=.</span><span class="mi">5</span><span class="p">)</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)):</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neighbors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">>=</span> <span class="mi">10</span><span class="p">:</span>
<span class="n">g</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">([(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">neighbors</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">comp</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">connected_component_subgraphs</span><span class="p">(</span><span class="n">g</span><span class="p">)):</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">with_labels</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">node_size</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="n">node_color</span><span class="o">=</span><span class="p">[</span><span class="o">.</span><span class="mi">5</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span><span class="p">,</span> <span class="n">edge_color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">width</span><span class="o">=.</span><span class="mi">5</span><span class="p">)</span>
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<p>Although interesting, this equivalence of course does not yield a more efficient algorithm, because it still relies on the neighborhood-finding helper (in our case a \(k\)-d tree), which dominates the overall complexity by being \(O(n \log n)\).</p>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-34390184517681207502014-03-20T12:49:00.000-04:002014-03-21T09:22:07.060-04:00K-means vs Louvain
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<p>I'm currently reading <a href="http://www.amazon.com/Data-Smart-Science-Transform-Information-ebook/dp/B00F0WRXI0/ref=sr_1_1?ie=UTF8&qid=1395270836&sr=8-1&keywords=data+smart">Data Smart</a>, a particularly good and entertaining book about machine learning and data science. The somewhat surprising twist in its approach is that it does almost entirely without code, and instead implements and illustrates everything in terms of spreadsheet operations. This has produced an interesting contradiction in me: the coding purist is somewhat put off by what he always perceived as a clunky and underpowered computing paradigm, while the pragmatist is intrigued by the possible productivity gains that might result from taking the time to learn properly about such an ubiquitous tool (which I'll admit I never did). That said, the author does an excellent job at teaching complex algorithms with (not so simple) spreadsheets, which after a while almost feels like a "declarative" way of modeling problems, a refeshing take I think for people used to a more procedural way of thinking.</p>
<p>At some point though the use of a solver becomes inevitable for such problems (which almost always imply an optimization component), and this is another aspect of the book that I found surprisingly enlightening. Instead of delving in the intricacies of particular algorithms, it provides a unified and abstract methodology, useful to solve a large class of problems, and allows to get a deeper feel of the way they're related. The price there is to pay is that using an embedded solver can often be less efficient than a specialized algorithm, and in my case, since I only have access to LibreOffice, the pain is particularly acute for certain problems.</p>
<p>In the second chapter of the book, we learn about <a href="http://en.wikipedia.org/wiki/K-means"><span class="math">\(k\)</span>-means</a> clustering using a toy dataset in which we have the selections made by 100 clients among 32 distinct wine deals. The clients get clustered in the 32-dimension space spanned by their wine tastes, which is easy to understand in terms of abstract geometry, but rather hard to visualize. <a href="http://en.wikipedia.org/wiki/Cosine_similarity">Cosine similarity</a> is introduced as a metric making more sense and yielding better results than <a href="http://en.wikipedia.org/wiki/Euclidean_distance">Euclidean distance</a> in this particular context. Then a few chapters later, the same problem is revisited, but this time from the perspective of graph theory, from which a very clever clustering method has been devised, based on the concept of <a href="http://en.wikipedia.org/wiki/Modularity_(networks)">modularity</a> (clustering in this context is rather called <a href="http://en.wikipedia.org/wiki/Community_structure">community detection</a>). A graph is first constructed from the cosine similarity matrix, which can be literally interpreted as an <a href="http://en.wikipedia.org/wiki/Adjacency_matrix">adjacency matrix</a>. We then cluster the nodes of this graph according to whether they share an edge or not, but with the adjustment that highly probable connections are less important, and vice versa. This can be solved with an algorithm called the <a href="http://perso.uclouvain.be/vincent.blondel/research/louvain.html">Louvain method</a>.</p>
<p>Since I decided to follow along using Python, I thought it would be nice to use the graph visualization to compare the results of <span class="math">\(k\)</span>-means clustering against those of modularity maximization. Using a couple of very powerful Python libraries (Numpy, Scikit-Learn, NetworkX and Matplotlib), this is really easy. Let's first download and extract the data directly from the source:</p>
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics.pairwise</span> <span class="kn">import</span> <span class="n">cosine_similarity</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">KMeans</span>
<span class="kn">import</span> <span class="nn">urllib</span><span class="o">,</span> <span class="nn">zipfile</span><span class="o">,</span> <span class="nn">StringIO</span><span class="o">,</span> <span class="nn">xlrd</span><span class="o">,</span> <span class="nn">community</span>
<span class="c"># download zipped data file</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">urllib</span><span class="o">.</span><span class="n">urlopen</span><span class="p">(</span><span class="s">'http://media.wiley.com/product_ancillary/6X/11186614/DOWNLOAD/ch02.zip'</span><span class="p">)</span>
<span class="c"># unzip it</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">zipfile</span><span class="o">.</span><span class="n">ZipFile</span><span class="p">(</span><span class="n">StringIO</span><span class="o">.</span><span class="n">StringIO</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">read</span><span class="p">()))</span>
<span class="c"># extract the Excel file</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">xlrd</span><span class="o">.</span><span class="n">open_workbook</span><span class="p">(</span><span class="n">file_contents</span><span class="o">=</span><span class="n">z</span><span class="o">.</span><span class="n">read</span><span class="p">(</span><span class="s">'ch02/WineKMC.xlsx'</span><span class="p">))</span>
<span class="n">matrix_sheet</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">sheet_by_name</span><span class="p">(</span><span class="s">'Matrix'</span><span class="p">)</span>
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<p>Next let's read its content to feed a <span class="math">\(32 \times 100\)</span> numpy matrix, which we then transpose to obtain a dataset of customers and their wine preferences:</p>
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<div class="highlight"><pre><span class="c"># input data</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">32</span><span class="p">,</span> <span class="mi">100</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">33</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">107</span><span class="p">):</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">matrix_sheet</span><span class="o">.</span><span class="n">cell_value</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
<span class="k">if</span> <span class="n">v</span><span class="p">:</span> <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">T</span> <span class="c"># 100 x 32</span>
<span class="n">D</span>
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array([[ 0., 0., 0., ..., 1., 0., 0.],
[ 0., 0., 0., ..., 0., 0., 0.],
[ 0., 0., 0., ..., 0., 0., 0.],
...,
[ 1., 0., 0., ..., 0., 1., 0.],
[ 0., 0., 0., ..., 0., 0., 0.],
[ 0., 0., 0., ..., 0., 1., 1.]])
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<p>We compute the cosine similarities of the clients, which will become an adjacency matrix, after having been constrained by the <span class="math">\(r\)</span>-neighborhood (with <span class="math">\(r=0.5\)</span>):</p>
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<div class="highlight"><pre><span class="n">S</span> <span class="o">=</span> <span class="n">cosine_similarity</span><span class="p">(</span><span class="n">D</span><span class="p">)</span> <span class="c"># 100 x 100</span>
<span class="n">S</span><span class="p">[</span><span class="n">S</span> <span class="o"><=</span> <span class="mf">0.49999</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span> <span class="c"># using < 0.5 is not working well here for some reason</span>
<span class="n">S</span><span class="p">[</span><span class="n">S</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">S</span>
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array([[ 1., 0., 0., ..., 0., 0., 0.],
[ 0., 1., 0., ..., 0., 0., 0.],
[ 0., 0., 1., ..., 0., 0., 0.],
...,
[ 0., 0., 0., ..., 1., 0., 0.],
[ 0., 0., 0., ..., 0., 1., 0.],
[ 0., 0., 0., ..., 0., 0., 1.]])
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<p>We construct the graph from this matrix, and apply community detection on it:</p>
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<div class="highlight"><pre><span class="c"># S as adjacency matrix</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_numpy_matrix</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="c"># Louvain method for community detection</span>
<span class="n">partition</span> <span class="o">=</span> <span class="n">community</span><span class="o">.</span><span class="n">best_partition</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
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<p>We can draw the results:</p>
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<div class="highlight"><pre><span class="n">rcParams</span><span class="p">[</span><span class="s">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">8</span>
<span class="n">pos</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">spring_layout</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="s">'bgrcmykw'</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">com</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">partition</span><span class="o">.</span><span class="n">values</span><span class="p">())):</span>
<span class="n">list_nodes</span> <span class="o">=</span> <span class="p">[</span><span class="n">nodes</span> <span class="k">for</span> <span class="n">nodes</span> <span class="ow">in</span> <span class="n">partition</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span>
<span class="k">if</span> <span class="n">partition</span><span class="p">[</span><span class="n">nodes</span><span class="p">]</span> <span class="o">==</span> <span class="n">com</span><span class="p">]</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx_nodes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">list_nodes</span><span class="p">,</span>
<span class="n">node_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">node_color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">);</span>
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<p>Finally we have a way to visualize and compare the results obtained with <span class="math">\(k\)</span>-means:</p>
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In [6]:
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<div class="highlight"><pre><span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">partition</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span> <span class="c"># use number of clusters found by MM (8)</span>
<span class="n">kmeans</span> <span class="o">=</span> <span class="n">KMeans</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
<span class="n">kmeans</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="n">list_nodes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">kmeans</span><span class="o">.</span><span class="n">labels_</span> <span class="o">==</span> <span class="n">i</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx_nodes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">list_nodes</span><span class="p">,</span>
<span class="n">node_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">node_color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">nx</span><span class="o">.</span><span class="n">draw_networkx_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">);</span>
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</div>
</div>
</div>
</div>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-85457553138811858592013-12-12T18:38:00.000-05:002013-12-12T18:38:52.201-05:00Robustified Linear Regression<div class="text_cell_render border-box-sizing rendered_html">
This post is part of a series, see also:
</div>
<ol>
<li><a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html" target="_blank">Linear Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/logistic-regression-101.html" target="_blank">Logistic Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/softmax-regression-101.html" target="_blank">Softmax Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/neural-network-101.html" target="_blank">Neural Network 101</a></li>
</ol>
<div class="text_cell_render border-box-sizing rendered_html">
<p>Let's revisit our toy <a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html">linear regression problem</a>, and in particular, look at what happens when the data contains a lot of outliers.</p>
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In [1]:
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<div class="highlight"><pre><span class="n">rcParams</span><span class="p">[</span><span class="s">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">8</span>
<span class="n">domain</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">dataset</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">intercept</span> <span class="o">+</span> <span class="n">noise</span>
<span class="k">return</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">dataset</span><span class="p">(</span><span class="mi">250</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="c"># add extra noise</span>
<span class="n">extra_noise</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">([</span>
<span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">100</span><span class="p">),</span>
<span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">60</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="p">])</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">vstack</span><span class="p">([</span><span class="n">points</span><span class="p">,</span> <span class="n">extra_noise</span><span class="p">])</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="mf">2.5</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="s">'r--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'Generator'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
</pre></div>
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</div>
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<p>If we reuse our <a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html">least square gradient descent</a> method, we see that it does not perform as well anymore.</p>
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<div class="highlight"><pre><span class="k">def</span> <span class="nf">least_square_linear_regression</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.00001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">):</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">initial_theta</span>
<span class="n">prev_mse</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s">'inf'</span><span class="p">)</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">(</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
<span class="n">mse</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="k">if</span> <span class="n">mse</span> <span class="o">-</span> <span class="n">prev_mse</span> <span class="o"><</span> <span class="n">stop</span><span class="p">:</span> <span class="k">break</span>
<span class="n">prev_mse</span> <span class="o">=</span> <span class="n">mse</span>
<span class="k">return</span> <span class="n">theta</span>
<span class="n">initial_theta</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">ls_theta</span> <span class="o">=</span> <span class="n">least_square_linear_regression</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">)</span>
<span class="k">print</span> <span class="n">ls_theta</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="n">ls_theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">ls_theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="s">'g--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'Least Square'</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
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[ 0.67349646 1.13530362]
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<p>To mitigate the problem, we can try a completely different way of searching for the right set of parameters, not based on either least squares or gradient descent: the <a href="http://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator">Theil-Sen estimator</a>. This non-parametric method is very simple: set the slope <span class="math">\(m\)</span> of the model as the median of the slopes resulting from each pair of training points (i.e. <span class="math">\((y_j − y_i)/(x_j − x_i))\)</span> for every pair <span class="math">\((i, j)\)</span>). Once set, find the intercept <span class="math">\(b\)</span> in a similar way, as the median of <span class="math">\(y_i − m x_i\)</span> for every <span class="math">\(i\)</span>. The results show that the solution obtained is really more robust, as it's less sensitive to outliers. The naive version of this algorithm, which is used here, while nice because it holds in two lines of Python, wouldn't work very well however on a bigger dataset, because of its quadratic running time (due to the fact that we're enumerating the <span class="math">\({{n}\choose{2}}\)</span> combinations). There are however more efficient algorithms, based on sorting, that can do it in <span class="math">\(O(n \log n)\)</span>.</p>
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<div class="highlight"><pre><span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</span>
<span class="c"># WARNING! Naive code below!</span>
<span class="k">def</span> <span class="nf">theil_sen_linear_regression</span><span class="p">(</span><span class="n">points</span><span class="p">):</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">median</span><span class="p">([(</span><span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">/</span> <span class="p">(</span><span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">median</span><span class="p">([</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">m</span> <span class="o">*</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">points</span><span class="p">])</span>
<span class="k">return</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="n">ts_theta</span> <span class="o">=</span> <span class="n">theil_sen_linear_regression</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="k">print</span> <span class="n">ts_theta</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="n">ls_theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">ls_theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="s">'r--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'Least Square'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="n">ts_theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">ts_theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="s">'g--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'Theil-Sen'</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
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(-1.498047457480939, 2.2686431432094376)
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Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-88832184830535932632013-10-16T11:34:00.000-04:002013-12-12T18:18:11.068-05:00Neural Network 101<div class="text_cell_render border-box-sizing rendered_html">
As this seems to be becoming a series of related posts, see also:
</div>
<ol>
<li><a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html" target="_blank">Linear Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/logistic-regression-101.html" target="_blank">Logistic Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/softmax-regression-101.html" target="_blank">Softmax Regression 101</a></li>
<li>Neural Network 101</li>
</ol>
<div class="text_cell_render border-box-sizing rendered_html">
<h2 id="A-Cloud-of-Points">A Cloud of Points<a class="anchor-link" href="#A-Cloud-of-Points">¶</a></h2>
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<p>As powerful as it is, a logistic regression model will only ever be able to solve <em>linearly separable</em> problems. No need to even try on problems like the one below (even though it's quite easy), because we saw that its <span class="math">\(\theta\)</span> parameters correspond directly to the parameters of a linear equation in general form.</p>
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<div class="highlight"><pre><span class="n">rcParams</span><span class="p">[</span><span class="s">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">8</span>
<span class="k">def</span> <span class="nf">dataset</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="n">xy</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">low</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">classes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">xy</span><span class="p">:</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
<span class="n">classes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span> <span class="k">if</span> <span class="n">d</span> <span class="o"><</span> <span class="mf">0.75</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">xy</span><span class="p">,</span> <span class="n">classes</span><span class="p">])</span>
<span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">dataset</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
<span class="n">class0</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)]</span>
<span class="n">class1</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class0</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class1</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">);</span>
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">
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<h2 id="Introducing-Non-Linearity">Introducing Non-Linearity<a class="anchor-link" href="#Introducing-Non-Linearity">¶</a></h2>
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<p>Although it's not 100% exact, one way to understand the classic neural network (or multilayer perceptron, as it is also sometimes called) is as an extension to the logistic regression model. If we recast our terminology in terms of <em>neurons</em>, <em>layers</em> and <em>weights</em> (where a neuron performs two successive computations: (1) the dot product of the incoming weights and the vector from the layer below it, and (2) the logistic squashing of the result), then LR can be seen as the top part (in the dashed rectangle) of the neural network schema shown below. The bottom part, an additional layer of weights and logistic units, is what introduces non-linearity in the model, and thus allows it to solve problems like the one above.</p>
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<div class="highlight"><pre><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
<span class="n">Image</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">'NN.png'</span><span class="p">)</span>
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<h2 id="Training-to-Get-Better">Training to Get Better<a class="anchor-link" href="#Training-to-Get-Better">¶</a></h2>
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<p>In this simple example, since our only output neuron is logistic, we can interpret its value as a probability (of being a member of class 0 or 1), and once again endow the model with probabilistic semantics (although it is not mandatory). We can again use the negative log-likelihood as our error function</p>
<p><span class="math">\[NLL = - \sum_{i}^{n} t^{(i)} \log y^{(i)} + (1 - t^{(i)}) \log (1 - y^{(i)}) \]</span></p>
<p>where <span class="math">\(y^{(i)}\)</span> corresponds to the output of the neural network when fed with the <span class="math">\(i\)</span>-th training example, and <span class="math">\(t^{(i)}\)</span> the <em>target</em>, its real class membership (0 or 1).</p>
<p>The training of a neural network is a little more involved, because the gradient of the error function needs to be <em>back-propagated</em> in successive computations</p>
<p><span class="math">\[ \frac{\partial NLL}{\partial y} = y - t\]</span></p>
<p><span class="math">\[ \frac{\partial NLL}{\partial net_y} = y \cdot (1 - y) \cdot \frac{\partial NLL}{\partial y} \]</span></p>
<p><span class="math">\[ \frac{\partial NLL}{\partial h} = \frac{\partial NLL}{\partial net_y} \cdot W_{hy} \]</span></p>
<p><span class="math">\[ \frac{\partial NLL}{\partial W_{hy}} = \frac{\partial NLL}{\partial net_y} \cdot h \]</span></p>
<p><span class="math">\[ \frac{\partial NLL}{\partial net_y} = h \cdot (1 - h) \cdot \frac{\partial NLL}{\partial h} \]</span></p>
<p><span class="math">\[ \frac{\partial NLL}{\partial W_{xh}} = \frac{\partial NLL}{\partial net_h} \cdot x \]</span></p>
<p>To finally yield the two weight updating formulas</p>
<p><span class="math">\[ W_{hy} := W_{hy} - \alpha \frac{\partial NLL}{\partial W_{hy}} \]</span></p>
<p><span class="math">\[ W_{xh} := W_{xh} - \alpha \frac{\partial NLL}{\partial W_{xh}} \]</span></p>
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<div class="highlight"><pre><span class="k">def</span> <span class="nf">logistic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
<span class="k">class</span> <span class="nc">NeuralNetwork</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_hiddens</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">W_xh</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n_hiddens</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">W_hy</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_hiddens</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_hiddens</span> <span class="o">=</span> <span class="n">n_hiddens</span>
<span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span>
<span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">n_iters</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">((</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">),</span> <span class="n">x</span><span class="p">))</span> <span class="c"># add input bias</span>
<span class="bp">self</span><span class="o">.</span><span class="n">t</span> <span class="o">=</span> <span class="n">t</span>
<span class="n">errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_iters</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">forward</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
<span class="n">nll</span> <span class="o">=</span> <span class="o">-</span><span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span>
<span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">))</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">n</span>
<span class="n">classif</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">))</span>
<span class="n">errors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nll</span><span class="p">,</span> <span class="n">classif</span><span class="p">))</span>
<span class="k">return</span> <span class="n">errors</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">h</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">((</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">),</span> <span class="c"># add hidden bias</span>
<span class="n">logistic</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">W_xh</span><span class="p">))))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">=</span> <span class="n">logistic</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">W_hy</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">backward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">d_nll_d_y</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">)</span>
<span class="n">d_nll_d_net_y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">d_nll_d_y</span>
<span class="n">d_nll_d_h</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">d_nll_d_net_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">W_hy</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="n">d_nll_d_W_hy</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">d_nll_d_net_y</span><span class="p">)</span>
<span class="n">d_nll_d_net_h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">h</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">h</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:])</span> <span class="o">*</span> <span class="n">d_nll_d_h</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:]</span>
<span class="n">d_nll_d_W_xh</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">d_nll_d_net_h</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">W_hy</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">d_nll_d_W_hy</span>
<span class="bp">self</span><span class="o">.</span><span class="n">W_xh</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">d_nll_d_W_xh</span>
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<p>The number of hidden units controls the representational power of the model. If it is too low, it will not be able to capture the complexity of the training data, as the example below shows.</p>
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<div class="highlight"><pre><span class="n">rcParams</span><span class="p">[</span><span class="s">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">4</span>
<span class="n">nn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">errors</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">points</span><span class="p">[:,:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,[</span><span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
<span class="n">_</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">classif_errors</span> <span class="o">=</span> <span class="n">asarray</span><span class="p">(</span><span class="n">errors</span><span class="p">)[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">classif_errors</span><span class="p">)),</span> <span class="n">classif_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'Classification error'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">points_nn</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">((</span><span class="n">points</span><span class="p">[:,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]],</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">y</span><span class="p">)))</span>
<span class="n">class0_nn</span> <span class="o">=</span> <span class="n">points_nn</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points_nn</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)]</span>
<span class="n">class1_nn</span> <span class="o">=</span> <span class="n">points_nn</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points_nn</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">class0_nn</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0_nn</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">class1_nn</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1_nn</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">);</span>
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</div>
</div>
</div>
</div>
</div>
<div class="text_cell_render border-box-sizing rendered_html">
<p>But if it is set right, there's no limit (in theory) to the complexity of the function that the network can learn.</p>
</div>
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In [15]:
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<div class="highlight"><pre><span class="n">nn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">errors</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">points</span><span class="p">[:,:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,[</span><span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
<span class="n">_</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">classif_errors</span> <span class="o">=</span> <span class="n">asarray</span><span class="p">(</span><span class="n">errors</span><span class="p">)[:,</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">classif_errors</span><span class="p">)),</span> <span class="n">classif_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'Classification error'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">points_nn</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">((</span><span class="n">points</span><span class="p">[:,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]],</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">y</span><span class="p">)))</span>
<span class="n">class0_nn</span> <span class="o">=</span> <span class="n">points_nn</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points_nn</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)]</span>
<span class="n">class1_nn</span> <span class="o">=</span> <span class="n">points_nn</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points_nn</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">class0_nn</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0_nn</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">class1_nn</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1_nn</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">);</span>
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">
</div>
</div>
</div>
</div>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-48367613156124551882013-10-12T10:59:00.002-04:002013-10-16T11:35:10.207-04:00Softmax Regression 101<div class="text_cell_render border-box-sizing rendered_html">
As this seems to be becoming a series of related posts, see also:
</div>
<ol>
<li><a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html" target="_blank">Linear Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/logistic-regression-101.html" target="_blank">Logistic Regression 101</a></li>
<li>Softmax Regression 101</li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/neural-network-101.html" target="_blank">Neural Network 101</a></li>
</ol>
<h2 id="A-Cloud-of-Points">
A Cloud of Points<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#A-Cloud-of-Points">¶</a></h2>
<div class="text_cell_render border-box-sizing rendered_html">
If logistic regression allows to solve binary classification problems, softmax regression, our third example of a generalized linear model, allows to solve multiclass ones. As before, we build an artificial dataset, this time by first picking <span class="math">\(k\)</span> random points, each corresponding to a class. Next, we sample points uniformly, assigning each of them to the class to which it is closest to the center.</div>
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<div class="prompt input_prompt">
In [1]:
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<div class="highlight">
<pre><span class="k">def</span> <span class="nf">dataset</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="n">xy</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">low</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">centers</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">low</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">classes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">xy</span><span class="p">:</span>
<span class="n">classes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">argmin</span><span class="p">([</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">centers</span><span class="p">]))</span>
<span class="k">return</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">xy</span><span class="p">,</span> <span class="n">classes</span><span class="p">]),</span> <span class="n">centers</span>
<span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span> <span class="mi">4</span>
<span class="n">points</span><span class="p">,</span> <span class="n">centers</span> <span class="o">=</span> <span class="n">dataset</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="s">'gbrcmykrcm'</span>
<span class="n">markers</span> <span class="o">=</span> <span class="s">'osd^v1234x'</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="n">class_i</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">i</span><span class="p">)]</span>
<span class="n">marker</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span> <span class="o">+</span> <span class="n">markers</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">markers</span><span class="p">)]</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class_i</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class_i</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">centers</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">centers</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]],</span> <span class="n">marker</span><span class="p">,</span> <span class="n">ms</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
</pre>
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<h2 id="Guessing-an-Initial-Model">
Guessing an Initial Model<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Guessing-an-Initial-Model">¶</a></h2>
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As before we will guess initial values for the <span class="math">\(\theta\)</span> parameters for our model, which will be, in the multiclass setting, a matrix of <span class="math">\(k\)</span> rows, each one having 3 columns, and thus corresponding to the parameters of a linear equation in the general form (<span class="math">\(ax + by + c = 0\)</span>), as with logistic regression.</div>
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In [2]:
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<pre><span class="n">initial_theta</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
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This time however, we will postpone the question of making geometric sense of those parameters, as it is a bit more involved than with logistic regression (we'll come back to it in the end, when the model is trained).</div>
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<h2 id="Training-to-Get-Better">
Training to Get Better<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Training-to-Get-Better">¶</a></h2>
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As softmax regression is still (as all generalized linear models are) a probabilistic method, this time we will use the <em>softmax</em> function to squash the activation of class <span class="math">\(i\)</span> into a probability value<br />
<span class="math">\[ h(x, i) = \frac{e^{x_i}}{\sum_{j}^{k} e^{x_j}} \]</span><br />
The error function for softmax regression is a generalization of the NLL we've been using for logistic regression<br />
<span class="math">\[NLL = - \sum_{i}^{n} \log \prod_{j}^{k} h_{\theta_j}(\vec{x^{(i)}})^{1\{c^{(i)}=j\}} \]</span><br />
The derivative of the NLL, with respect to <span class="math">\(\theta\)</span> again follows the familiar form, with the difference that it is now two-dimensional<br />
<span class="math">\[ \frac{\partial NLL}{\partial \theta_{ij}} = (c - h_{\theta_{i}}(\vec{x})) \cdot \vec{x_{i,j}} \]</span></div>
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<pre><span class="k">def</span> <span class="nf">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">theta</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">sum_exps</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">asarray</span><span class="p">([</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)])),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">sum_exps</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="n">n</span><span class="p">,)</span>
<span class="c"># k x n, where each col should sum to 1</span>
<span class="k">return</span> <span class="n">asarray</span><span class="p">([</span><span class="n">exp</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span> <span class="o">/</span> <span class="n">sum_exps</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)])</span>
<span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">):</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">initial_theta</span><span class="p">)</span>
<span class="n">bxy</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span> <span class="c"># first component is intercept (i.e. all 1s)</span>
<span class="n">thetas</span> <span class="o">=</span> <span class="p">[</span><span class="n">initial_theta</span><span class="p">]</span>
<span class="n">nll_errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">classif_errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:]</span> <span class="c"># copy</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">(</span><span class="n">bxy</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">nll_errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="nb">sum</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">c</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)]))</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
<span class="n">classif_errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sum</span><span class="p">((</span><span class="n">argmax</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)))</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nll_errors</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">nll_errors</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">nll_errors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o"><</span> <span class="n">stop</span><span class="p">:</span> <span class="k">break</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="n">ci</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span> <span class="o">==</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
<span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">(</span><span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">ci</span><span class="p">)</span> <span class="c"># intercept component</span>
<span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">ci</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
<span class="n">theta</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">ci</span><span class="p">)</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
<span class="n">thetas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="k">return</span> <span class="n">thetas</span><span class="p">,</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="n">classif_errors</span>
<span class="n">thetas</span><span class="p">,</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="n">classif_errors</span> <span class="o">=</span> <span class="n">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">)</span>
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Let's verify that the NLL and the classification errors are getting minimized.</div>
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<pre><span class="n">_</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">'Training iterations'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'NLL'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nll_errors</span><span class="p">)),</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">'Training iterations'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'Classification error'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">classif_errors</span><span class="p">)),</span> <span class="n">classif_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
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To visualize the decision lines, we note that even though we have <span class="math">\(k\)</span> instances of (3-component) <span class="math">\(\theta_i\)</span> parameters (which it would be thus tempting to use directly), they don't actually correspond to what we are looking for. Rather, we must consider that as each class must be distinguished from all the other classes, we must be looking for <span class="math">\({{k}\choose{2}}\)</span> decision boundaries, each yielding a pairwise comparison (note that comparing class <span class="math">\(i\)</span> with class <span class="math">\(j\)</span> is the same as the opposite, hence the use of <em>combinations</em>). For each comparison of class <span class="math">\(i\)</span> with class <span class="math">\(j\)</span>, the decision line parameters can be retrieved as <span class="math">\(\theta_{ij} = \theta_i - \theta_j\)</span>.<br /><br />
Even though I invested a good amount of time trying to understand why this last aspect works and what it means, I haven't found a satisfying answer, at least in the form of a good geometric intuition. If anyone can provide additional insight, please jump in.</div>
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<pre><span class="k">def</span> <span class="nf">general_to_slope_intercept</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
<span class="n">slope</span> <span class="o">=</span> <span class="o">-</span><span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">theta</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="n">intercept</span> <span class="o">=</span> <span class="o">-</span><span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">theta</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="k">return</span> <span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span>
<span class="n">best_theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="n">class_i</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">i</span><span class="p">)]</span>
<span class="n">marker</span> <span class="o">=</span> <span class="n">colors</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">colors</span><span class="p">)]</span> <span class="o">+</span> <span class="n">markers</span><span class="p">[</span><span class="n">i</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">markers</span><span class="p">)]</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class_i</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class_i</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="p">)</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span> <span class="o">=</span> <span class="n">general_to_slope_intercept</span><span class="p">(</span><span class="n">best_theta</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</span>
<span class="k">for</span> <span class="n">comb</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">best_theta</span><span class="p">[</span><span class="n">comb</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">-</span> <span class="n">best_theta</span><span class="p">[</span><span class="n">comb</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span> <span class="o">=</span> <span class="n">general_to_slope_intercept</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">intercept</span><span class="p">,</span>
<span class="s">'r-'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">axis</span><span class="p">((</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">));</span>
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" />
</div>
</div>
</div>
</div>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-7353969034149095422013-10-07T15:50:00.003-04:002013-10-16T11:36:10.768-04:00Logistic Regression 101<div class="text_cell_render border-box-sizing rendered_html">
As this seems to be becoming a series of related posts, see also:</div>
<ol>
<li><a href="http://cjauvin.blogspot.ca/2013/10/linear-regression-101.html" target="_blank">Linear Regression 101</a></li>
<li>Logistic Regression 101</li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/softmax-regression-101.html" target="_blank">Softmax Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/neural-network-101.html" target="_blank">Neural Network 101</a></li>
</ol>
<h2 id="A-Cloud-of-Points">
A Cloud of Points<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#A-Cloud-of-Points">¶</a></h2>
<div class="text_cell_render border-box-sizing rendered_html">
We will next tackle the problem of binary supervised classification, using the logistic regression method. The artificial dataset we will use this time is built by sampling random 2D points in the plane, and assigning them a class (0 or 1) with a probability that is proportional to their distance from a line with given slope and intercept parameters. In other words, if we consider a class 0 point on the left of the dividing line (and the same for class 1 on the right) as correctly classified, the points far from the line (on either side) are likely to be correctly classified, while those nearer are more likely to be misclassified. The goal will be to build a classifier able to distinguish the two classes for a given point, and use it to classify any new point which wasn't part of the training set.</div>
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In [1]:
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<pre><span class="k">def</span> <span class="nf">dataset</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">):</span>
<span class="n">xy</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">low</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">classes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">xy</span><span class="p">:</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">intercept</span><span class="p">)</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">slope</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">classes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span> <span class="k">if</span> <span class="n">d</span> <span class="o"><</span> <span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">xy</span><span class="p">,</span> <span class="n">classes</span><span class="p">])</span>
<span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">dataset</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">class0</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)]</span>
<span class="n">class1</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">where</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class0</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class1</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">);</span>
</pre>
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<h2 id="Guessing-an-Initial-Model">
Guessing an Initial Model<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Guessing-an-Initial-Model">¶</a></h2>
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As we did for linear regression, we first guess an initial set of <span class="math">\(\theta\)</span> parameters for our model. This time it has an extra component, since our inputs are characterized by two coordinates (<span class="math">\(x\)</span> and <span class="math">\(y\)</span>), instead of one. It is still a linear model though, and <span class="math">\(\theta\)</span> can be interpreted as the parameters of an equation of the general form (<span class="math">\(ax + by + c = 0\)</span>): <span class="math">\(\theta_1 x + \theta_2 y + \theta_0 = 0\)</span>. We need a simple function to convert between the general and slope-intercept form, for drawing purposes, but of course they're equivalent.</div>
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<pre><span class="k">def</span> <span class="nf">general_to_slope_intercept</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
<span class="n">slope</span> <span class="o">=</span> <span class="o">-</span><span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">theta</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="n">intercept</span> <span class="o">=</span> <span class="o">-</span><span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">theta</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="k">return</span> <span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span>
<span class="n">initial_theta</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span> <span class="o">=</span> <span class="n">general_to_slope_intercept</span><span class="p">(</span><span class="n">initial_theta</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class0</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class1</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">intercept</span><span class="p">,</span>
<span class="s">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'initial guess'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">axis</span><span class="p">((</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">legend</span><span class="p">();</span>
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<h2 id="Introducing-Probabilities">
Introducing Probabilities<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Introducing-Probabilities">¶</a></h2>
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Even though the graphical representation we've been using may suggest otherwise, what we're really dealing with this time is class membership, and only indirectly planar position. And since class membership, our target variable, is binary, rather than scalar (as was the case with linear regression), one way to think about it is in terms of the probability, for a given point, to belong to a certain class<br />
<span class="math">\[P(C = k | \vec{x}; \theta)\]</span><br />
in other words, given its 2D coordinates (note the vector notation, to emphasize that <span class="math">\(x\)</span> here corresponds to 2D coordinates, not a scalar feature as with linear regression) and the current <span class="math">\(\theta\)</span> parameters, what's the probability for a certain point to be classified as <span class="math">\(k\)</span>, where <span class="math">\(k \in \{0,1\}\)</span>. This is what we'll try to model.<br />
It turns out that there's a simple non-linear function that squashes its input into the range <span class="math">\([0, 1]\)</span>: the logistic function<br />
<span class="math">\[ h(x) = \frac{1}{1 + e^{-x}} \]</span><br />
Although this function, in itself, has nothing to do with probabilities, there's nothing that prevent us from using it to endow our model with probabilistic semantics. In particular, we can turn the attributes of a point <span class="math">\(\vec{x}\)</span> into a probability estimate, just by passing its dot product with <span class="math">\(\theta\)</span> through the logistic function, which will squash it appropriately<br />
<span class="math">\[P(C = k | \vec{x}; \theta) = h(\theta^T \vec{x}) \]</span></div>
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<pre><span class="k">def</span> <span class="nf">logistic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">logistic</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">)),</span> <span class="s">'b-'</span><span class="p">);</span>
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<h2 id="Training-to-Get-Better">
Training to Get Better<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Training-to-Get-Better">¶</a></h2>
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As with linear regression, we will find the best parameters for the model by minimizing an error function. This time however the function is probabilistic, and is called the negative log-likelihood<br />
<span class="math">\[NLL = - \sum_{i}^{n} c^{(i)} \log h_{\theta}(\vec{x}^{(i)}) + (1 - c^{(i)}) \log (1 - h_{\theta}(\vec{x}^{(i)}) \]</span><br />
where <span class="math">\(c^{(i)}\)</span> is the class (0 or 1) of the <span class="math">\(i\)</span>-th point, <span class="math">\(\vec{x}^{(i)}\)</span> its 2D coordinates, and <span class="math">\(h_{\theta}\)</span> denotes our model itself, consisting in the squashed dot product of the coordinates with its current state, <span class="math">\(\theta\)</span>.<br />
As with linear regression, we use the <em>batch gradient descent</em> method to minimize this function, by taking iterative steps in the direction of its derivative (gradient), which has a surprisingly familiar form<br />
<span class="math">\[ \frac{\partial NLL}{\partial \theta_j} = (c - h_{\theta}(\vec{x})) \cdot \vec{x}_j \]</span><br />
Indeed the error function derivatives for both linear and logistic regressions are identical, and the reason for this is that they can be both thought as being specific instances of a broader class called <em>generalized linear models</em>. Following that, there's a way to reinterpret the MSE of the linear regression, also in probabilistic terms.<br />
The training algorithm for logistic regression is thus very close to the one for linear regression, with the exception of the <span class="math">\(h\)</span> function, which is now logistic, and so introduces non-linearity in the model<br />
<span class="math">\[\theta_j := \theta_j - \alpha \sum_{i}^{n} (h_{\theta}(\vec{x}_j^{(i)}) - c^{(i)}) \cdot \vec{x}_j^{(i)}\]</span><br />
Despite this non-linearity, it is worth noting that the optimal decision boundary found by training is still a 2D line, as logistic regression is fundamentally a linear modeling method. In particular, when we apply the "general to slope-intercept" form transformation for the optimal <span class="math">\(\theta\)</span>, we find approximate values of the original parameters that we used to generate the dataset.</div>
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<pre><span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">):</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
<span class="n">bxy</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
<span class="n">thetas</span> <span class="o">=</span> <span class="p">[</span><span class="n">initial_theta</span><span class="p">]</span>
<span class="n">nll_errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">classif_errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">logistic</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">bxy</span><span class="p">,</span> <span class="n">theta</span><span class="p">))</span>
<span class="n">nll_errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="nb">sum</span><span class="p">(</span><span class="n">c</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">h</span><span class="p">))</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
<span class="n">classif_errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sum</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="o">!=</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)))</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nll_errors</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">nll_errors</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">nll_errors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o"><</span> <span class="n">stop</span><span class="p">:</span> <span class="k">break</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">(</span><span class="n">h</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="c"># intercept component</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
<span class="n">thetas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="k">return</span> <span class="n">thetas</span><span class="p">,</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="n">classif_errors</span>
<span class="n">thetas</span><span class="p">,</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="n">classif_errors</span> <span class="o">=</span> <span class="n">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">)</span>
<span class="n">best_theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c"># last one</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class0</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class0</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'bo'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">class1</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">class1</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r^'</span><span class="p">)</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span> <span class="o">=</span> <span class="n">general_to_slope_intercept</span><span class="p">(</span><span class="n">best_theta</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'best_theta slope and intercept = </span><span class="si">%f</span><span class="s">, </span><span class="si">%f</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">intercept</span><span class="p">,</span>
<span class="s">'g-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'trained model'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">axis</span><span class="p">((</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">legend</span><span class="p">();</span>
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<pre>best_theta slope and intercept = 2.546265, -1.042059
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We can plot the successive values of the NLL error to confirm that it was minimized.</div>
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In [5]:
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<pre><span class="n">xlabel</span><span class="p">(</span><span class="s">'Training iterations'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s">'NLL'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nll_errors</span><span class="p">)),</span> <span class="n">nll_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">);</span>
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But we can also look at the evolution of another error function, the classification error, which simply tracks, while training, the number of misclassified points. Although we're not directly minimizing this function, it can be thought as a proxy to the NLL, with which it is highly correlated. It is also more intuitive, and thus easier to interpret and make sense of. The interesting thing about this function in our case is that although it first steadily decreases, it rapidly stabilizes itself at around 100, meaning that no matter how hard we train, we always have many points that are not correctly classified. By looking at the dataset it is obvious that there is no line that could perfectly separate the two classes (we actually built the dataset that very way), and logistic regression simply does its best, finding the best compromise.</div>
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<pre><span class="n">xlabel</span><span class="p">(</span><span class="s">'Training iterations'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s">'Classification error'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">classif_errors</span><span class="p">)),</span> <span class="n">classif_errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
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Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-39602416057117470542013-10-05T11:47:00.001-04:002013-10-16T11:36:26.340-04:00Linear Regression 101<div class="text_cell_render border-box-sizing rendered_html">
As it looks like I found myself a new job as a data scientist (freelancing time is over!), it's time to brush up on machine learning theory and practice. Let's start with the basics: linear regression. This seems to be becoming a series of related posts, see also:
</div>
<ol>
<li>Linear Regression 101</li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/logistic-regression-101.html" target="_blank">Logistic Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/softmax-regression-101.html" target="_blank">Softmax Regression 101</a></li>
<li><a href="http://cjauvin.blogspot.ca/2013/10/neural-network-101.html" target="_blank">Neural Network 101</a></li>
</ol>
<h2 id="A-Cloud-of-Points">
A Cloud of Points<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#A-Cloud-of-Points">¶</a></h2>
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Let's create an artificial dataset by adding some gaussian noise to points sampled along a linear function with known slope and intercept parameters. The goal will be to find (or <em>fit</em>) a model by allowing it to "discover" those parameters, with the help of those points only. With this model, it will be possible to predict the <span class="math">\(y\)</span> value of a new <span class="math">\(x\)</span> value, which wasn't part of the training dataset.</div>
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In [3]:
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<pre><span class="k">def</span> <span class="nf">dataset</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">([</span><span class="n">n</span><span class="p">])</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">intercept</span> <span class="o">+</span> <span class="n">noise</span>
<span class="k">return</span> <span class="n">column_stack</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
<span class="n">points</span> <span class="o">=</span> <span class="n">dataset</span><span class="p">(</span><span class="mi">500</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">);</span>
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<h2 id="Guessing-an-Initial-Model">
Guessing an Initial Model<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Guessing-an-Initial-Model">¶</a></h2>
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Next we guess an initial value for <span class="math">\(\theta\)</span>, our model parameters (with component <span class="math">\(\theta_0\)</span> as the intercept and <span class="math">\(\theta_1\)</span> as the slope). Because it is a random guess, it is initially very unlikely to be a good model of course.</div>
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<pre><span class="n">initial_theta</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">initial_theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">initial_theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="s">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'initial guess'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
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<h2 id="Training-to-Get-Better">
Training to Get Better<a class="anchor-link" href="http://www.blogger.com/blogger.g?blogID=1673244706324941001#Training-to-Get-Better">¶</a></h2>
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We train the parameters of the model by minimizing its error function, the mean squared error<br />
<span class="math">\[MSE = \frac{1}{n}\sum_{i=1}^n (h_{\theta}(x^{(i)}) - y^{(i)})^2\]</span><br />
where <span class="math">\(h_{\theta}(x) = \theta_0 + \theta_1 x\)</span>, i.e. the value that the model associates to <span class="math">\(x\)</span>, given the current state of <span class="math">\(\theta\)</span>. The interpretation of this function is very intuitive: it's the average discrepancy between what our model says (<span class="math">\(h\)</span>) and the training data (<span class="math">\(y\)</span>). The better our model becomes, the lower this function should be. There are many ways to perform this minimization (including a closed-form solution which directly yields the answer, but is a little harder to derive), but here we will use a technique called <em>batch gradient descent</em>. The idea is to compute the derivative of the error function (or gradient) with respect to <span class="math">\(\theta\)</span>, and iteratively take a step in the direction of steepest descent, i.e. where its value is lowered the most. The derivative of the error function is<br />
<span class="math">\[\frac{\partial MSE}{\partial \theta_j} = (h_{\theta}(x) - y) \cdot x_j\]</span><br />
Note that the error derivative is defined in terms of the components of <span class="math">\(\theta\)</span>, of which there are two (the slope and intercept), while <span class="math">\(x\)</span> is a scalar. The way to treat this is to consider an implicit additional input component (or "bias") corresponding to the intercept, which has a constant value of 1. The training algorithm will iteratively update <span class="math">\(\theta\)</span> in proportion <span class="math">\(\alpha\)</span> of the gradient (to avoid taking too large steps that would make us possibly overshoot the target), and will do so in batch, meaning that it will be accumulated over the entire training set before the update occurs<br />
<span class="math">\[\theta_j := \theta_j - \alpha \sum_{i}^{n} (h_{\theta}(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)}\]</span><br />
As we accumulate the MSE values while the training goes, we can stop when we find that the improvement (in terms of error reduction) between two iterations is below a certain threshold.</div>
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<pre><span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">):</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
<span class="n">thetas</span> <span class="o">=</span> <span class="p">[</span><span class="n">initial_theta</span><span class="p">]</span>
<span class="n">errors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">(</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="c"># intercept component</span>
<span class="n">theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
<span class="n">thetas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sum</span><span class="p">((</span><span class="n">h</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">))</span> <span class="c"># MSE</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">errors</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">errors</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">errors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o"><</span> <span class="n">stop</span><span class="p">:</span> <span class="k">break</span>
<span class="k">return</span> <span class="n">thetas</span><span class="p">,</span> <span class="n">errors</span>
<span class="n">thetas</span><span class="p">,</span> <span class="n">errors</span> <span class="o">=</span> <span class="n">train</span><span class="p">(</span><span class="n">points</span><span class="p">,</span> <span class="n">initial_theta</span><span class="p">)</span>
<span class="n">best_theta</span> <span class="o">=</span> <span class="n">thetas</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c"># last one</span>
<span class="k">print</span> <span class="s">'best_theta ='</span><span class="p">,</span> <span class="n">best_theta</span>
<span class="n">plot</span><span class="p">(</span><span class="n">points</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">'.'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">best_theta</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">best_theta</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="s">'g-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'trained model'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
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<pre>best_theta = [-0.91688322 2.46956016]
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We can see that the trained <span class="math">\(\theta\)</span> (slope of 2.496 and intercept of -1.04) is not so far from the original values we picked (2.5 and -1), which is a good signe. Finally, we can plot the error values, to convince ourselves that the training indeed minimized them in a consistent way.</div>
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<pre><span class="n">xlabel</span><span class="p">(</span><span class="s">'Training iterations'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s">'MSE'</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">errors</span><span class="p">)),</span> <span class="n">errors</span><span class="p">,</span> <span class="s">'r-'</span><span class="p">)</span>
<span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
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Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-56817031479161121462013-09-05T09:39:00.001-04:002013-09-05T13:07:11.129-04:00Siamese Dream<p>A <a href="http://en.wikipedia.org/wiki/Magic_square">magic square</a> of odd order can be efficiently constructed with a simple method brought from the travels of the French mathematician <a href="http://en.wikipedia.org/wiki/Simon_de_la_Loub%C3%A8re">Simon de la Loubère</a> to what was then Thailand (Siam), in the 17th century. I like to think of a time when you brought things like that from overseas trips, and not only a gazillion photos and some cheap souvenirs.</p>
<p>The method is easier to undertand with a visual <a href="http://en.wikipedia.org/wiki/File:SiameseMethod.gif">example</a>:</p>
<div class="separator" style="clear: both; text-align: center;"><a href="http://en.wikipedia.org/wiki/File:SiameseMethod.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-FnuPyaDZNY4/Uih6mMNg7RI/AAAAAAAAFTc/bLyR9ayXCS4/s1600/SiameseMethod.gif" /></a></div>
<p>A Python implementation is also very easy, especially when using Numpy, with which the <code>pos</code> pointer variable, being a <code>numpy.array</code>, can be both updated with one-liner arithmetic operators, and used as a two-dimensional index into the matrix (when cast as a <code>tuple</code>):</p>
<div class="code-bg"><pre>
<span class="keyword">import</span> numpy <span class="keyword">as</span> np
<span class="keyword">def</span> <span class="function-name">de_la_loubere</span>(n):
<span class="py-variable-name">m</span> = np.zeros((n, n), dtype=int)
<span class="py-variable-name">pos</span> = np.asarray([<span class="py-number">0</span>, n/<span class="py-number">2</span>]) <span class="comment"># initial
</span> <span class="py-variable-name">m</span>[tuple(pos)] = <span class="py-number">1</span>
<span class="py-variable-name">i</span> = <span class="py-number">2</span>
<span class="keyword">while</span> i <= n ** <span class="py-number">2</span>:
<span class="keyword">if</span> m[tuple((pos + [-<span class="py-number">1</span>, <span class="py-number">1</span>]) % n)] != <span class="py-number">0</span>:
<span class="py-variable-name">pos</span> += [<span class="py-number">1</span>, <span class="py-number">0</span>] <span class="comment"># blocked: go down
</span> <span class="keyword">else</span>:
<span class="py-variable-name">pos</span> += [-<span class="py-number">1</span>, <span class="py-number">1</span>] <span class="comment"># go up/right
</span> <span class="py-variable-name">pos</span> %= n <span class="comment"># wrap
</span> <span class="py-variable-name">m</span>[tuple(pos)] = i
<span class="py-variable-name">i</span> += <span class="py-number">1</span>
<span class="keyword">return</span> m
<span class="py-variable-name">n</span> = <span class="py-number">15</span>
<span class="py-variable-name">m</span> = de_la_loubere(n)
<span class="comment-delimiter"># </span><span class="comment">verify that it's really magic
</span><span class="py-variable-name">magic</span> = n * (n ** <span class="py-number">2</span> + <span class="py-number">1</span>) / <span class="py-number">2</span> <span class="comment"># 65
</span><span class="keyword">assert</span> (np.sum(m, axis=<span class="py-number">0</span>) == np.repeat(magic, n)).all()
<span class="keyword">assert</span> (np.sum(m, axis=<span class="py-number">1</span>) == np.repeat(magic, n)).all()
<span class="keyword">assert</span> m.trace() == magic
<span class="keyword">assert</span> np.fliplr(m).trace() == magic
<span class="keyword">print</span> m
<span class="comment-delimiter"># </span><span class="comment">[[122 139 156 173 190 207 224 1 18 35 52 69 86 103 120]</span>
<span class="comment-delimiter"># </span><span class="comment">[138 155 172 189 206 223 15 17 34 51 68 85 102 119 121]</span>
<span class="comment-delimiter"># </span><span class="comment">[154 171 188 205 222 14 16 33 50 67 84 101 118 135 137]</span>
<span class="comment-delimiter"># </span><span class="comment">[170 187 204 221 13 30 32 49 66 83 100 117 134 136 153]</span>
<span class="comment-delimiter"># </span><span class="comment">[186 203 220 12 29 31 48 65 82 99 116 133 150 152 169]</span>
<span class="comment-delimiter"># </span><span class="comment">[202 219 11 28 45 47 64 81 98 115 132 149 151 168 185]</span>
<span class="comment-delimiter"># </span><span class="comment">[218 10 27 44 46 63 80 97 114 131 148 165 167 184 201]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 9 26 43 60 62 79 96 113 130 147 164 166 183 200 217]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 25 42 59 61 78 95 112 129 146 163 180 182 199 216 8]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 41 58 75 77 94 111 128 145 162 179 181 198 215 7 24]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 57 74 76 93 110 127 144 161 178 195 197 214 6 23 40]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 73 90 92 109 126 143 160 177 194 196 213 5 22 39 56]</span>
<span class="comment-delimiter"># </span><span class="comment">[ 89 91 108 125 142 159 176 193 210 212 4 21 38 55 72]</span>
<span class="comment-delimiter"># </span><span class="comment">[105 107 124 141 158 175 192 209 211 3 20 37 54 71 88]</span>
<span class="comment-delimiter"># </span><span class="comment">[106 123 140 157 174 191 208 225 2 19 36 53 70 87 104]]</span>
</pre></div>
Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-28067607237663243852013-05-15T23:03:00.002-04:002013-05-16T08:48:56.943-04:00Impossibly Lean Audit System for Postgres with hstore<p>
I really like Postgres! I recently discovered a new trick that plays especially well with the access control pattern I <a href="http://cjauvin.blogspot.ca/2013/04/impossibly-lean-access-control-with.html">recently described</a>: a simple (but rather powerful) audit/logging pattern to track the changes made to a database (i.e. <code>insert</code>s, <code>update</code>s and <code>delete</code>s), and most importantly, allow to query them in a flexible way.
</p>
<p>
It's easy to set up a <code>trigger</code> to record changes. However, if you follow this <a href="http://wiki.postgresql.org/wiki/Audit_trigger">basic pattern</a>, you end up with your update data stored as unstructured <code>text</code>, which is not the ideal format to query. You could modify that scheme to use as many audit tables as you want to log (thus each reproducing the structure of its target table), but there's a much <a href="http://wiki.postgresql.org/wiki/Audit_trigger_91plus">nicer solution</a>, using PG's <code>hstore</code> data type.
</p>
<p>
<code>hstore</code> is PG's answer to <a href="http://en.wikipedia.org/wiki/Nosql">NoSQL</a>: it's a string-based key-value data type, which you can embed inside your relational schema, within which it interoperates with other data types and SQL constructs seamlessly. There's also the newer <code>JSON</code> type, which plays a similar role, but since its API is not yet as mature as <code>hstore</code>'s, we're not going to use it here.
</p>
<p>
So here's the audit table:
</p>
<div class="code-bg"><pre>
<span class="keyword">create</span> extension hstore;
<span class="keyword">create</span> <span class="keyword">table</span> <span class="function-name">audit</span> (
audit_id serial <span class="keyword">primary</span> <span class="keyword">key</span>,
table_name text <span class="keyword">not</span> <span class="keyword">null</span>,
user_name text <span class="keyword">not</span> <span class="keyword">null</span>,
action_timestamp <span class="type">timestamp</span> <span class="keyword">not</span> <span class="keyword">null</span> <span class="keyword">default</span> <span class="builtin">current_timestamp</span>,
action text <span class="keyword">not</span> <span class="keyword">null</span> <span class="keyword">check</span> (<span class="keyword">action</span> <span class="keyword">in</span> (<span class="string">'i'</span>,<span class="string">'d'</span>,<span class="string">'u'</span>)),
old_values hstore,
new_values hstore,
updated_cols text[],
query text
);
</pre></div>
<p>
And here's the audit trigger that goes with it:
</p>
<div class="code-bg"><pre>
<span class="keyword">create</span> <span class="keyword">or</span> replace <span class="keyword">function</span> <span class="function-name">if_modified_func</span>() <span class="keyword">returns</span> <span class="keyword">trigger</span> <span class="keyword">as</span> $body$
<span class="keyword">begin</span>
if tg_op = <span class="string">'UPDATE'</span> <span class="keyword">then</span>
<span class="keyword">insert</span> <span class="keyword">into</span> audit (<span class="keyword">table_name</span>, user_name, <span class="keyword">action</span>, old_values, new_values, updated_cols, query)
<span class="keyword">values</span> (tg_table_name::text, <span class="builtin">current_user</span>::text, <span class="string">'u'</span>, hstore(<span class="keyword">old</span>.*), hstore(<span class="keyword">new</span>.*),
akeys(hstore(<span class="keyword">new</span>.*) - hstore(<span class="keyword">old</span>.*)), current_query());
<span class="keyword">return</span> <span class="keyword">new</span>;
elsif tg_op = <span class="string">'DELETE'</span> <span class="keyword">then</span>
<span class="keyword">insert</span> <span class="keyword">into</span> audit (<span class="keyword">table_name</span>, user_name, <span class="keyword">action</span>, old_values, query)
<span class="keyword">values</span> (tg_table_name::text, <span class="builtin">current_user</span>::text, <span class="string">'d'</span>, hstore(<span class="keyword">old</span>.*), current_query());
<span class="keyword">return</span> <span class="keyword">old</span>;
elsif tg_op = <span class="string">'INSERT'</span> <span class="keyword">then</span>
<span class="keyword">insert</span> <span class="keyword">into</span> audit (<span class="keyword">table_name</span>, user_name, <span class="keyword">action</span>, new_values, query)
<span class="keyword">values</span> (tg_table_name::text, <span class="builtin">current_user</span>::text, <span class="string">'i'</span>, hstore(<span class="keyword">new</span>.*), current_query());
<span class="keyword">return</span> <span class="keyword">new</span>;
<span class="keyword">end</span> if;
<span class="keyword">end</span>;
$body$
<span class="keyword">language</span> plpgsql;
</pre></div>
<p>
Notice how the rows get converted (<code>hstore(old.*)</code> and <code>hstore(new.*)</code>). Now suppose that we have a <code>book</code> table that we'd like to audit:
</p>
<div class="code-bg"><pre>
<span class="keyword">create</span> <span class="keyword">table</span> <span class="function-name">book</span> (
book_id serial <span class="keyword">primary</span> <span class="keyword">key</span>,
title text,
author text,
n_pages <span class="type">int</span>
);
<span class="keyword">create</span> <span class="keyword">trigger</span> book_audit <span class="keyword">after</span> <span class="keyword">insert</span> <span class="keyword">or</span> <span class="keyword">update</span> <span class="keyword">or</span> <span class="keyword">delete</span> <span class="keyword">on</span> book <span class="keyword">for</span> <span class="keyword">each</span> <span class="type">row</span> <span class="keyword">execute</span> <span class="keyword">procedure</span> <span class="function-name">if_modified_func</span>();
</pre></div>
<p>
If we insert a book in it, we can see the audit mechanism in action:
</p>
<div class="code-bg"><pre>
<span class="keyword">insert</span> <span class="keyword">into</span> book (title, n_pages) <span class="keyword">values</span> (<span class="string">'PG is Great'</span>, 250);
<span class="keyword">select</span> action, table_name, new_values -> <span class="string">'title'</span> <span class="keyword">as</span> title <span class="keyword">from</span> audit;
action | table_name | title
--------+------------+-------------
i | book | PG is Great
</pre></div>
<p>
The third retrieved value (<code>title</code>) demontrates <code>hstore</code>'s syntax for value retrieval from a key (using the <code>-></code> operator), which is just an example among the bunch of <a href="http://www.postgresql.org/docs/9.2/static/hstore.html#HSTORE-OP-TABLE">operators and functions offered</a>.
</p>
<p>If we perform an update on the book:</p>
<div class="code-bg"><pre>
<span class="keyword">update</span> book <span class="keyword">set</span> author = <span class="string">'Christian Jauvin'</span>, n_pages = 300 <span class="keyword">where</span> book_id = 1;
<span class="keyword">select</span> action, table_name, updated_cols <span class="keyword">from</span> audit;
action | table_name | updated_cols
--------+------------+------------------
i | book |
u | book | {author,n_pages}
</pre></div>
<p>
The third column now sports the columns that have been updated, using the <code>updated_cols</code> mechanism, implemented as a PG <a href="http://www.postgresql.org/docs/9.2/static/arrays.html"><code>array</code></a> (another very nice data structure) along with the <code>akeys</code> function, with which we collect the keys of the <code>hstore</code> structure resulting from the <code>hstore(new.*)</code> "minus" <code>hstore(old.*)</code> operation performed in the "update" part of the trigger function.
</p>
<p>It's also easy to perform "before and after" type queries:</p>
<div class="code-bg"><pre>
<span class="keyword">select</span> old_values -> <span class="string">'n_pages'</span> <span class="keyword">as</span> <span class="keyword">before</span>, new_values -> <span class="string">'n_pages'</span> <span class="keyword">as</span> <span class="keyword">after</span> <span class="keyword">from</span> audit <span class="keyword">where</span> audit_id = 2;
before | after
--------+-------
250 | 300
</pre></div>
<p>Don't forget however that <code>hstore</code> values are stored as strings, so this for instance wouldn't work:</p>
<div class="code-bg"><pre>
<span class="keyword">select</span> old_values -> <span class="string">'n_pages'</span> + new_values -> <span class="string">'n_pages'</span> <span class="keyword">from</span> audit <span class="keyword">where</span> audit_id = 2;
HINT: No operator matches the given name and argument type(s). You might need to add explicit type casts.
</pre></div>
<p>whereas this would:</p>
<div class="code-bg"><pre>
<span class="keyword">select</span> (old_values -> <span class="string">'n_pages'</span>)::<span class="type">int</span> + (new_values -> <span class="string">'n_pages'</span>)::<span class="type">int</span> <span class="keyword">from</span> audit <span class="keyword">where</span> audit_id = 2;
</pre></div>
<p>
One final aspect to note: I find that this pattern works well with my <a href="http://cjauvin.blogspot.ca/2013/04/impossibly-lean-access-control-with.html">access control pattern</a>, where every application user has its own PG <code>role</code>, on behalf of which the database operations are performed. In this scenario, the user/role gets automatically logged by the trigger (using the <code>current_user</code> variable). In contrast, an access control system implemented at the level of the application (i.e. with some kind of <code>user</code> table and all database operations performed on behalf of a single PG role) wouldn't enjoy that much simplicity I believe.
</p>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com2tag:blogger.com,1999:blog-1673244706324941001.post-42607957858613823412013-04-01T22:06:00.001-04:002013-04-08T12:27:09.902-04:00Impossibly Lean Access Control with Postgres + FlaskOne of the most common feature for a database-backed web application is to implement some kind of access control mechanism to regulate what different types of users (admins, guests, etc) can do. There are of course many ways to implement such a pattern: you can do it client-side, by modifying your UI dynamically (e.g. graying out certain widgets) according to the identity of the user, or server-side (as part of the so-called business rules maybe). Often a mix of both approaches will be used.
<br /><br />
In this article, I will to show a way to implement such a pattern in a very simple way, using <a href="http://www.postgresql.org/" target="_blank">Postgres</a>, <a href="http://flask.pocoo.org/" target="_blank">Flask</a>, <a href="http://pythonhosted.org/Flask-Login/" target="_blank">Flask-Login</a> and <a href="https://github.com/cjauvin/little_pger" target="_blank">little_pger</a>, a very thin SQL wrapper just above <a href="http://initd.org/psycopg/docs" target="_blank">psycopg2</a> which I created.
<br /><br />
Because the idea has received some criticism, I must say that (1) it is only a proof of concept (i.e. probably not robust enough for production in its current form) and (2) I don't think that such a pattern would scale well to a big userbase (it's rather meant for small-scale, intranet-style webapps with a limited number of users, which is what I've been mainly developing). The idea for the pattern was inspired by this blog post, that I read a while ago:
<br /><br />
<a href="http://database-programmer.blogspot.ca/2009/02/comprehensive-database-security-model.html">http://database-programmer.blogspot.ca/2009/02/comprehensive-database-security-model.html</a>
<br /><br />
while my skepticism toward ORMs in general stems from this even older (and admittedly controversial) post:
<br /><br />
<a href="http://database-programmer.blogspot.ca/2008/06/why-i-do-not-use-orm.html">http://database-programmer.blogspot.ca/2008/06/why-i-do-not-use-orm.html</a>
<br /><br />
I say my pattern is "impossibly lean" because it introduces a minimal amount of code and concepts (not even the shadow of an ORM will be found here), by leveraging PG's role access management subsystem. These powerful features are already available the moment you create a PG database, so why would you need to reinvent the wheel?
<br /><br />
Let's start by supposing that our application has a very simple database, with only one table:
<div class="code-bg">
<pre>
<span class="keyword">create</span> database pg_flask_ac;
\connect pg_flask_ac
<span class="keyword">create</span> <span class="keyword">table</span> <span class="function-name">foo</span> (
foo_id serial <span class="keyword">primary</span> <span class="keyword">key</span>
);
</pre>
</div>
Suppose also that we have two users for it: an almighty <i>admin</i> (who can do whatever pleases him), and a shy <i>guest</i> (only allowed to look around). Although we could create an additional SQL "user" table, with a column dedicated to the representation of their privilege levels, another solution is to simply create corresponding PG roles:
<div class="code-bg">
<pre>
<span class="keyword">create</span> <span class="keyword">role</span> joe_admin login superuser password <span class="string">'<secret>'</span>;
<span class="keyword">create</span> <span class="keyword">role</span> joe_guest password <span class="string">'<secret>'</span>;
<span class="keyword">grant</span> <span class="keyword">select</span> <span class="keyword">on</span> <span class="keyword">all</span> tables <span class="keyword">in</span> <span class="keyword">schema</span> <span class="keyword">public</span> <span class="keyword">to</span> joe_guest;
</pre>
</div>
<code>joe_admin</code>, being a <code>superuser</code> can perform any SQL operation, while <code>joe_guest</code> is restricted to only being able to select things.. that seems fine, but is there a way to leverage these SQL-level constraints into our application? Sure we can, and here's all it takes to implement the pattern:
<div class="code-bg">
<pre>
<span class="keyword">from</span> flask <span class="keyword">import</span> *
<span class="keyword">from</span> flask.ext.login <span class="keyword">import</span> *
<span class="keyword">import</span> psycopg2, psycopg2.extras
<span class="keyword">import</span> little_pger <span class="keyword">as</span> pg
<span class="py-variable-name">app</span> = Flask(__name__)
<span class="py-variable-name">app.secret_key</span> = <span class="string">'a secret!'</span>
<span class="py-variable-name">login_manager</span> = LoginManager()
login_manager.init_app(app)
<span class="keyword">class</span> <span class="py-class-name">User</span>(UserMixin):
<span class="keyword">def</span> <span class="function-name">__init__</span>(<span class="py-pseudo-keyword">self</span>, oid):
<span class="py-pseudo-keyword">self</span>.id = oid
<span class="comment-delimiter"># </span><span class="comment">called first: here we connect as an admin (with the login privilege)
</span><span class="py-decorators">@app.before_request</span>
<span class="keyword">def</span> <span class="function-name">before_request</span>():
<span class="py-variable-name">g.db</span> = psycopg2.connect(<span class="string">"dbname=pg_flask_ac user=joe_admin"</span>,
<span class="py-variable-name">connection_factory</span>=psycopg2.extras.RealDictConnection)
<span class="comment-delimiter"># </span><span class="comment">called next: from the admin connection, we fetch the name of the
</span><span class="comment-delimiter"># </span><span class="comment">target role (identified with the unique (o)id encrypted in the session cookie)
</span><span class="py-decorators">@login_manager.user_loader</span>
<span class="keyword">def</span> <span class="function-name">load_user</span>(id):
<span class="py-variable-name">cur</span> = g.<span class="py-number">db</span>.cursor()
<span class="py-variable-name">rn</span> = pg.select1(cur, <span class="string">'pg_authid'</span>, <span class="string">'rolname'</span>, where={<span class="string">'oid'</span>:<span class="py-builtins"> id</span>})
<span class="keyword">if</span> rn:
<span class="comment"># executed on behalf of the admin
</span> cur.execute(<span class="string">'set role %s'</span>, [rn])
<span class="comment"># from this point any command performed via g.db.cursor() will be
</span> <span class="comment"># on the target role's behalf
</span> <span class="keyword">return</span> User(id)
<span class="keyword">return</span> <span class="py-pseudo-keyword">None</span>
</pre>
</div>
The interesting part is the interplay between <code>before_request</code> (Flask) and <code>load_user</code> (Flask-Login) which are both called automatically, before serving any view request. The first simply sets an admin connection from which the second will be able to switch user (by first fetching the appropriate role name in the <code>pg_authid</code> table with the <code>id</code> stored in the session cookie and then issuing the <code>set role</code> SQL command using it).
We can view this mechanism in action by adding two views with different access policies:
<div class="code-bg">
<pre>
<span class="comment-delimiter"># </span><span class="comment">this read-only view should be available to anyone
</span><span class="py-decorators">@app.route</span>(<span class="string">'/look_at_something'</span>)
<span class="py-decorators">@login_required</span>
<span class="keyword">def</span> <span class="function-name">look_at_something</span>():
<span class="keyword">return</span> jsonify({<span class="string">'success'</span>: <span class="py-pseudo-keyword">True</span>, <span class="string">'count'</span>: pg.count(g.<span class="py-number">db</span>.cursor(), <span class="string">'foo'</span>)})
<span class="comment-delimiter"># </span><span class="comment">this view should only be available to an admin; it will raise an exception
</span><span class="comment-delimiter"># </span><span class="comment">(at the SQL level) if accessed by anyone else
</span><span class="py-decorators">@app.route</span>(<span class="string">'/change_something'</span>)
<span class="py-decorators">@login_required</span>
<span class="keyword">def</span> <span class="function-name">change_something</span>():
pg.insert(g.<span class="py-number">db</span>.cursor(), <span class="string">'foo'</span>)
g.<span class="py-number">db</span>.commit()
<span class="keyword">return</span> jsonify({<span class="string">'success'</span>: <span class="py-pseudo-keyword">True</span>})
</pre>
</div>
which we can then drive using some Flask testing code:
<div class="code-bg">
<pre>
<span class="keyword">if</span><span class="py-builtins"> __name__</span> == <span class="string">'__main__'</span>:
<span class="keyword">for</span> user <span class="keyword">in</span> [<span class="string">'joe_admin'</span>, <span class="string">'joe_guest'</span>]:
<span class="keyword">for</span> view <span class="keyword">in</span> [<span class="string">'/look_at_something'</span>, <span class="string">'/change_something'</span>]:
<span class="keyword">with</span> app.test_request_context(view):
app.preprocess_request()
<span class="py-variable-name">oid</span> = pg.select1(g.<span class="py-number">db</span>.cursor(), <span class="string">'pg_authid'</span>, <span class="string">'oid'</span>,
<span class="py-variable-name">where</span>={<span class="string">'rolname'</span>: user})
login_user(User(oid))
<span class="keyword">print</span> <span class="string">'%s %s..'</span> % (user, view),
<span class="keyword">try</span>:
app.dispatch_request()
<span class="keyword">print</span> <span class="string">'and succeeds!'</span>
<span class="py-exception-name">except</span>:
<span class="keyword">print</span> <span class="string">'but fails..'</span>
<span class="comment-delimiter"># </span><span class="comment">joe_admin /look_at_something.. and succeeds!
</span><span class="comment-delimiter"># </span><span class="comment">joe_admin /change_something.. and succeeds!
</span><span class="comment-delimiter"># </span><span class="comment">joe_guest /look_at_something.. and succeeds!
</span><span class="comment-delimiter"># </span><span class="comment">joe_guest /change_something.. but fails..</span>
</pre>
</div>
Of course an access violation exception by itself wouldn't be terribly useful, so an easy way to use that pattern in a real environment would be to intercept it using Flask's <code>errorhandler</code> decorator, in order to return, for instance, a JSON formatted error message, along with a 403 response (upon which the client should presumably react appropriately):
<div class="code-bg">
<pre>
<span class="py-decorators">@app.errorhandler</span>(psycopg2.ProgrammingError)
<span class="keyword">def</span> <span class="function-name">db_access_error</span>(<span class="py-number">e</span>):
<span class="py-variable-name">msg</span> = <span class="string">"This user doesn't have the required privilege to perform this action."</span>
<span class="keyword">return</span> jsonify({<span class="string">'success'</span>: <span class="py-pseudo-keyword">False</span>, <span class="string">'message'</span>: msg}), <span class="py-number">403</span>
</pre>
</div>
Finally, note that we didn't make use of the role passwords up to this point, because we assumed that a user had already been logged in, but of course it must occur at some point in the process, and here's a possible way to do it (which implies knowing about the particular way PG encrypts passwords in the <code>pg_authid</code> table):
<div class="code-bg">
<pre>
<span class="py-decorators">@app.route</span>(<span class="string">"/login"</span>, methods=[<span class="string">'POST'</span>])
<span class="keyword">def</span> <span class="function-name">login</span>():
<span class="py-variable-name">user</span> = request.form(<span class="string">'user'</span>)
<span class="py-variable-name">pw</span> = request.form(<span class="string">'pw'</span>)
<span class="py-variable-name">cur</span> = g.<span class="py-number">db</span>.cursor()
cur.execute(<span class="string">"""select * from pg_authid where rolname = %s and
rolpassword = 'md5' || md5(%s || %s)"""</span>,
[user, pw, user])
<span class="py-variable-name">u</span> = cur.fetchone()
<span class="keyword">if</span> u:
login_user(User(u[<span class="string">'oid'</span>]))
<span class="keyword">return</span> jsonify({<span class="string">'success'</span>: u <span class="keyword">is</span> <span class="keyword">not</span> <span class="py-pseudo-keyword">None</span>})
</pre>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-4448860898656555542013-03-12T17:40:00.000-04:002013-03-13T23:03:56.479-04:00Suspension of Parser DisbeliefAt last, it's done: I have expanded the <a href="http://cjauvin.blogspot.ca/2013/02/choose-your-own-closure.html" target="_blank">53-section prototype</a> of my <a href="http://github.com/cjauvin/gamebook.js" target="_blank">gamebook.js</a> engine into a fully playable, complete implementation of <a href="http://en.wikipedia.org/wiki/Fire_on_the_water" target="_blank">Fire on the Water</a>, the second gamebook in the <a href="http://en.wikipedia.org/wiki/Lone_Wolf_(gamebooks)" target="_blank">Lone Wolf </a>series, created by<a href="http://en.wikipedia.org/wiki/Joe_Dever" target="_blank"> Joe Dever</a> in the 80s. To the obvious question (Why start with the <i>second</i> book?), I don't really have a good answer, apart from the fact that FotW was one of the earliest gamebooks I've read (in French, as <i>La traversée infernale</i>), and always one of my favorites (I think in large part because of the maritime theme).<br />
<br />
So to recap, <b>gamebook.js</b> is an experimental crossbreed between two classic genres: <a href="http://en.wikipedia.org/wiki/Interactive_fiction" target="_blank">interactive fiction</a> (IF) and <a href="http://en.wikipedia.org/wiki/Gamebook" target="_blank">gamebooks</a>. Instead of navigating an explicit menu of choices (as with a classical gamebook), those are rather willfully concealed after each section, and your job is to "reveal" them by typing any command you want (using clues from the text) which the parser then tries to match. The idea is to expand, as much as possible, the feeling of "freedom" while exploring the gamebook world, as well as offering a novel (and hopefully fun) way to interact with it.<br />
<br />
So far so good.. but given the typical small number of choices for a section, what's the point really in having a parser, and how does it work anyway? It's true that it obviously couldn't be as sophisticated as a complex IF engine, but I think the one I've created works well in a gamebook setting, and listing its relevant features is probably the best way to show why:<br />
<ul>
<li>A <a href="http://en.wikipedia.org/wiki/Stemming" target="_blank">stemmer</a> is used, to reduce word inflections to common, more easily matchable forms (e.g <i>stemmer = stemming = stemmed = stem</i>)</li>
<li>A meticulously hand-assembled table of synonyms is also used</li>
<li>The (web-based) textual environment supports contextual word autocompletion (thus offering hints, which can be turned off)</li>
<li>The ambiguity of certain choices can be enhanced or lessened with (again) tediously, manually curated sets of words (although the default is to match the words (and their synonyms) found in the text of the choices)</li>
<li>Although complex sentence structures are not supported (as it would be useless in most cases), compound expressions can be used (e.g. "go inside") to enhance meaning (or illusion thereof)</li>
<li>The Action Chart and item management subsystem is seamlessly integrated in the textual interface (so after winning a combat, you can "use your Healing Potion", for instance, or "eat a Meal", if required)</li>
</ul>
So the trick is actually to suspend your "parser disbelief", and allow yourself to compose commands as they naturally come, and more often than not (I contend) the flow and coherence of the story will make it work seamlessly.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://projectaon.org/staff/christian/gamebook.js" target="_blank">
<img border="0" src="http://1.bp.blogspot.com/-0_MRsiJF7zI/UT-TCIIDypI/AAAAAAAADPs/ztavK3WMyoA/s1600/button.png" />
</a></div>
<div>
<h3>
Some Technical Aspects (for those interested)</h3>
<div>
<ul>
<li>The engine is <a href="https://github.com/cjauvin/gamebook.js" target="_blank">open</a> and implemented in 100% JavaScript (it should run at least in recent versions of Chrome, Firefox and Safari), using a few excellent external packages (the <a href="http://terminal.jcubic.pl/" target="_blank">jQuery Terminal</a> plugin, which I <a href="https://github.com/cjauvin/jquery.terminal/tree/word-completion" target="_blank">modified a bit</a>, and a JS <a href="https://github.com/kristopolous/Porter-Stemmer" target="_blank">implementation</a> of the classic <a href="http://tartarus.org/martin/PorterStemmer/" target="_blank">Porter Stemmer</a> algorithm)</li>
<li>The content is fully contained in a single, highly structured <a href="http://www.projectaon.org/staff/christian/gamebook.js/fotw.html" target="_blank">JSON file</a>, which must be hosted on <a href="http://www.projectaon.org/en/Main/Home" target="_blank">Project Aon</a> for <a href="http://www.projectaon.org/en/Main/License" target="_blank">legal reasons</a></li>
<li>The extraction of content from the original Project Aon <a href="http://www.projectaon.org/en/xml/02fotw.xml" target="_blank">XML file</a> is semi-automated using a <a href="https://github.com/cjauvin/gamebook.js/blob/master/xml2json.py" target="_blank">Python script</a>, which weaves it with another customized, handcrafted "override" file, to yield a single final JSON file (in theory, this means that expanding to other LW books should be fairly easy, although the amount of work that must go in the "override" file should not be underestimated, as it would be <i>very</i> hard to fully automate everything)</li>
<li>The JSON content file (almost a <a href="http://en.wikipedia.org/wiki/Domain-specific_language" target="_blank">DSL</a> in itself) has an elaborate set of structures to implement the (sometimes <a href="http://www.projectaon.org/en/xhtml-simple/lw/02fotw.htm#footnotz" target="_blank">complex</a>, borderline ambiguous) ruleset (boolean requirement operators for certain choices (i.e. must have X amount of gold, or possess the Kai Discipline of Y), item description structures, etc.)</li>
<li>The architecture of the engine is adequately decoupled: the general logic (applicable to any LW gamebook) is contained in a <a href="https://github.com/cjauvin/gamebook.js/blob/master/gamebook.js" target="_blank">single module</a> (~1500 lines), while the book-specific rules and behaviors are kept in a <a href="https://github.com/cjauvin/gamebook.js/blob/master/fotw.js" target="_blank">separate module</a> (in this case, ~500 lines of FotW-specific code) with a "pluggable" interface (where any section needing a special treatment simply has an entry in a dictionary data structure)</li>
</ul>
</div>
</div>
Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com2tag:blogger.com,1999:blog-1673244706324941001.post-38127005691954027412013-02-05T10:17:00.000-05:002013-02-05T10:17:00.507-05:00Choose your own Closure<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-w0g5iM7x1dk/URAxWf1K9aI/AAAAAAAADPA/4wJonSrx0Tw/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-w0g5iM7x1dk/URAxWf1K9aI/AAAAAAAADPA/4wJonSrx0Tw/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome.png" /></a></div>
<br />
In another bout of <a href="http://cjauvin.blogspot.ca/2012/06/hello-6510.html" target="_blank">nostalgic computing</a>, I recently had the idea for a twist in the classic <a href="http://en.wikipedia.org/wiki/Gamebook" target="_blank">gamebook</a> mechanics: instead of navigating an explicit menu of options, you play via a (web-based) text console in which you are free to type any command, using clues from the text of the current section. The engine then tries to match your input with one of the predefined options, yielding a gameplay more akin to <a href="http://en.wikipedia.org/wiki/Interactive_fiction">interactive fiction</a>.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-5ieJALIVWGo/URAWdTWUCGI/AAAAAAAADOU/etdGKJWNB2s/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-5ieJALIVWGo/URAWdTWUCGI/AAAAAAAADOU/etdGKJWNB2s/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome.png" /></a></div>
<br />
Given that the options of a typical gamebook section are somewhat limited, it would be useless for the command parser to be very sophisticated. For it to be fun however (and not just an exercise in guessing), it has to be relatively flexible, which is achieved using different techniques such as synonym matching, spelling tolerance (based on the <a href="http://en.wikipedia.org/wiki/Edit_distance" target="_blank">edit distance</a>) and some other tricks. When stuck, it's always possible to reveal the section options:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-Sik-XyPxD5g/URAW3iEWFYI/AAAAAAAADOc/XCDY9VwkZv0/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-Sik-XyPxD5g/URAW3iEWFYI/AAAAAAAADOc/XCDY9VwkZv0/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-1.png" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
Inventory and stats are fully implemented, and can be handled with textual commands:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-REfgAA-oq0c/URAitCu1iOI/AAAAAAAADOs/5vuwD1fmSfc/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-REfgAA-oq0c/URAitCu1iOI/AAAAAAAADOs/5vuwD1fmSfc/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-2.png" /></a></div>
<br />
as well as combats, of course:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-pjrCJtP864M/URAlSyHaU3I/AAAAAAAADO0/4HANKlaEs3M/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-pjrCJtP864M/URAlSyHaU3I/AAAAAAAADO0/4HANKlaEs3M/s1600/Screenshot-gamebook.js+-+Fire+on+the+Water+-+Google+Chrome-3.png" /></a></div>
<br />
<h4>
Project Aon</h4>
<div>
The engine currently implements only the 53 first sections (in <a href="http://www.projectaon.org/en/svg/lw/02fotw.svgz" target="_blank">story</a>, rather than numeric order) of the <a href="http://en.wikipedia.org/wiki/Fire_on_the_water" target="_blank">Fire on the Water</a> gamebook (second in the <a href="http://en.wikipedia.org/wiki/Lone_Wolf_(gamebooks)" target="_blank">Lone Wolf</a> series, always one of my favorites), whose <a href="http://www.projectaon.org/en/Main/FireOnTheWater" target="_blank">electronic version</a> is available through <a href="http://www.projectaon.org/en/Main/Home" target="_blank">Project Aon</a>.</div>
<div>
<br /></div>
<div>
It's written in 100% JavaScript, and should run in most modern browsers (though I think there are currently some issues with Firefox, unfortunately). Although the code of the engine is <a href="https://github.com/cjauvin/gamebook.js" target="_blank">open and available on GitHub</a>, the gamebook content (as well as some of its logic, fully contained in a <a href="http://projectaon.org/staff/christian/gamebook.js/fotw.html" target="_blank">single JSON file</a>) can only be hosted on Project Aon, given their <a href="http://www.projectaon.org/en/Main/FAQ#faq8" target="_blank">copyright restrictions</a>.</div>
<br />
It's a work in progress (the current version is more than likely to have bugs) and if there is interest for it, I will certainly implement the remaining 297 sections of FotW. I could also study the question of making the engine more general, to allow the support of other gamebook series (although the current demo is tightly coupled to the LW mechanics, it would be relatively easy to modify). I plan to write soon another post to describe and explain some implementation details.<br />
<h2>
<a href="http://projectaon.org/staff/christian/gamebook.js" target="_blank">>>> Try it! <<<</a></h2>
Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-59821114426519109582012-12-09T13:39:00.000-05:002013-01-13T16:04:35.981-05:00Find True Love (on a dating site, using Python)<div class="text_cell_render border-box-sizing rendered_html">
<p>Imagine that after having accumulated a roster of potential candidates on a dating site, you embark on the task of meeting them, one by one, in order to find your unique soul mate. Since it could be insulting, for any particular candidate, to be chosen as a result of an a posteriori analysis of the entire list, you have to make your choice on the spot, after each meeting. In other words, if you reject a candidate, you cannot call him or her back later. But if, on the contrary, you think that the candidate is the one, you have to stop searching (and live with your choice). When making a choice, you are (only) allowed to take into account the candidates met so far. If there are $n$ candidates, you can then make your choice right after the first, wait until you met them all, or anything in between. But is there an optimal policy, i.e. a moment to stop where your chances of having found true love is maximized? This is the <a href="http://en.wikipedia.org/wiki/Secretary_problem">secretary problem</a>, and there is indeed a solution.</p>
<p>Let's first implement a guessing function, which takes a list of numerical values (e.g. an "attractiveness score", or if you are Bayesian-minded like <a href="http://www.amazon.com/dp/159420411X">some</a>, a prior belief in a good match with each candidate) and a cutoff point, which can be anywhere inside this list (start, end, or in between):</p>
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<div class="highlight"><pre><span class="k">def</span> <span class="nf">guess_max</span><span class="p">(</span><span class="n">values</span><span class="p">,</span> <span class="n">cutoff</span><span class="p">):</span>
<span class="n">before</span> <span class="o">=</span> <span class="n">values</span><span class="p">[:</span><span class="n">cutoff</span><span class="p">]</span>
<span class="n">max_before</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">before</span><span class="p">)</span> <span class="k">if</span> <span class="n">before</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
<span class="n">after</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="n">cutoff</span><span class="p">:]</span>
<span class="k">return</span> <span class="nb">next</span><span class="p">((</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">after</span> <span class="k">if</span> <span class="n">v</span> <span class="o">></span> <span class="n">max_before</span><span class="p">),</span> <span class="n">values</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</div>
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<p>If we create a list of 1000 candidates (with random attractiveness values, between 0 and 1), we can use this function with different cutoff strategies, to see what happens:</p>
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">random</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">candidates</span> <span class="o">=</span> <span class="p">[</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
<span class="nb">max</span><span class="p">(</span><span class="n">candidates</span><span class="p">)</span> <span class="c"># real answer</span>
</pre></div>
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<div class="prompt output_prompt">Out[2]:</div>
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<pre>0.99992538253578345</pre>
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</div>
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<div class="highlight"><pre><span class="n">guess_max</span><span class="p">(</span><span class="n">candidates</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="c"># choose first</span>
</pre></div>
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<pre>0.33486707420570316</pre>
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</div>
</div>
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<div class="prompt input_prompt">In [4]:</div>
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<div class="highlight"><pre><span class="n">guess_max</span><span class="p">(</span><span class="n">candidates</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="c"># choose last</span>
</pre></div>
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<pre>0.046756671317170762</pre>
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<div class="highlight"><pre><span class="n">guess_max</span><span class="p">(</span><span class="n">candidates</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span> <span class="c"># stop at midpoint</span>
</pre></div>
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<pre>0.99992538253578345</pre>
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<p>These results are of course not conclusive, because the candidates could be in any order, and our policy must thus be studied in a probabilistic way. Let's do this by simply retrying a policy a certain number of times (each time reshuffling our set of candidates), and count the number of times it's right:</p>
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<div class="highlight"><pre><span class="k">def</span> <span class="nf">est_prob_of_being_right</span><span class="p">(</span><span class="n">values</span><span class="p">,</span> <span class="n">cutoff</span><span class="p">,</span> <span class="n">n_trials</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
<span class="n">max_val</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
<span class="n">n_found</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">trial</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">n_trials</span><span class="p">):</span>
<span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
<span class="n">n_found</span> <span class="o">+=</span> <span class="mi">1</span> <span class="k">if</span> <span class="n">guess_max</span><span class="p">(</span><span class="n">values</span><span class="p">,</span> <span class="n">cutoff</span><span class="p">)</span> <span class="o">==</span> <span class="n">max_val</span> <span class="k">else</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">n_found</span> <span class="o">/</span> <span class="n">n_trials</span>
</pre></div>
</div>
</div>
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<p>If we do this in a systematic way, by sweeping the range of possible cutoffs:</p>
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<div class="highlight"><pre><span class="n">cutoffs</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="c"># try a cutoff value every 10</span>
<span class="n">probs</span> <span class="o">=</span> <span class="p">[</span><span class="n">est_prob_of_being_right</span><span class="p">(</span><span class="n">candidates</span><span class="p">,</span> <span class="n">cutoff</span><span class="p">)</span> <span class="k">for</span> <span class="n">cutoff</span> <span class="ow">in</span> <span class="n">cutoffs</span><span class="p">]</span>
<span class="n">plot</span><span class="p">(</span><span class="n">cutoffs</span><span class="p">,</span> <span class="n">probs</span><span class="p">)</span>
</pre></div>
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<div class="prompt output_prompt">Out[7]:</div>
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<pre>[<matplotlib.lines.Line2D at 0x7f128027d7d0>]</pre>
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<p>the coarse and surprisingly shaped probability distribution that results suggests that the optimal policy is to stop searching after you have met about $40\%$ of the candidates, in line with the exact solution, which is to <em>stop rejecting</em> after the $n/e$ first candidates have been seen, and proceed from there to pick the first most promising one so far, with whom the probability of true love will be highest, at $1/e$ ($\approx36.8\%$). If you want to understand why, <a href="http://www.math.uah.edu/stat/urn/Secretary.html">these</a> <a href="http://www.math.upenn.edu/~ted/210F10/References/Secretary.pdf">explain</a> it well.</p>
</div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com3tag:blogger.com,1999:blog-1673244706324941001.post-48596242515389713792012-10-27T15:00:00.001-04:002012-12-17T17:01:38.418-05:00A Tribute to an Unsung PatternI've always been a big fan of the <a href="http://en.wikipedia.org/wiki/Autocompletion">autocomplete</a> UI pattern. Although it was invented way before, I guess it would be fair to say that its web incarnation was made ubiquitous in the wake of the <a href="http://en.wikipedia.org/wiki/Web_2.0">Web 2.0</a> revolution. It certainly wasn't its most important innovation, but I think this simple idea deserves more praise than it usually receives, because it has a deep impact on the way users interact with an underlying, often unknown data model (the Google search field is certainly the most powerful example). I'd like to pay a small tribute to this modest pattern by showing a possible implementation, using some of my favorite tools: Python, ExtJS and PostgreSQL. Many tutorials already exist for this of course, but I'd like to focus on the particular problem of being maximally tolerant with the user's input (i.e. not impose any structure or order on it), which I solve in a compact way with <a href="http://github.com/cjauvin/little_pger">little_pger</a>, a small Python "pseudo-ORM" module.
<h4>The Data</h4>
Say we'd like to create an autocomplete field for the list of Ubuntu releases, stored on our Postgres server:
<br /><br />
<!-- <a href="https://gist.github.com/3965131#file-ubuntu-sql" target="_blank">https://gist.github.com/3965131#file-ubuntu-sql</a> -->
<script src="https://gist.github.com/4262959.js"></script>
<h4>The Widget</h4>
Our JavaScript client widget will be an instance of the incredibly versatile <a href="http://dev.sencha.com/deploy/ext-4.1.0-gpl/examples/form/combos.html">ExtJS combobox</a>:
<br /><br />
<!-- <a href="https://gist.github.com/3965131#file-combo-js" target="_blank">https://gist.github.com/3965131#file-combo-js</a> -->
<script src="https://gist.github.com/4262874.js"></script>
Note that the <code>Model</code> data structure (required by the combobox, and which mirrors our database table) conflates the three fields (<code>adjective</code>, <code>animal</code> and <code>version</code>) into a single <code>release</code> field, with a <code>convert</code> function to specify the layout of the data that the user will see.
<h4>The Backend</h4>
The widget is fed by our <a href="http://flask.pocoo.org/">Python Flask WSGI</a> backend, with which it communicates using JSON:
<br /><br />
<!-- <a href="https://gist.github.com/3965131#file-backend-py" target="_blank">https://gist.github.com/3965131#file-backend-py</a> -->
<script src="https://gist.github.com/4262845.js"></script>
<h4>Making It a Little Smarter</h4>
To access the database, the backend code uses <a href="https://github.com/cjauvin/little_pger">little_pger</a>, a small module I wrote, which acts as a very thin (but Pythonic!) layer above <a href="http://www.initd.org/psycopg/">Psycopg2</a> and SQL. I use it as a replacement for an ORM (as I don't really like them) and thus call it a "pseudo-ORM". In this context, it does us a nice favor by solving the problem of searching through all the fields of the table, by AND-matching, in any order, the user-supplied tokens (e.g. "lynx 10.04 lucid" should match). For this, the <code>where</code> parameter of the <code>little_pger.select</code> function offers a nice syntactic flexibility, as those examples show:
<br /><br />
<!-- <a href="https://gist.github.com/3965131#file-where-py" target="_blank">https://gist.github.com/3965131#file-where-py</a> -->
<script src="https://gist.github.com/4262882.js"></script>
So in our context, the trick is to use the last pattern (i.e. <code>set</code>-based), with the concatenated (<code>||</code>) fields we are interested in searching, which could successfully match a search for "lynx 04 lucid" with this SQL query, for example:
<br /><br />
<!-- <a href="https://gist.github.com/3965131#file-query-sql" target="_blank">https://gist.github.com/3965131#file-query-sql</a> -->
<script src="https://gist.github.com/4262891.js"></script>
A more complete version of this tribute's code is available on <a href="https://github.com/cjauvin/autocomplete-tribute">GitHub</a>.
<br /><br />Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-57791346990356387472012-10-19T13:32:00.000-04:002012-10-19T13:32:34.187-04:00Stack Overflow Tag Similarity VisualizationFor the recent Kaggle Stack Overflow machine learning contest, I have created this <a href="https://www.kaggle.com/c/predict-closed-questions-on-stack-overflow/prospector#199">visualization submission</a>, where the words found in questions with the most frequent tags have been used to compute their semantic similarity. The result is a matrix where nearby columns (representing their word use patterns) should correspond to similar, or related tags.
<br /><br />
To do this, the $t$ most frequent tags have been extracted from a subset of the training examples, along with the $w$ most frequent words found in the "title" and "body" parts of questions tagged with at least one of these. This results in a $t \times w$ matrix where each column corresponds to a "tag vector" of w words. The tag vector components are computed using the <a href="http://en.wikipedia.org/wiki/TF_IDF">tf-idf</a> statistic, which tries to quantify the importance of a word by weighting its occurrence frequency for a particular tag ($tf$ term) against its frequency across the overall set of tags ($idf$ term). The problem with this representation is that neither the order of the columns nor the rows convey any information. Is there a way we could somehow reorder at least the tag vectors (i.e. columns)?
<br /><br />
From the previous matrix, we can compute a new $t \times t$ matrix where each cell corresponds to the pairwise <a href="http://en.wikipedia.org/wiki/Cosine_similarity">cosine similarity</a> between two tags, which is an estimate of their degree of semantic relatedness:
<br /><br />
<a href="http://i.imgur.com/3H5kq.png" imageanchor="1" style=""><img border="0" height="488" width="800" src="http://i.imgur.com/3H5kq.png" /></a>
<br /><br />
Using this similarity matrix, we can reorder the columns of the first matrix: column/tag 1 should be similar to column/tag 2, which in turn should be similar to column/tag 3, and so on. The ideal reordering would maximize the sum of similarities across the whole chain of tags, but as I'm pretty sure that finding it is a NP-complete problem (thus intractable even for such a small matrix), I had to settle on a suboptimal greedy solution. Still, some interesting patterns emerge from the result, as one can see a gradient of tag relatedness, ranging from operating systems, Microsoft and web technologies, C-family languages, Apple technologies, web again, databases, and some general purpose programming languages.
<br /><br />
<a href="http://i.imgur.com/D30WH.png" imageanchor="1" style=""><img border="0" height="488" width="800" src="http://i.imgur.com/D30WH.png" /></a>
<br /><br />
Finally, as it seems that my submission is not too popular for some reason, I really wouldn't mind a quick <a href="https://www.kaggle.com/c/predict-closed-questions-on-stack-overflow/prospector#199">thumbs up</a> on the Kaggle page, if you think it's a reasonable idea!
<br /><br />
Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com2tag:blogger.com,1999:blog-1673244706324941001.post-28767213030570278072012-08-27T16:58:00.000-04:002012-08-27T16:58:33.534-04:00An EC2 SouvenirAs I don't run Python programs requiring 25 gigabytes of memory very often, I took this souvenir snapshot from the console of my <a href="http://aws.amazon.com/ec2/instance-types/">EC2 High-Memory Double Extra Large Instance</a> while it was trying to make sense of a big dataset with a <a href="http://scikit-learn.org">scikit-learn</a> linear <a href="http://en.wikipedia.org/wiki/Support_vector_machine">SVM</a>:
<br /><br />
<center>
<a href="http://1.bp.blogspot.com/-eptM2CWpFww/UDveC2Sc-7I/AAAAAAAADHA/YzqR-1dmkp8/s1600/python_25GB_svm_ec2_souvenir.png" imageanchor="1" style=""><img border="0" width="75%" src="http://1.bp.blogspot.com/-eptM2CWpFww/UDveC2Sc-7I/AAAAAAAADHA/YzqR-1dmkp8/python_25GB_svm_ec2_souvenir.png" /></a>
</center>
<br />
Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com2tag:blogger.com,1999:blog-1673244706324941001.post-5892742582156684802012-06-24T15:47:00.000-04:002012-06-24T15:47:51.953-04:00from ikea import *I just finished assembling two BILLY Ikea bookcases (with their extension units) in a row (to capitalize on the (somewhat minimal) learning and memory inertia involved), and I got to admit that I found the process rather enjoyable, and almost <a href="http://www.python.org/dev/peps/pep-0020/">pythonic</a> in nature. I don't know if it has always been the case (I suspect it must be the result of a long evolution, with lots of trial and error), but I found the assembly instructions to be so clear, so minimal (with their visual-only language), so logical (with positive, as well as <i>negative</i> advices, very importantly) that it's almost impossible to make hard to unwind (or worst, unrecoverable) errors, if you take time to study them carefully. I had to write about it, and so I highly recommend Ikea assembling for a relaxing moment, away from more complicated and abstract intellectual problems.
<br /><br />Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-78464746863585833912012-06-18T14:41:00.000-04:002014-02-01T09:54:15.998-05:00Hello 6510!The way I see it, modern-day <a href="http://en.wikipedia.org/wiki/Commodore_64">C=64</a> hacking is the epitome of "autotelic computing" (a fancy new word I <a href="http://www.economist.com/blogs/babbage/2012/06/super-star-programmers">just</a> <a href="http://www.tempobook.com/2011/10/25/thrust-drag-and-the-10x-effect/">learned</a>, which seemed to capture so perfectly one of my deepest intellectual ideals, I couldn't resist renaming my blog).
<br /><br />
Like so many programmers of the current generation, I've been a long time C=64 hacker, very autotelic to boot. As soon as I realized that <a href="http://en.wikipedia.org/wiki/Commodore_BASIC">BASIC V2</a> was not very appropriate for game programming, I <i>had</i> to learn 6510 assembly. But for a French-speaking ~12 year-old, this is no easy task. With the help of a forgotten library book I was very lucky to find, I was able achieve a few proofs of concept, but not much more (sadly no Super Mario, King's Quest, <a href="http://en.wikipedia.org/wiki/Un_squadron">U.N. Squadron</a> or <a href="http://en.wikipedia.org/wiki/Power_Drift">Power Drift</a> clone, contrary to my wildest dreams).
<br /><br />
Nevertheless, I've been following with interest the many aspects of the nostalgia-fueled C=64 subculture that has been thriving exponentially with the internet in the last two decades: numerous <a href="http://vice-emu.sourceforge.net">emulators</a>, game repositories, <a href="http://www.cc65.org">compilers</a>, <a href="http://en.wikipedia.org/wiki/MOS_Technology_SID">SID</a>-based chiptune revival, etc. Fascinating stuff, driven mostly by the unflinching love of this old machine!
<br /><br />
Recently I got an idea for an autotelic project, that I won't disclose now, because it's not very likely to ever materialize (or should I say <i>logicalize</i>?). But to gently push it towards reality anyway, I thought I'd share and document my very first <a href="http://en.wikipedia.org/wiki/Hello_world_program">Hello World</a> attempt with modern C=64 tools.
<br /><br />
Why would you torture yourself with a bare-metal (or emulated) assembler when you can use such a powerful tool as the aptly named <a href="http://theweb.dk/KickAssembler/Main.php">Kick Assembler</a>? Java-based, command-line driven, sporting some augmented syntax and parsing capabilities, a higher-level scripting engine, etc. In fact this fine piece of engineering made me realize my own "shopping habits" when contemplating the adoption of a new software package: whenever I can find, within the first five pages of the manual, how to achieve the initial task I have in mind, I'm instantly sold. And that's precisely what happened with this one, as this assembly code snippet (adapted very slightly from <a href="http://en.wikipedia.org/wiki/Jim_Butterfield">Jim Butterfield</a>'s excellent book <i>Learning Machine Code Programming on the Commodore 64</i>) shows (that book, and better English courses perhaps, could have possibly stirred my game programming career otherwise):
<div class="code-bg"><pre>
:BasicUpstart2(start) <span class="comment-delimiter">// </span><span class="comment">autostart macro
</span>
<span class="function-name">start</span>: <span class="keyword">ldx</span> #$00 <span class="comment-delimiter">// </span><span class="comment">put 0 in X register
</span><span class="function-name">loop</span>: <span class="keyword">lda</span> $4000,x <span class="comment-delimiter">// </span><span class="comment">put X-indexed char of string (stored at $4000) in acc
</span> <span class="keyword">jsr</span> $ffd2 <span class="comment-delimiter">// </span><span class="comment">call CHROUT kernal routine to print single char in acc
</span> <span class="keyword">inx</span> <span class="comment-delimiter">// </span><span class="comment">increment X
</span> <span class="keyword">cpx</span> #12 <span class="comment-delimiter">// </span><span class="comment">did we reach the 12th char?
</span> <span class="keyword">bne</span> loop <span class="comment-delimiter">// </span><span class="comment">if not, loop again
</span> <span class="keyword">rts</span> <span class="comment-delimiter">// </span><span class="comment">if yes, terminate
</span>
<span class="keyword">.pc</span> = $4000 <span class="string">"data"</span>
<span class="keyword">.text</span> <span class="string">"HELLO WORLD!"</span>
</pre></div>
which is very easy to pipe directly in an emulator (I use <a href="http://vice-emu.sourceforge.net">x64</a> on Ubuntu):
<div class="code-bg"><pre>
$ java -jar KickAss.jar hello.asm -execute x64
</pre></div>
yielding the nostalgic and familiar (but feeling so <i>cramped</i> nowadays, don't you find?):
<br /><br />
<center>
<a href="http://3.bp.blogspot.com/--JyOUaajZEw/T99x9QDQMbI/AAAAAAAADFs/o3pi2QkO430/s1600/c64.png" imageanchor="1" style=""><img border="0" src="http://3.bp.blogspot.com/--JyOUaajZEw/T99x9QDQMbI/AAAAAAAADFs/o3pi2QkO430/c64.png" /></a>
</center>
<br />
This post may (or may not..) be followed by posts in the same vein.
<br /><br />
<b>Update</b>: my original idea was way more ambitious, but I ended up writing this <a href="https://github.com/cjauvin/tetris-464">Tetris clone</a>, to learn C=64 assembly. Without <a href="http://www.lemon64.com/forum/viewtopic.php?t=43001">help</a> from the <a href="http://www.lemon64.com">Lemon64</a> community, things would have been undeniably <a href="http://www.lemon64.com/forum/viewtopic.php?t=42953">harder</a>. If someone is interested, I could consider writing about it, tutorial-style.
<br /><br />
<center>
<a href="http://3.bp.blogspot.com/-H7kFU3Gzuco/UBAa7RtXjeI/AAAAAAAADGg/N-3URxHWqkE/s1600/tetris-c64.png" imageanchor="1" style=""><img border="0" src="http://3.bp.blogspot.com/-H7kFU3Gzuco/UBAa7RtXjeI/AAAAAAAADGg/N-3URxHWqkE/tetris-c64.png" /></a>
</center>
<br /><br />Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com1tag:blogger.com,1999:blog-1673244706324941001.post-48607784075141364812012-06-17T13:21:00.001-04:002012-07-15T15:20:15.054-04:00Two Tales of Data Clerking<h4>
I - A couple of years ago..</h4>
I was tasked with the creation of a web-based medical database application to manage TB case data. As the data had spatial components, the choice of a database tech (<a href="http://www.postgresql.org/">PostgreSQL</a>, with its <a href="http://postgis.refractions.net/">PostGIS</a> extension) was easy. For the client-side tech (mostly data entry forms coming from the paper world), the choice was harder. After having explored and reviewed many frameworks, I finally settled for the powerful <a href="http://www.sencha.com/products/extjs/">Ext JS</a> library. Lastly, as I wanted to code the server-side logic in Python, I decided to go with <a href="http://www.modpython.org/">mod_python</a>, which wasn't then in the almost <a href="http://en.wikipedia.org/wiki/Mod_python">defunct state</a> it is in today. I played for a while with the idea of a more integrated environment like <a href="https://www.djangoproject.com/">Django</a>, but I decided to go without after having realized that the mildly complicated database logic I had in mind was not well supported by an <a href="http://en.wikipedia.org/wiki/Object-relational_mapping">ORM</a>. In its place, I devised <a href="https://github.com/cjauvin/little_pger">little_pger</a>, a very simple, <a href="http://initd.org/psycopg/">psycopg2</a>-based "proto-ORM", wrapping common SQL queries in a <a href="http://cjauvin.blogspot.ca/2011/01/helper-module-for-postgresql-and.html">convenient and pythonic interface</a> (at least to my taste!).
<br />
<br />
Having made all these choices, I was ready to build the application. I did, and in retrospect, it's very clear that the most time-consuming aspect, by far, was the delicate user interface widgets I had to create to maximize user happiness and data consistency (with all the power Ext JS yields, it's hard to resist):
<ul>
<li>Different kinds of autocompletion text fields</li>
<li>Panels with variable number of sub-forms</li>
<li>Address validation widget</li>
<li>Etc..</li>
</ul>
<br />
<center>
<a href="http://3.bp.blogspot.com/-3gJ1EVM7v8o/T93xbIsTKbI/AAAAAAAADEY/p4wLDskH6G0/s1600/tbdb_ui.png" imageanchor="1" style=""><img border="0" height="195" width="320" src="http://3.bp.blogspot.com/-3gJ1EVM7v8o/T93xbIsTKbI/AAAAAAAADEY/p4wLDskH6G0/s320/tbdb_ui.png" /></a>
<span style="padding-left: 50px;"></span>
<a href="http://2.bp.blogspot.com/-WNgNnzWw4JY/T939RBiWR4I/AAAAAAAADE0/YiSdXSoit5I/s1600/tbdb_ui2.png" imageanchor="1" style=""><img border="0" height="195" width="320" src="http://2.bp.blogspot.com/-WNgNnzWw4JY/T939RBiWR4I/AAAAAAAADE0/YiSdXSoit5I/s320/tbdb_ui2.png" /></a>
</center>
<br />
The initial versions of the app were so heavy that I had to revise the whole UI architecture.. but then a nice thing happened: the JavaScript performance explosion brought by the browser war.. suddenly this wasn't a concern anymore!
<br />
<br />
This system has been working very reliably (from an Ubuntu VM) since then. It has required slightly more user training and adjustments than I'd assumed (sophisticated UI semantics is always clearer in the designer's mind) but it is still being used, and its users are still (seemingly) happy, and so this could have been the end of an happy, although rather uneventful story, if it wasn't for the fact that I actually spent some time meditating upon what I'd do differently, being offered the opportunity.
<br />
<br />
For instance, learning about <a href="http://en.wikipedia.org/wiki/Web_Server_Gateway_Interface">WSGI</a>, but above all, the very elegant <a href="http://flask.pocoo.org/">Flask</a> microframework, made me want to be part of the fun too. So I refactored my application in terms of it (which was mostly painless). So long mod_python..
<br />
<br />
By then however, a kind of "UI fatigue" had begun to set in, and when I was asked to quickly code an addon module, I admit it: I did it with Django (and no one really noticed).
<br />
<br />
Time passed, I worked on many other projects and then it came again.
<br />
<h4>
II - A couple of months ago..</h4>
I was asked to create a medical database to hold vaccination survey data. By that time, I had developed another condition: "database fatigue" (AKA obsessive-compulsive normalization), which is probably better explained by these schema diagrams from my previous app:
<br />
<br />
<center>
<a href="http://2.bp.blogspot.com/-4oLUPuLcmGU/T9ybJ1YwgKI/AAAAAAAADDM/82Eg3W7fjPM/s1600/tbdb_schema_hierch.png" imageanchor="1"><img border="0" height="261" src="http://2.bp.blogspot.com/-4oLUPuLcmGU/T9ybJ1YwgKI/AAAAAAAADDM/82Eg3W7fjPM/s320/tbdb_schema_hierch.png" style="border: 1px grey solid;" width="320" /></a>
<span style="padding-left: 50px;"></span>
<a href="http://2.bp.blogspot.com/-y6pVgk3uWXA/T9ydFatm6vI/AAAAAAAADDk/jlS7E53Yvp8/s1600/tbdb_schema_circ.png" imageanchor="1"><img border="0" height="306" src="http://2.bp.blogspot.com/-y6pVgk3uWXA/T9ydFatm6vI/AAAAAAAADDk/jlS7E53Yvp8/s320/tbdb_schema_circ.png" style="border: 1px grey solid;" width="320" /></a>
</center>
<br />
And so, even though I still strongly believed in the power of relational databases, as the <a href="http://en.wikipedia.org/wiki/Nosql">NoSQL</a> paradigm was gaining traction, <a href="http://www.mongodb.org/">MongoDB</a> seemed like a fun alternative to try.
<br />
<br />
This time however, as I lacked the courage (or ignorance) I had when facing the prospect of "webifying" 47 pages of paper forms (though it was only a mere 13 pages for this new task), I decided to explore other options. The solution I came up with is based on a kind of "end user outsourcing": since the paper form was already created as a PDF document, all I did was superimpose dynamic fields on it, and added a big "Submit" button on top (attached to a bit of JS that sends the form payload over HTTP). Of course this is not web-based anymore (rather HTTP-based), and it makes for a less sophisticated user experience (e.g. no autocompletion fields), but that's the price to pay for almost pain-free UI development. So long, <a href="http://dev.sencha.com/deploy/ext-4.0.0/examples/layout-browser/layout-browser.html">grid and accordion layouts</a>..
<br />
<br />
<center>
<a href="http://2.bp.blogspot.com/-uCPJbaq5xAw/T94FIlp7NqI/AAAAAAAADFE/wKVSTARM0GE/s1600/mc_form1.png" imageanchor="1" style=""><img border="0" height="195" width="320" src="http://2.bp.blogspot.com/-uCPJbaq5xAw/T94FIlp7NqI/AAAAAAAADFE/wKVSTARM0GE/s320/mc_form1.png" /></a>
<span style="padding-left: 50px;"></span>
<a href="http://2.bp.blogspot.com/-WfMCUGyoweo/T94FN9jmJSI/AAAAAAAADFQ/j60wXF_O7nY/s1600/mc_form2.png" imageanchor="1" style=""><img border="0" height="195" width="320" src="http://2.bp.blogspot.com/-WfMCUGyoweo/T94FN9jmJSI/AAAAAAAADFQ/j60wXF_O7nY/s320/mc_form2.png" /></a>
</center>
<br />
The really nice thing about it is hidden from the user's view: the <i>entire</i> application server reduces to this Python function (resting of course on the immense shoulders of <a href="http://flask.pocoo.org/">Flask</a> and <a href="http://pypi.python.org/pypi/pymongo">PyMongo</a>), which receives the form data (as a POST dict) and stores it directly in the MongoDB database, as a schemaless document:
<br />
<div class="code-bg">
<pre><span class="keyword">from</span> flask <span class="keyword">import</span> *
<span class="keyword">from</span> pymongo <span class="keyword">import</span> *
application = Flask(<span class="string">'form_app'</span>)
@application.route(<span class="string">'/submit'</span>, methods=[<span class="string">'POST'</span>])
<span class="keyword">def</span> <span class="function-name">submit</span>():
request.parameter_storage_class = <span class="py-builtins">dict</span>
db = Connection().form_app
db.forms.update({<span class="string">'case_id'</span>: request.form[<span class="string">'case_id'</span>]},
request.form, upsert=<span class="py-pseudo-keyword">True</span>, safe=<span class="py-pseudo-keyword">True</span>)
<span class="keyword">return</span> Response(<span class="string">"""%FDF-1.2 1 0 obj &lt;&lt;
/FDF &lt;&lt; /Status (The form was successfully submitted, thank you!) >>
>> endobj trailer &lt;&lt; /Root 1 0 R >>
%%EOF
"""</span>, mimetype=<span class="string">'application/vnd.fdf'</span>)
</pre>
</div>
Although for a shorter period of time, this system has also been working in quite a satisfying way since it went online. Its users seem happy, and even though it's less sexy than a web UI, there are merits to this distributed, closer-to-paper approach. The MongoDB piece also makes it a joy (and a breeze) to shuffle data around (though I'm not sure what is due to sheer novelty however).
<br /><br />
Of course these database app designs are so far apart that it could seem almost dishonest to compare them. But the goal was not to compare, just to show that it's interesting and sometimes fruitful to explore different design options, when facing similar problems.
<br /><br />Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-62807859453721526362012-05-31T19:39:00.000-04:002012-06-04T10:51:32.137-04:00Reverse Engineering the Birthday ParadoxRecently, I've been thinking about
the <a href="http://en.wikipedia.org/wiki/Birthday_paradox">Birthday
Paradox</a> (not to be confused with
this <a href="http://cjauvin.blogspot.com/2011/11/birthday-simulator.html">similarly
named problem</a>). The first time was while reviewing the concept
of <a href="http://en.wikipedia.org/wiki/Birthday_attack">birthday
attacks</a> for the final exam of
the <a href="https://www.coursera.org/course/crypto">online
cryptography class</a> I'm currently taking. The second time was while
reading
this <a href="http://web.stonehill.edu/compsci/History_Math/math-read.htm">intriguing
exploration</a> of the processes at play when struggling to understand
a math problem.
<br /><br />
The usual way to think about the problem is to consider the
probability $\overline{p}$ of the opposite event we're interested in:
that there is <i>no</i> birthday"collision" at all within a group of
$k$ people. This, we're said (without really explaining why), is
easier to calculate, and it follows from it that we can retrieve the
probability $p$ of our goal event (i.e. that there is <i>at least
one</i> "collision") as $1 - \overline{p}$.
<br /><br />
The above-mentioned essay discusses at length this idea and the
underlying principles, but it also casually challenges the reader with
an interesting side problem: find a way to calculate
$p$ <i>directly</i>, that is, without relying on the probability of
the opposite event. Being a programmer, I decided to tackle this
problem not from the principles, but from the opposite direction, by
trying to derive understanding from "procedural tinkering".
<br /><br />
In the first version of this article, I had derived a way of screening the
<a href="http://en.wikipedia.org/wiki/Permutation#In_combinatorics">$k$-permutations</a>
out of the
<a href="http://en.wikipedia.org/wiki/Cartesian_product#Cartesian_square_and_Cartesian_power">cartesian power set</a> $N^k$. I thought this was the answer, but someone on
the Cryptography Class discussion forum helped me understand that this
was actually only a rearrangement of the indirect computation (i.e. $p
= 1 - \overline{p}$). The correct way to compute $p$ directly should
rather involve the sum of:
<ul>
<li>the chance of a collision occurring <i>only</i> on the second choice</li>
<li>the chance of a collision occurring <i>only</i> on the third choice</li>
<li>...</li>
<li>the chance of a collision occurring <i>only</i> on the $k$-th choice</li>
</ul>
This seemed to make sense, but as I wanted to study this proposition
in more detail on a simplified problem instance, I wrote this Python
program:
<div class="code-bg"><pre>
<span class="keyword">from</span> __future__ <span class="keyword">import</span> division
<span class="keyword">from</span> itertools <span class="keyword">import</span> product
n = 10
k = 5
prev_colls = <span class="py-builtins">set</span>()
p_coll = 0
<span class="keyword">def</span> <span class="function-name">findPrevColl</span>(p, j):
<span class="keyword">for</span> i <span class="keyword">in</span> <span class="py-builtins">range</span>(2, j):
<span class="keyword">if</span> p[:i] <span class="keyword">in</span> prev_colls:
<span class="keyword">return</span> <span class="py-pseudo-keyword">True</span>
<span class="keyword">return</span> <span class="py-pseudo-keyword">False</span>
<span class="keyword">for</span> j <span class="keyword">in</span> <span class="py-builtins">range</span>(2, k+1):
n_colls = 0
count = 0
<span class="keyword">for</span> p <span class="keyword">in</span> product(<span class="py-builtins">range</span>(n), repeat=j):
<span class="keyword">if</span> <span class="py-builtins">len</span>(set(p)) < j <span class="keyword">and</span> <span class="keyword">not</span> findPrevColl(p, j):
n_colls += 1
prev_colls.add(p)
count += 1
<span class="py-builtins">print</span> <span class="string">'at k = %d, P_k = %f'</span> % (j, n_colls / count)
p_coll += n_colls / count
<span class="py-builtins">print</span> <span class="string">'sum(P_k) = %f'</span> % p_coll
<span class="comment-delimiter"># </span><span class="comment">verify result with the "indirect" formula
</span><span class="keyword">import</span> operator
<span class="keyword">from</span> numpy.testing <span class="keyword">import</span> assert_approx_equal
assert_approx_equal(p_coll, 1 - <span class="py-builtins">reduce</span>(operator.mul, <span class="py-builtins">range</span>(n-k+1, n+1)) / n ** k)
<span class="comment-delimiter"># </span><span class="comment">at k = 2, P_k = 0.100000
</span><span class="comment-delimiter"># </span><span class="comment">at k = 3, P_k = 0.180000
</span><span class="comment-delimiter"># </span><span class="comment">at k = 4, P_k = 0.216000
</span><span class="comment-delimiter"># </span><span class="comment">at k = 5, P_k = 0.201600
</span><span class="comment-delimiter"># </span><span class="comment">sum(P_k) = 0.697600</span>
</pre></div>
which works by enumerating, for every $k$, all the possible trials (of
length $k$) and looking for a "new" collision with each one, "new"
meaning that no subset (of length $< k$) of this particular trial has
ever been causing a collision. The probabilities for each $k$ are then
summed to yield the overall, direct probability of at least one
collision. Note that this code relies on the <code>itertools</code>
module's <a href="http://docs.python.org/library/itertools.html#itertools.product">product</a>
function to generate the cartesian powers of $k$ (i.e. all possible
trials of length $k$) and the
<code>set</code> data structure for easy past and current collision
detection. Once fully convinced that this was working, the obvious
next step was to derive an analytic formulation for it. By studying
some actual values from my program, I figured out that the probability
for a collision occurring <i>only</i> on the $k$-th choice should be:
\[ P_{only}(n, k) = \frac{\left(\frac{n! \cdot (k-1)}{(n-k+1)!}\right)}{n^k}\]
meaning that the total, direct probability of at least one collision
is the sum:
\[ P_{any}(n, k) = \sum_k{\frac{\left(\frac{n! \cdot (k-1)}{(n-k+1)!}\right)}{n^k}}\]
As this wasn't still fully satisfying, because it doesn't yield an
intuitive understanding of what's happening, the same helpful person
on the Cryptography forum offered an equivalent, but much better
rewrite:
\[ P_{only}(n, k) = \left(\frac{n}{n} \cdot \frac{n-1}{n} \cdot \cdot \cdot \frac{n-k+2}{n}\right) \cdot \left(\frac{k-1}{n}\right)\]
which can be easily understood by imagining a bucket of $n$ elements:
the chance of a collision happening exactly at choice $k$ is the probability that
there was <i>no</i> collision with the first $k-1$ choices
($\frac{n}{n} \cdot \frac{n-1}{n} \cdot \cdot \cdot$, increasing as
the bucket fills up) multiplied by the probability of a collision at
$k$, which will happen $k-1$ times out of $n$, if we assume that the
bucket is already filled with $k-1$ different values (by virtue of the
no-previous-collision assumption).
<br /><br />
Although the conclusion of my previous attempt was that it's often
possible to derive mathematical understanding from "procedural
tinkering" (i.e. from programming to math), I'm not so sure anymore,
as this second version is definitely a counterexample.
<br /><br />Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-5828261263043484852012-04-24T11:46:00.000-04:002012-05-22T10:13:23.830-04:00Lean Geocoding<a href="http://en.wikipedia.org/wiki/Geocoding">Geocoding</a> is the art of turning street addresses into geographical coordinates (e.g. lat/long). In this article we will explore some of the essential ideas upon which a geocoder can be built, along with a simple example assembled from freely available components.
<h4>The Data Model</h4>
A geocoder is comprised of two main components: a road network database (with an associated <i>data model</i>) and a set of queries acting on it. The database is spatial in the sense that it contains a representation of the road network as a set of geometric features (i.e. typically lines). The representation mechanism works at the street segment level: a <i>row</i> corresponds to the subset of a street defining a block (usually mostly linear, but not always, as in most North American cities, for instance):
<br /><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-tsnNubRJ8JA/T3kLeTSmk_I/AAAAAAAACoU/dNnYQvVT3D8/s1600/street_segment.png" imageanchor="1" style=""><img border="0" height="271" width="400" src="http://3.bp.blogspot.com/-tsnNubRJ8JA/T3kLeTSmk_I/AAAAAAAACoU/dNnYQvVT3D8/s400/street_segment.png" /></a></div>
In addition to a geometric attribute (describing its shape), a given street segment is associated to four numeric (i.e. non-spatial) attributes: <code>from_left</code>, <code>to_left</code>, <code>from_right</code> and <code>to_right</code>. These numbers correspond to the ranging values of the typical <a href="http://en.wikipedia.org/wiki/House_numbering">house numbering</a> scheme, having the odd numbers on one side of the street, and the even ones on the other. If we add its name as an additional string attribute, a street in this model can be thus defined (in most cases) as the set of segments sharing it:
<div class="code-bg"><pre>
name | from_left | to_left | from_right | to_right | astext(the_geom)
------------+-----------+---------+------------+----------+---------------------------------------------------
[...]
Jean-Talon | 1000 | 1024 | 1001 | 1035 | LINESTRING(-73.625538467 45.524777946,-73.625[...]
Jean-Talon | 1001 | 1025 | 1000 | 1080 | LINESTRING(-73.6126363129999 45.5417330830001[...]
Jean-Talon | 1101 | 1185 | 1110 | 1150 | LINESTRING(-73.612026869 45.5425537690001,-73[...]
Jean-Talon | 1201 | 1247 | 1210 | 1244 | LINESTRING(-73.611316541 45.543310246,-73.610[...]
Jean-Talon | 1213 | 1245 | 200 | 260 | LINESTRING(-71.2765479409999 46.8713327090001[...]
Jean-Talon | 1247 | 1273 | 270 | 298 | LINESTRING(-71.2756217529999 46.8717741090001[...]
Jean-Talon | 1251 | 1293 | 1250 | 1294 | LINESTRING(-73.610724326 45.543951109,-73.610[...]
Jean-Talon | 1275 | 1299 | 300 | 360 | LINESTRING(-71.274682868 46.872224756,-71.273[...]
Jean-Talon | 1383 | 1383 | 1360 | 1388 | LINESTRING(-73.609408209 45.545375275,-73.608[...]
Jean-Talon | 1385 | 1385 | 1390 | 1408 | LINESTRING(-73.608972737 45.545850934,-73.608[...]
[...]
</pre></div>
<h4>The Database</h4>
As part of the 2011 Census, Statistics Canada offers (free of charge!) the <a href="http://www12.statcan.gc.ca/census-recensement/2011/geo/RNF-FRR/index-eng.cfm">Road Network File</a>, modeling the full country. Let's first import the shapefile version (<code>grnf000r11a_e.shp</code>) of the dataset into a <a href="http://postgis.refractions.net">PostGIS</a> spatial database:
<div class="code-bg"><pre>
$ createdb -T template_postgis geocoding
$ shp2pgsql -W latin1 -s 4269 -S grnf000r11a_e.shp rnf | psql -d geocoding
</pre></div>
Note that the <code>-s 4269</code> option refers to the <a href="http://en.wikipedia.org/wiki/Map_projection">mapping projection system</a> <a href="http://spatialreference.org/ref/epsg/4269">used</a> (and specified in the manual), while <code>-S</code> enforces the generation of simple, rather than MULTI geometries for spatial data. Let's also cast some columns to a numeric type and rename them, to make our life simpler later:
<div class="code-bg"><pre>
$ psql -d geocoding -c 'alter table rnf alter afl_val type int using afl_val::int
$ psql -d geocoding -c 'alter table rnf rename afl_val to from_left'
$ psql -d geocoding -c 'alter table rnf alter atl_val type int using atl_val::int'
$ psql -d geocoding -c 'alter table rnf rename atl_val to to_left'
$ psql -d geocoding -c 'alter table rnf alter afr_val type int using afr_val::int'
$ psql -d geocoding -c 'alter table rnf rename afr_val to from_right'
$ psql -d geocoding -c 'alter table rnf alter atr_val type int using atr_val::int'
$ psql -d geocoding -c 'alter table rnf rename atr_val to to_right'
</pre></div>
<h4>Address Interpolation</h4>
Since it wouldn't make sense to store an exhaustive list of all the existing address locations in which we'd perform a simple lookup (there are too many, and their configuration is rapidly evolving), our database is actually a compressed model of reality. To perform the geocoding of an actual address, we need an additional tool, <a href="http://en.wikipedia.org/wiki/Interpolation">interpolation</a>, with which we can find any new point on a line on the basis of already defined points. For instance, given a street segment ranging from 1000 to 2000, it takes a very simple calculation to determine that a house at 1500 should be about halfway. For a piecewise linear segment, the calculation is a little bit more involved (although as intuitively clear), but thanks to PostGIS, the details are hidden from us with the help of the <code>ST_Line_Interpolate_Point</code> function. Here is the result of interpolating address 6884 on a segment ranging (on the left) from 6878 to 6876:
<br /><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-coH3SfGp_UA/T3ojqyRGdbI/AAAAAAAACps/fU0YDdj1NQY/s1600/louishemon_geo.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://2.bp.blogspot.com/-coH3SfGp_UA/T3ojqyRGdbI/AAAAAAAACps/fU0YDdj1NQY/s400/louishemon_geo.png" /></a></div>
<h4>The Query</h4>
At this point, we have everything we need to build a basic geocoding query, which we wrap inside a SQL function (mostly for syntactic purposes):
<div class="code-bg"><pre>
<span class="keyword">create</span> <span class="keyword">function</span> <span class="function-name">geocode</span>(street_nb <span class="type">int</span>, street_name text) <span class="keyword">returns</span> setof geometry <span class="keyword">as</span> $$
<span class="keyword">select</span> st_line_interpolate_point(the_geom,
<span class="keyword">case</span> <span class="keyword">when</span> $1 % 2 = from_left % 2 <span class="keyword">and</span>
$1 <span class="keyword">between</span> least(from_left, to_left) <span class="keyword">and</span> greatest(from_left, to_left)
<span class="keyword">then</span> <span class="keyword">case</span> <span class="keyword">when</span> from_left = to_left <span class="keyword">then</span> 0
<span class="keyword">else</span> ($1 - least(from_left, to_left)) /
(greatest(from_left, to_left) - least(from_left, to_left))::<span class="type">real</span>
<span class="keyword">end</span>
<span class="keyword">when</span> $1 % 2 = from_right % 2 <span class="keyword">and</span>
$1 <span class="keyword">between</span> least(from_right, to_right) <span class="keyword">and</span> greatest(from_right, to_right)
<span class="keyword">then</span> <span class="keyword">case</span> <span class="keyword">when</span> from_right = to_right <span class="keyword">then</span> 0
<span class="keyword">else</span> ($1 - least(from_right, to_right)) /
(greatest(from_right, to_right) - least(from_right, to_right))::<span class="type">real</span>
<span class="keyword">end</span>
<span class="keyword">end</span>)
<span class="keyword">from</span> rnf
<span class="keyword">where</span> <span class="keyword">name</span> = $2 <span class="keyword">and</span>
((from_left % 2 = $1 % 2 <span class="keyword">and</span> $1 <span class="keyword">between</span> least(from_left, to_left) <span class="keyword">and</span> greatest(from_left, to_left))
<span class="keyword">or</span>
(from_right % 2 = $1 % 2 <span class="keyword">and</span> $1 <span class="keyword">between</span> least(from_right, to_right) <span class="keyword">and</span> greatest(from_right, to_right)))
$$ <span class="keyword">language</span> <span class="keyword">sql</span> immutable;
<span class="keyword">select</span> astext(geocode(1234, <span class="string">'Jean-Talon'</span>)); <span class="comment">-- POINT(-73.6108985068823 45.5437626198824)</span>
</pre></div>
The <code>where</code> clause finds the target segment, while the nested <code>case</code> expression as a whole calculates the fraction to be fed (along with the segment geometry) to the <code>ST_Line_Interpolate_Point</code> function (i.e. the proportion of the target segment length at which the address point should be located). Both work with the <code>between</code>, <code>least</code> and <code>greatest</code> functions (without which the query would be even more verbose), the last two needed in particular because the order of the range values cannot be assumed. The outer <code>case</code> expression finds the side of the street, depending on the range value parity (easily checked with <code>%</code>, the modulo operator), while the inner one checks for the special case where the range is comprised of only the target address (which would yield a division by zero, if not properly handled). Finally, the function can be declared <code>immutable</code> for optimal performance, since it should have no effect on the database.
<h4>Street Name Spelling Tolerance</h4>
The mechanism described above works well if we happen to use the exact same spelling for street names as the database (for instance, if the input UI is some kind of autocompletion field tapping directly into it), but it breaks rapidly if we introduce real-world inputs (e.g. via a free text field without any constaints). For instance, the query above wouldn't work with even a slight variation (e.g. "1234 Jean Talon", where just the hyphen is missing), because the string matching performed is not flexible at all: either the name is right or it isn't. Moreoever, in a real world system, the very notion of a spelling error is vague, because it spans a wide spectrum ranging from expectable variations, like the use of abbreviations ("Saint-Jerome" vs. "St-Jerome"), use of letter case ("SAINT-JEROME"), accents ("Saint-Jérôme", at least in some languages), extra specifiers ("av. Saint-Jerome E.") to downright spelling errors ("Saint-Jerrome"). Although we cannot solve this problem entirely, the use of certain string transformation and matching techniques can help, by introducing some tolerance in the process. The simple mechanism I will describe has two components. The street names (both from the query and the database) are first <i>normalized</i> (or standardized), to ease the process of comparing them (by effectively reducing the space of possible name representations). The comparison of the normalized names is then performed using an <i>edit distance</i> metric, with the goal of quantifying the <i>similarity</i> of two names, to determine whether they match or not (within a certain tolerance parameter).
<h5>Normalization</h5>
The most obvious normalizing transformations are: (1) replacing any non-alphanumeric character by a single one, (2) converting to upper case and (3) removing accents (which is particularly relevant in Canada, given the French speaking Quebec province, but also unfortunately requires a <a href="http://www.laudatio.com/wordpress/2008/11/05/postgresql-83-to_ascii-utf8/">small hack</a>, at least with my particular version of PG). We can also use regular expressions to detect and normalize certain common naming patterns. All the required transformations can be encapsulated in such a PL/pgSQL function, for instance:
<div class="code-bg"><pre>
<span class="keyword">create</span> <span class="keyword">function</span> <span class="function-name">normalize</span>(in text) <span class="keyword">returns</span> text <span class="keyword">as</span> $$
<span class="keyword">declare</span>
s text;
<span class="keyword">begin</span>
s = to_ascii(convert_to(<span class="builtin">upper</span>(<span class="builtin">trim</span>($1)), <span class="string">'latin1'</span>), <span class="string">'latin1'</span>);
s = regexp_replace(s, E<span class="string">'\\W+'</span>, <span class="string">'%'</span>, <span class="string">'g'</span>);
s = regexp_replace(s, E<span class="string">'^SAINTE%|^SAINT%|^STE%|^ST%'</span>, <span class="string">'S*%'</span>, <span class="string">'g'</span>);
<span class="keyword">return</span> s;
<span class="keyword">end</span>;
$$ <span class="keyword">language</span> plpgsql;
<span class="keyword">select</span> normalize(<span class="string">'St-Jérôme'</span>); <span class="comment">-- S*%JEROME</span>
</pre></div>
which we can now use to precompute the normalized database street names:
<div class="code-bg"><pre>
$ psql -d geocoding -c 'alter table rnf add column name_norm text'
$ psql -d geocoding -c 'update rnf set name_norm = normalize(name)'
</pre></div>
<h5>Similarity Matching</h5>
The last step is performed by using an <a href="http://en.wikipedia.org/wiki/Edit_distance">edit distance</a> metric to define what is an acceptable matching level (or similarity) between two names. A particular instance of such a metric is the well-known <a href="http://en.wikipedia.org/wiki/Levenshtein_distance">Levenshtein distance</a>, which is implemented by the <a href="http://www.postgresql.org/docs/8.3/static/fuzzystrmatch.html">fuzzystrmatch</a> PG module, and can be interpreted roughly as the "number of edits needed to transform a string A into string B". Here is an updated version of our geocoding function (with an additional parameter for the edit distance deviation tolerance), with which slight errors can now be forgiven, thanks to our flexible name spelling mechanism:
<div class="code-bg"><pre>
<span class="keyword">create</span> <span class="keyword">function</span> <span class="function-name">geocode</span>(street_nb <span class="type">int</span>, street_name text, tol <span class="type">int</span>) <span class="keyword">returns</span> setof geometry <span class="keyword">as</span> $$
<span class="keyword">select</span> st_line_interpolate_point(the_geom,
<span class="keyword">case</span> <span class="keyword">when</span> $1 % 2 = from_left % 2 <span class="keyword">and</span>
$1 <span class="keyword">between</span> least(from_left, to_left) <span class="keyword">and</span> greatest(from_left, to_left)
<span class="keyword">then</span> <span class="keyword">case</span> <span class="keyword">when</span> from_left = to_left <span class="keyword">then</span> 0
<span class="keyword">else</span> ($1 - least(from_left, to_left)) /
(greatest(from_left, to_left) - least(from_left, to_left))::<span class="type">real</span>
<span class="keyword">end</span>
<span class="keyword">when</span> $1 % 2 = from_right % 2 <span class="keyword">and</span>
$1 <span class="keyword">between</span> least(from_right, to_right) <span class="keyword">and</span> greatest(from_right, to_right)
<span class="keyword">then</span> <span class="keyword">case</span> <span class="keyword">when</span> from_right = to_right <span class="keyword">then</span> 0
<span class="keyword">else</span> ($1 - least(from_right, to_right)) /
(greatest(from_right, to_right) - least(from_right, to_right))::<span class="type">real</span>
<span class="keyword">end</span>
<span class="keyword">end</span>)
<span class="keyword">from</span> rnf
<span class="keyword">where</span> levenshtein(normalize($2), name_norm) <= $3 <span class="keyword">and</span>
((from_left % 2 = $1 % 2 <span class="keyword">and</span> $1 <span class="keyword">between</span> least(from_left, to_left) <span class="keyword">and</span> greatest(from_left, to_left))
<span class="keyword">or</span>
(from_right % 2 = $1 % 2 <span class="keyword">and</span> $1 <span class="keyword">between</span> least(from_right, to_right) <span class="keyword">and</span> greatest(from_right, to_right)))
$$ <span class="keyword">language</span> <span class="keyword">sql</span> immutable;
<span class="keyword">select</span> astext(geocode(1234, <span class="string">'Jean Tallon'</span>, 3)); <span class="comment">-- POINT(-73.6108985068823 45.5437626198824)</span>
</pre></div>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com0tag:blogger.com,1999:blog-1673244706324941001.post-58753063532618765282012-02-17T17:00:00.001-05:002014-04-26T10:26:45.645-04:00An Eulerian Hoax<!-- <script src="https://github.com/downloads/processing-js/processing-js/processing-1.3.6.min.js"></script> -->
<script src="http://cdnjs.cloudflare.com/ajax/libs/processing.js/1.4.7/processing.min.js"></script>
<p>Let's end the suspense.. there was a big catch with the <a href="http://cjauvin.blogspot.com/2012/02/implicit-bridges.html">previous puzzle</a>: it was impossible to solve!</p>
<p>As you may have suspected, the reason is mathematical, and it boils down to the fact that the puzzle actually corresponds to the <a href="http://en.wikipedia.org/wiki/Dual_graph">dual graph</a> of another graph, this one lacking an essential property: an <a href="http://en.wikipedia.org/wiki/Eulerian_path">Eulerian path</a>. Even though it may appear different at first, this is actually quite similar to the well-known problem of the <a href="http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg">Seven Bridges of Königsberg</a>, which was solved by Euler in 1735.</p>
<p>To explore what it means in more details, let's first consider the first, easy problem and its dual graph. The dual graph (in red) models the topology of the regions implicitly formed by the edge network of the blue graph: it tells which region connects to which. Notice that the space surrounding the blue graph, being a single region, corresponds to the top red node, reachable by all the red nodes lying in a region with "outside facing (blue) walls":</p>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="http://4.bp.blogspot.com/-8HDk27IVNJY/Tz6DUfLyDTI/AAAAAAAACcI/_ucTxDKmmcs/easygraphs.png" />
</div>
<p>Starting at the top red node (i.e. "outside", in terms of the dual graph), there exists a path (that can be followed with the arrows) with which we will eventually follow every red edges, without visiting any one twice: this is an <i>Eulerian path</i> (since this path is starting and ending at the same node, it's actually more precisely an <i>Eulerian circuit</i>). A path in this graph corresponds to a "pencil path" across the blue edges (or "walls") of the puzzle graph:</p>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="http://1.bp.blogspot.com/-qdD1fPZzBPo/Tz6HNwKNtyI/AAAAAAAACcU/4Ouon7GqLmc/easygraphpath.png" />
</div>
<p>Now while trying to solve the previous, <a href="http://cjauvin.blogspot.com/2012/02/implicit-bridges.html">impossible puzzle</a> (in blue), you were in reality trying to find an Eulerian path within its dual graph (in red):</p>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="http://4.bp.blogspot.com/-Qd8Uh_38arY/Tz6OIOQfkjI/AAAAAAAACcg/8a8pRWSQPbc/impossiblegraph.png" />
</div>
<p>which was proven impossible by Euler in this particular case, because it does not satisfy the (necessary and sufficient) condition of having zero or two (red) nodes of degree two. Specifically, as can be seen, it has four nodes of odd degree.</p>
<p>Just for fun, let's do the opposite, and try to cast the graph of the Königsberg problem into a form suitable for another interactive puzzle applet. From the simple and compact graph that can be derived from the bridge and island structure, it's not immediately clear how to do this (at least not in terms of the straight segments that the applet requires, as far as I can see):</p>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="http://2.bp.blogspot.com/-0JJUEtQMvk8/Tz7WkL9dnsI/AAAAAAAACdo/GAjrutNiuBI/konigsberg11.png" />
</div>
<p>But if we simply rearrange the edge between the top and bottom nodes, it becomes easy:</p>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="http://3.bp.blogspot.com/-F7yHH9mRet0/T0GAw74ovpI/AAAAAAAACd4/2BaI1Qc28wE/s400/konigsberg2.png" />
</div>
<p>which yields this (impossible, you've been warned this time!) puzzle (click anywhere on the grid to place the cursor, and drag any of its ends through the segments; you can press "r" to reset, "b" or right-click to move back, or use the yellow and blue buttons at the top):</p>
<script type="text/processing" data-processing-target="mycanvas3">
Grid grid = null;
int width = 600;
int height = 500;
int cell_size = 5;
Cursor[] cursors = null;
int last_picked_cursor = -1;
int n_steps_back = 10;
void setup() {
size(width, height);
grid = new Grid(height / cell_size, width / cell_size);
int nr = grid.n_rows;
int nc = grid.n_cols;
// konigsberg
grid.addAnchor(1 + nr/4 + 2, 1 + nc/8);
grid.addAnchor(1 + nr/4 + 2, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + nr/4 + 2, 1 + nc/8 + 2 * nc/8*3);
grid.addAnchor(1 + 3 * nr/4 + 2 - 5, 1 + nc/8);
grid.addAnchor(1 + 3 * nr/4 + 2 - 5, 1 + nc/8 + 2 * nc/8*3);
grid.addWall(0, 1);
grid.addWall(1, 2);
grid.addWall(2, 4);
grid.addWall(4, 3);
grid.addWall(3, 0);
grid.addWall(3, 1);
grid.addWall(1, 4);
grid.finalizeWallAnchors();
grid.findZones();
//grid.showZones();
// button border
ArrayList<PVector> reset_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
reset_btn_data.add(new PVector(i, j));
}
}
}
// button cell data
reset_btn_data.add(new PVector(2, 4));
reset_btn_data.add(new PVector(2, 5));
reset_btn_data.add(new PVector(2, 7));
reset_btn_data.add(new PVector(3, 3));
reset_btn_data.add(new PVector(3, 6));
reset_btn_data.add(new PVector(3, 7));
reset_btn_data.add(new PVector(4, 2));
reset_btn_data.add(new PVector(4, 5));
reset_btn_data.add(new PVector(4, 6));
reset_btn_data.add(new PVector(4, 7));
reset_btn_data.add(new PVector(5, 2));
reset_btn_data.add(new PVector(6, 3));
reset_btn_data.add(new PVector(7, 4));
reset_btn_data.add(new PVector(7, 5));
reset_btn_data.add(new PVector(7, 6));
grid.addButton(reset_btn_data, new PVector(6, 6),
new PVector(200, 200, 0), new PVector(255, 255, 0));
ArrayList<PVector> back_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
back_btn_data.add(new PVector(i, j));
}
}
}
back_btn_data.add(new PVector(2, 4));
back_btn_data.add(new PVector(3, 3));
back_btn_data.add(new PVector(4, 2));
back_btn_data.add(new PVector(4, 3));
back_btn_data.add(new PVector(4, 4));
back_btn_data.add(new PVector(4, 5));
back_btn_data.add(new PVector(4, 6));
back_btn_data.add(new PVector(4, 7));
back_btn_data.add(new PVector(5, 3));
back_btn_data.add(new PVector(6, 4));
grid.addButton(back_btn_data, new PVector(6, grid.n_cols - 14),
new PVector(0, 0, 200), new PVector(0, 0, 255));
}
void draw() {
grid.updateButtonTimer();
}
void mousePressed() {
if (mouseButton == LEFT) {
PVector mouse_gc = grid.getCell(mouseX, mouseY);
if (cursors == null && grid.get(mouse_gc).isStartable()) {
cursors = new Cursor[2];
cursors[0] = new Cursor(mouse_gc);
cursors[1] = new Cursor(mouse_gc);
}
if (cursors != null) {
if (cursors[0].pick(mouse_gc)) {
last_picked_cursor = 0;
cursors[0].setActive(true);
cursors[1].setActive(false);
} else if (cursors[1].pick(mouse_gc)) {
last_picked_cursor = 1;
cursors[0].setActive(false);
cursors[1].setActive(true);
} else {
grid.buttonClicked(mouse_gc);
}
}
} else if (mouseButton == RIGHT) {
if (cursors != null) {
grid.pushButton(1);
}
}
}
//void mouseMoved() {
void mouseDragged() {
if (cursors != null) {
if (cursors[0].is_picked) {
cursors[0].move(grid.getCell(mouseX, mouseY));
} else if (cursors[1].is_picked) {
cursors[1].move(grid.getCell(mouseX, mouseY));
}
}
}
void mouseReleased() {
if (cursors != null) {
cursors[0].is_picked = false;
cursors[1].is_picked = false;
}
}
void keyPressed() {
if (key == 'r') {
grid.pushButton(0);
} else if (key == 'b') {
if (cursors != null) {
grid.pushButton(1);
}
}
}
boolean eq(PVector v1, PVector v2) {
return (v1.x == v2.x && v1.y == v2.y && v1.z == v2.z);
}
boolean adj(PVector v1, PVector v2) {
return abs(v1.x - v2.x) <= 1 && abs(v1.y - v2.y) <= 1;
}
class Grid {
Cell[][] cells;
int n_rows, n_cols;
ArrayList<ArrayList> wall_gcs;
ArrayList<Integer> wall_states;
ArrayList<PVector> anchor_gcs;
ArrayList<ArrayList> button_data;
int btn_timer_started_at, btn_timer_delay, btn_timer_id;
Grid(int nr, int nc) {
n_rows = nr;
n_cols = nc;
wall_gcs = new ArrayList();
wall_states = new ArrayList();
anchor_gcs = new ArrayList();
cells = new Cell[n_rows + 2][n_cols + 2]; // add 6-cell padding to each side
button_data = new ArrayList();
btn_timer_started_at = -1;
btn_timer_delay = 50;
btn_timer_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
cells[i][j] = new Cell((j-1)*cell_size, (i-1)*cell_size);
}
}
}
Cell get(PVector gc) {
return cells[int(gc.x)][int(gc.y)];
}
Cell get(int i, int j) {
return cells[i][j];
}
PVector getCell(int x, int y) {
return new PVector(int(y / cell_size) + 1, int(x / cell_size) + 1);
}
// 8x8 button with 10x10 frame
void addButton(ArrayList<PVector> data, PVector topleft_gc, PVector col, PVector pcol) {
int btn_id = button_data.size();
int top = int(topleft_gc.x);
int left = int(topleft_gc.y);
// button background must also be set
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
get(top + i, left + j).setButton(btn_id, null, null);
}
}
ArrayList<PVector> data_tl = new ArrayList(); // data repositioned at topleft
for (int i = 0; i < data.size(); i++) {
PVector gc = data.get(i);
gc.add(topleft_gc);
get(gc).setButton(btn_id, col, pcol);
data_tl.add(gc);
}
button_data.add(data_tl);
}
void updateButton(int btn_id, boolean is_pushed) {
ArrayList<PVector> data = button_data.get(btn_id);
for (int i = 0; i < data.size(); i++) {
grid.get(data.get(i)).pushButton(is_pushed);
}
}
int buttonClicked(PVector mouse_gc) {
int btn_id = get(mouse_gc).button_id;
if (btn_id >= 0) {
pushButton(btn_id);
}
return btn_id;
}
void pushButton(int btn_id) {
updateButton(btn_id, true);
startButtonTimer(btn_id);
}
void startButtonTimer(int btn_id) {
btn_timer_started_at = millis();
btn_timer_id = btn_id;
}
void updateButtonTimer() {
if (btn_timer_started_at > 0) {
if (millis() - btn_timer_started_at >= btn_timer_delay) {
updateButton(btn_timer_id, false);
if (btn_timer_id == 0) {
cursors = null;
reset();
} else if (btn_timer_id == 1) {
cursors[last_picked_cursor].back(n_steps_back);
}
btn_timer_started_at = -1;
btn_timer_id = -1;
}
}
}
// a wall links an anchor to an other (they must be on same row or col)
void addWall(int a1, int a2) {
ArrayList wall_cells = new ArrayList();
PVector a1_gc = anchor_gcs.get(a1);
PVector a2_gc = anchor_gcs.get(a2);
int wall_id = wall_gcs.size();
// vertical wall (i1==i2)
if (a1_gc.y == a2_gc.y) {
if (a1_gc.x > a2_gc.x) { // swap wrt i
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int j = int(a1_gc.y);
for (int i = int(a1_gc.x)+1; i < int(a2_gc.x); i++) {
get(i, j).setWall(wall_id);
wall_cells.add(new PVector(i, j));
}
// horizontal wall (j1==j2)
} else if (a1_gc.x == a2_gc.x) {
if (a1_gc.y > a2_gc.y) { // swap wrt j
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int i = int(a1_gc.x);
for (int j = int(a1_gc.y)+1; j < int(a2_gc.y); j++) {
get(i, j).setWall(wall_id);
wall_cells.add(new PVector(i, j));
}
// diagonal wall
} else {
//assert(abs(a1_gc.x - a2_gc.x) == abs(a1_gc.y - a2_gc.y)); // must be 45 degrees
if (a1_gc.y > a2_gc.y) { // swap wrt j
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int i, j;
for (int k = 0; k < int(abs(a2_gc.x - a1_gc.x)); k++) {
if (a1_gc.x < a2_gc.x) { // right/downward
i = int(a1_gc.x) + k + 1;
} else { // right/upward
i = int(a1_gc.x) - k;
}
j = int(a1_gc.y) + k;
get(i, j).setWall(wall_id);
wall_cells.add(new PVector(i, j));
if (a1_gc.x < a2_gc.x) { // right/downward
get(i, j+1).setWall(wall_id);
wall_cells.add(new PVector(i, j+1));
} else {
get(i, j+1).setWall(wall_id);
wall_cells.add(new PVector(i, j+1));
}
}
}
wall_gcs.add(wall_cells);
wall_states.add(-1);
}
void addAnchor(int i, int j) {
get(i, j).setAnchor(anchor_gcs.size());
anchor_gcs.add(new PVector(i, j));
}
void findZones() {
int zone_id = 0;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
if (get(i, j).isZoneFree()) {
zoneFill(new PVector(i, j), zone_id);
zone_id += 1;
}
}
}
//println("found " + zone_id + " zones");
}
// find distinct zones by flood filling
void zoneFill(PVector start_gc, int zone_id) {
ArrayList<PVector> frontier = new ArrayList();
frontier.add(start_gc);
while (frontier.size() > 0) {
PVector gc = frontier.remove(0);
if (!get(gc).isZoneFree()) continue;
int i = int(gc.x);
int j = int(gc.y);
get(i, j).zone_id = zone_id;
if (i > 0 && get(i-1, j).isZoneFree()) {
frontier.add(new PVector(i-1, j));
}
if (i < n_rows+1 && get(i+1, j).isZoneFree()) {
frontier.add(new PVector(i+1, j));
}
if (j > 0 && get(i, j-1).isZoneFree()) {
frontier.add(new PVector(i, j-1));
}
if (j < n_cols+1 && get(i, j+1).isZoneFree()) {
frontier.add(new PVector(i, j+1));
}
}
}
void showZones() {
int max_zone_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).setZoneColor();
max_zone_id = max(max_zone_id, get(i, j).zone_id);
}
}
println("found " + (max_zone_id+1) + " zones");
}
void setWallState(int wid, int state) {
ArrayList cells = wall_gcs.get(wid);
for (int i = 0; i < cells.size(); i++) {
get((PVector)cells.get(i)).setWallState(state);
}
wall_states.set(wid, state);
}
int getWallState(int wid) {
return wall_states.get(wid);
}
void updateWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
setWallState(i, wall_states.get(i));
}
}
void resetWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
wall_states.set(i, -1);
}
}
void reset() {
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).reset();
}
}
resetWallStates();
}
// for each wall, set first and last cell as anchors
void finalizeWallAnchors() {
int anchor_id = anchor_gcs.size();
for (int i = 0; i < wall_gcs.size(); i++) {
ArrayList<PVector> gcs = wall_gcs.get(i);
int last = gcs.size() - 1;
// vertical/horizontal wall
if (gcs.get(0).x == gcs.get(last).x || gcs.get(0).y == gcs.get(last).y) {
get(gcs.get(0)).setAnchor(anchor_id++);
get(gcs.get(gcs.size()-1)).setAnchor(anchor_id++);
} else {
for (int j = 0; j < 4; j++) {
get(gcs.get(j)).setAnchor(anchor_id++);
get(gcs.get(gcs.size()-(j+1))).setAnchor(anchor_id++);
}
}
}
}
}
class Cell {
PVector pos;
int wall_id, wall_state, zone_id, anchor_id, button_id;
boolean has_trace, has_cursor, is_button_pushed, is_cursor_active;
PVector button_color, button_push_color;
Cell(float x, float y) {
pos = new PVector(x, y);
wall_id = -1; // -1: no wall, 0--n: walls
wall_state = -1; // -1: not set, 0:bad, 1:good
zone_id = -1; // -1: no zone, 0--n: zones
anchor_id = -1; // -1: no anchor
button_id = -1; // -1: no button
has_trace = false;
has_cursor = false;
button_color = null;
button_push_color = null;
is_button_pushed = false;
is_cursor_active = false;
display();
}
boolean isStartable() {
return !isWall() && !isAnchor() && !isButton();
}
boolean isWall() {
return wall_id >= 0;
}
boolean isAnchor() {
return anchor_id >= 0;
}
void setAnchor(int id) {
anchor_id = id;
display();
}
void setButton(int id, PVector col, PVector push_col) {
button_id = id;
button_color = col;
button_push_color = push_col;
display();
}
boolean isButton() {
return button_id >= 0;
}
void pushButton(boolean b) {
is_button_pushed = b;
display();
}
boolean isZoneFree() {
return !isWall() && zone_id < 0 && !isAnchor();
}
void setTrace(boolean b) {
has_trace = b;
display();
}
void setWall(int wid) {
wall_id = wid;
display();
}
void setWallState(int state) {
wall_state = state;
display();
}
void setCursor(boolean c, boolean a) {
has_cursor = c;
is_cursor_active = a;
display();
}
void reset() {
has_trace = false;
has_cursor = false;
wall_state = -1;
display();
}
void display() {
stroke(75, 75, 75); // grey outline
if (has_cursor) {
if (is_cursor_active) {
fill(255, 255, 0); // yellow
} else {
fill(175, 175, 0); // fader yellow
}
} else if (has_trace) {
fill(0, 0, 255); // blue
} else if (isAnchor()) {
fill(127); // grey
} else if (isWall()) {
if (wall_state == -1) {
fill(225); // wall default: white
} else if (wall_state == 0) {
fill(255, 0, 0); // wall bad: red
} else if (wall_state == 1) {
fill(0, 255, 0); // wall ok: green
}
} else if (isButton() && button_color != null) {
if (is_button_pushed) {
fill(button_push_color.x, button_push_color.y, button_push_color.z);
} else {
fill(button_color.x, button_color.y, button_color.z);
}
} else {
fill(0); // black
}
rect(pos.x, pos.y, cell_size, cell_size);
}
void setZoneColor() {
if (zone_id >= 0) {
stroke(127);
fill(0, (zone_id + 3) * 25, 0);
rect(pos.x, pos.y, cell_size, cell_size);
}
}
}
class Cursor {
PVector curr_gc;
boolean is_picked, is_active;
int prev_zone_id;
int curr_wall_id;
ArrayList<PVector> trace_history;
Cursor(PVector new_gc) {
curr_gc = new_gc.get();
is_picked = false;
is_active = false;
prev_zone_id = -1;
curr_wall_id = -1;
trace_history = new ArrayList();
set(true);
//println("cursor (" + int(curr_gc.x) + "," + int(curr_gc.y) + ")");
}
void set(boolean b) {
int i = int(curr_gc.x);
int j = int(curr_gc.y);
grid.get(i-1, j).setCursor(b, is_active);
grid.get(i+1, j).setCursor(b, is_active);
grid.get(i, j).setCursor(b, is_active);
grid.get(i, j-1).setCursor(b, is_active);
grid.get(i, j+1).setCursor(b, is_active);
}
boolean pick(PVector mouse_gc) {
is_picked = (curr_gc.x-1 <= mouse_gc.x && mouse_gc.x <= curr_gc.x+1 &&
curr_gc.y-1 <= mouse_gc.y && mouse_gc.y <= curr_gc.y+1);
return is_picked;
}
void move(PVector new_gc) {
int new_i = int(new_gc.x);
int new_j = int(new_gc.y);
if (eq(curr_gc, new_gc)) { // below cell size threshold
return;
}
// detect boundaries and anchor cells
if (new_i < 1 || new_j < 1 || new_i > grid.n_rows ||
new_j > grid.n_cols || grid.get(new_gc).isAnchor() ||
grid.get(new_gc).isButton()) {
is_picked = false;
return;
}
// dist with curr cell; n=1 -> adjacent move (no need to interpolate)
int n = int(max(abs(new_gc.x - curr_gc.x), abs(new_gc.y - curr_gc.y)));
PVector mid_gc = curr_gc.get(); // mid_gc is a vector that we'll move stepwise in the direction of new_gc
ArrayList<PVector> mid_gcs = new ArrayList();
mid_gcs.add(mid_gc.get());
for (int i = 0; i < n-1; i++) { // interpolation
PVector d = new_gc.get();
d.sub(mid_gc);
d.normalize();
mid_gc.add(d);
mid_gc.x = round(mid_gc.x);
mid_gc.y = round(mid_gc.y);
if (grid.get(mid_gc).isAnchor()) {
is_picked = false;
return;
}
mid_gcs.add(mid_gc.get());
}
for (int i = 0; i < mid_gcs.size(); i++) {
mid_gc = mid_gcs.get(i);
grid.get(mid_gc).setTrace(true);
detectWallCrossing(mid_gc);
trace_history.add(mid_gc.get());
}
detectWallCrossing(new_gc);
set(false);
curr_gc = new_gc.get();
cursors[0].set(true);
cursors[1].set(true);
}
void detectWallCrossing(PVector new_gc) {
if (grid.get(new_gc).isWall()) {
if (curr_wall_id < 0) {
//println("entering wall " + grid.get(i, j).wall_id + " at (" + i + "," + j + ")");
}
curr_wall_id = grid.get(new_gc).wall_id;
if (grid.getWallState(curr_wall_id) != -1) { // if wall state is determined, we know
grid.setWallState(curr_wall_id, 0); // right away that it cannot be good anymore
}
} else {
if (curr_wall_id >= 0) { // was in wall, got out
//println("exiting wall " + curr_wall_id + " at (" + i + "," + j + ")");
//println("prev zone = " + prev_zone_id + " , curr zone = " + grid.get(new_gc).zone_id);
if (grid.get(new_gc).zone_id != prev_zone_id &&
grid.getWallState(curr_wall_id) == -1) {
grid.setWallState(curr_wall_id, 1);
} else {
grid.setWallState(curr_wall_id, 0);
}
curr_wall_id = -1;
}
prev_zone_id = grid.get(new_gc).zone_id;
}
}
void back(int n) {
int m = min(n, trace_history.size());
for (int i = 0; i < m; i++) {
grid.get(curr_gc).setTrace(false);
set(false);
PVector prev_gc = trace_history.get(trace_history.size()-1);
curr_gc = prev_gc;
set(true);
trace_history.remove(trace_history.size()-1);
}
grid.resetWallStates();
cursors[0].replayHistory();
cursors[1].replayHistory();
}
void replayHistory() {
curr_wall_id = -1;
prev_zone_id = -1;
for (int i = 0; i < trace_history.size(); i++) {
detectWallCrossing(trace_history.get(i));
grid.get(trace_history.get(i)).setTrace(true);
}
detectWallCrossing(curr_gc);
grid.updateWallStates();
}
void setActive(boolean b) {
is_active = b;
set(true);
}
}
</script>
<div class="separator" style="clear: both; text-align: center;" onselectstart="return false;" ondragstart="return false;">
<canvas id="mycanvas3"></canvas>
</div>
<p>(The applet was created with <a href="http://processingjs.org">Processing.js</a>)</p>
<p>I'd say this one is more suspect as a puzzle, in the sense that you can rapidly almost <i>feel</i> its impossibility.. in a way that the <a href="http://cjauvin.blogspot.com/2012/02/implicit-bridges.html">previous</a>, more complicated one was better able to conceal.</p>
<p>Maybe that explains why, even after my father had told me it was impossible, I kept trying to solve it stubbornly, fascinated by its paradoxical nature.</p>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com1tag:blogger.com,1999:blog-1673244706324941001.post-72887926875330054612012-02-13T21:35:00.002-05:002014-04-26T10:30:34.219-04:00Troubling Bridges<!-- <script src="https://github.com/downloads/processing-js/processing-js/processing-1.4.1.min.js"></script> -->
<script src="http://cdnjs.cloudflare.com/ajax/libs/processing.js/1.4.7/processing.min.js"></script>
<p>A long time ago, my father taught me a simple drawing game over which I would obsess for quite a while. It went like this: you first draw a certain shape (preferably with an ink pen), and you then try to cross through all the line segments at once, with a single stroke (i.e. without raising what is preferably a pencil) and without visiting a given segment twice.</p>
<p>I tried to recreate the spirit of this game (in memory of my father perhaps) as an experiment to learn the <a href="http://processingjs.org">Processing.js</a> platform. To test the game mechanics, you can try an easy puzzle first (click anywhere on the grid to place the cursor, and drag any of its ends through the segments, with the goal of turning them all green; you can press "r" to reset, "b" or right-click to move back, or use the yellow and blue buttons at the top):</p>
<script type="text/processing" data-processing-target="mycanvas1">
Grid grid = null;
int width = 600;
int height = 500;
int cell_size = 5;
Cursor[] cursors = null;
int last_picked_cursor = -1;
int n_steps_back = 10;
void setup() {
size(width, height);
grid = new Grid(height / cell_size, width / cell_size);
int nr = grid.n_rows;
int nc = grid.n_cols;
// easy
grid.addAnchor(1 + nr/4, 1 + nc/8);
grid.addAnchor(1 + nr/4, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addWall(0, 1);
grid.addWall(1, 2);
grid.addWall(0, 3);
grid.addWall(1, 4);
grid.addWall(2, 5);
grid.addWall(3, 4);
grid.addWall(4, 5);
grid.addWall(3, 6);
grid.addWall(4, 7);
grid.addWall(5, 8);
grid.addWall(6, 7);
grid.addWall(7, 8);
grid.finalizeWallAnchors();
grid.findZones();
//grid.showZones();
// button border
ArrayList<PVector> reset_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
reset_btn_data.add(new PVector(i, j));
}
}
}
// button cell data
reset_btn_data.add(new PVector(2, 4));
reset_btn_data.add(new PVector(2, 5));
reset_btn_data.add(new PVector(2, 7));
reset_btn_data.add(new PVector(3, 3));
reset_btn_data.add(new PVector(3, 6));
reset_btn_data.add(new PVector(3, 7));
reset_btn_data.add(new PVector(4, 2));
reset_btn_data.add(new PVector(4, 5));
reset_btn_data.add(new PVector(4, 6));
reset_btn_data.add(new PVector(4, 7));
reset_btn_data.add(new PVector(5, 2));
reset_btn_data.add(new PVector(6, 3));
reset_btn_data.add(new PVector(7, 4));
reset_btn_data.add(new PVector(7, 5));
reset_btn_data.add(new PVector(7, 6));
grid.addButton(reset_btn_data, new PVector(6, 6),
new PVector(200, 200, 0), new PVector(255, 255, 0));
ArrayList<PVector> back_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
back_btn_data.add(new PVector(i, j));
}
}
}
back_btn_data.add(new PVector(2, 4));
back_btn_data.add(new PVector(3, 3));
back_btn_data.add(new PVector(4, 2));
back_btn_data.add(new PVector(4, 3));
back_btn_data.add(new PVector(4, 4));
back_btn_data.add(new PVector(4, 5));
back_btn_data.add(new PVector(4, 6));
back_btn_data.add(new PVector(4, 7));
back_btn_data.add(new PVector(5, 3));
back_btn_data.add(new PVector(6, 4));
grid.addButton(back_btn_data, new PVector(6, grid.n_cols - 14),
new PVector(0, 0, 200), new PVector(0, 0, 255));
}
void draw() {
grid.updateButtonTimer();
}
void mousePressed() {
if (mouseButton == LEFT) {
PVector mouse_gc = grid.getCell(mouseX, mouseY);
if (cursors == null && grid.get(mouse_gc).isStartable()) {
cursors = new Cursor[2];
cursors[0] = new Cursor(mouse_gc);
cursors[1] = new Cursor(mouse_gc);
}
if (cursors != null) {
if (cursors[0].pick(mouse_gc)) {
last_picked_cursor = 0;
cursors[0].setActive(true);
cursors[1].setActive(false);
} else if (cursors[1].pick(mouse_gc)) {
last_picked_cursor = 1;
cursors[0].setActive(false);
cursors[1].setActive(true);
} else {
grid.buttonClicked(mouse_gc);
}
}
} else if (mouseButton == RIGHT) {
if (cursors != null) {
grid.pushButton(1);
}
}
}
//void mouseMoved() {
void mouseDragged() {
if (cursors != null) {
if (cursors[0].is_picked) {
cursors[0].move(grid.getCell(mouseX, mouseY));
} else if (cursors[1].is_picked) {
cursors[1].move(grid.getCell(mouseX, mouseY));
}
}
}
void mouseReleased() {
if (cursors != null) {
cursors[0].is_picked = false;
cursors[1].is_picked = false;
}
}
void keyPressed() {
if (key == 'r') {
grid.pushButton(0);
} else if (key == 'b') {
if (cursors != null) {
grid.pushButton(1);
}
}
}
boolean eq(PVector v1, PVector v2) {
return (v1.x == v2.x && v1.y == v2.y && v1.z == v2.z);
}
boolean adj(PVector v1, PVector v2) {
return abs(v1.x - v2.x) <= 1 && abs(v1.y - v2.y) <= 1;
}
class Grid {
Cell[][] cells;
int n_rows, n_cols;
ArrayList<ArrayList> wall_gcs;
ArrayList<Integer> wall_states;
ArrayList<PVector> anchor_gcs;
ArrayList<ArrayList> button_data;
int btn_timer_started_at, btn_timer_delay, btn_timer_id;
Grid(int nr, int nc) {
n_rows = nr;
n_cols = nc;
wall_gcs = new ArrayList();
wall_states = new ArrayList();
anchor_gcs = new ArrayList();
cells = new Cell[n_rows + 2][n_cols + 2]; // add 6-cell padding to each side
button_data = new ArrayList();
btn_timer_started_at = -1;
btn_timer_delay = 50;
btn_timer_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
cells[i][j] = new Cell((j-1)*cell_size, (i-1)*cell_size);
}
}
}
Cell get(PVector gc) {
return cells[int(gc.x)][int(gc.y)];
}
Cell get(int i, int j) {
return cells[i][j];
}
PVector getCell(int x, int y) {
return new PVector(int(y / cell_size) + 1, int(x / cell_size) + 1);
}
// 8x8 button with 10x10 frame
void addButton(ArrayList<PVector> data, PVector topleft_gc, PVector col, PVector pcol) {
int btn_id = button_data.size();
int top = int(topleft_gc.x);
int left = int(topleft_gc.y);
// button background must also be set
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
get(top + i, left + j).setButton(btn_id, null, null);
}
}
ArrayList<PVector> data_tl = new ArrayList(); // data repositioned at topleft
for (int i = 0; i < data.size(); i++) {
PVector gc = data.get(i);
gc.add(topleft_gc);
get(gc).setButton(btn_id, col, pcol);
data_tl.add(gc);
}
button_data.add(data_tl);
}
void updateButton(int btn_id, boolean is_pushed) {
ArrayList<PVector> data = button_data.get(btn_id);
for (int i = 0; i < data.size(); i++) {
grid.get(data.get(i)).pushButton(is_pushed);
}
}
int buttonClicked(PVector mouse_gc) {
int btn_id = get(mouse_gc).button_id;
if (btn_id >= 0) {
pushButton(btn_id);
}
return btn_id;
}
void pushButton(int btn_id) {
updateButton(btn_id, true);
startButtonTimer(btn_id);
}
void startButtonTimer(int btn_id) {
btn_timer_started_at = millis();
btn_timer_id = btn_id;
}
void updateButtonTimer() {
if (btn_timer_started_at > 0) {
if (millis() - btn_timer_started_at >= btn_timer_delay) {
updateButton(btn_timer_id, false);
if (btn_timer_id == 0) {
cursors = null;
reset();
} else if (btn_timer_id == 1) {
cursors[last_picked_cursor].back(n_steps_back);
}
btn_timer_started_at = -1;
btn_timer_id = -1;
}
}
}
// a wall links an anchor to an other (they must be on same row or col)
void addWall(int a1, int a2) {
ArrayList wall_cells = new ArrayList();
PVector a1_gc = anchor_gcs.get(a1);
PVector a2_gc = anchor_gcs.get(a2);
//assert(a1_gc.x == a2_gc.x || a1_gc.y == a2_gc.y);
if (abs(a1_gc.x - a2_gc.x) > abs(a1_gc.y - a2_gc.y)) {
if (a1_gc.x > a2_gc.x) { // swap
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int j = int(a1_gc.y);
for (int i = int(a1_gc.x)+1; i < int(a2_gc.x); i++) {
get(i, j).setWall(wall_gcs.size());
wall_cells.add(new PVector(i, j));
}
} else {
if (a1_gc.y > a2_gc.y) { // swap
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int i = int(a1_gc.x);
for (int j = int(a1_gc.y)+1; j < int(a2_gc.y); j++) {
get(i, j).setWall(wall_gcs.size());
wall_cells.add(new PVector(i, j));
}
}
wall_gcs.add(wall_cells);
wall_states.add(-1);
}
void addAnchor(int i, int j) {
get(i, j).setAnchor(anchor_gcs.size());
anchor_gcs.add(new PVector(i, j));
}
void findZones() {
int zone_id = 0;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
if (get(i, j).isZoneFree()) {
zoneFill(new PVector(i, j), zone_id);
zone_id += 1;
}
}
}
//println("found " + zone_id + " zones");
}
// find distinct zones by flood filling
void zoneFill(PVector start_gc, int zone_id) {
ArrayList<PVector> frontier = new ArrayList();
frontier.add(start_gc);
while (frontier.size() > 0) {
PVector gc = frontier.remove(0);
if (!get(gc).isZoneFree()) continue;
int i = int(gc.x);
int j = int(gc.y);
get(i, j).zone_id = zone_id;
if (i > 0 && get(i-1, j).isZoneFree()) {
frontier.add(new PVector(i-1, j));
}
if (i < n_rows+1 && get(i+1, j).isZoneFree()) {
frontier.add(new PVector(i+1, j));
}
if (j > 0 && get(i, j-1).isZoneFree()) {
frontier.add(new PVector(i, j-1));
}
if (j < n_cols+1 && get(i, j+1).isZoneFree()) {
frontier.add(new PVector(i, j+1));
}
}
}
void showZones() {
int max_zone_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).setZoneColor();
max_zone_id = max(max_zone_id, get(i, j).zone_id);
}
}
println("found " + (max_zone_id+1) + " zones");
}
void setWallState(int wid, int state) {
ArrayList cells = wall_gcs.get(wid);
for (int i = 0; i < cells.size(); i++) {
get((PVector)cells.get(i)).setWallState(state);
}
wall_states.set(wid, state);
}
int getWallState(int wid) {
return wall_states.get(wid);
}
void updateWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
setWallState(i, wall_states.get(i));
}
}
void resetWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
wall_states.set(i, -1);
}
}
void reset() {
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).reset();
}
}
resetWallStates();
}
// for each wall, set first and last cell as anchors
void finalizeWallAnchors() {
int anchor_id = anchor_gcs.size();
for (int i = 0; i < wall_gcs.size(); i++) {
ArrayList<PVector> gcs = wall_gcs.get(i);
get(gcs.get(0)).setAnchor(anchor_id++);
get(gcs.get(gcs.size()-1)).setAnchor(anchor_id++);
}
}
}
class Cell {
PVector pos;
int wall_id, wall_state, zone_id, anchor_id, button_id;
boolean has_trace, has_cursor, is_button_pushed, is_cursor_active;
PVector button_color, button_push_color;
Cell(float x, float y) {
pos = new PVector(x, y);
wall_id = -1; // -1: no wall, 0--n: walls
wall_state = -1; // -1: not set, 0:bad, 1:good
zone_id = -1; // -1: no zone, 0--n: zones
anchor_id = -1; // -1: no anchor
button_id = -1; // -1: no button
has_trace = false;
has_cursor = false;
button_color = null;
button_push_color = null;
is_button_pushed = false;
is_cursor_active = false;
display();
}
boolean isStartable() {
return !isWall() && !isAnchor() && !isButton();
}
boolean isWall() {
return wall_id >= 0;
}
boolean isAnchor() {
return anchor_id >= 0;
}
void setAnchor(int id) {
anchor_id = id;
display();
}
void setButton(int id, PVector col, PVector push_col) {
button_id = id;
button_color = col;
button_push_color = push_col;
display();
}
boolean isButton() {
return button_id >= 0;
}
void pushButton(boolean b) {
is_button_pushed = b;
display();
}
boolean isZoneFree() {
return !isWall() && zone_id < 0 && !isAnchor();
}
void setTrace(boolean b) {
has_trace = b;
display();
}
void setWall(int wid) {
wall_id = wid;
display();
}
void setWallState(int state) {
wall_state = state;
display();
}
void setCursor(boolean c, boolean a) {
has_cursor = c;
is_cursor_active = a;
display();
}
void reset() {
has_trace = false;
has_cursor = false;
wall_state = -1;
display();
}
void display() {
stroke(75, 75, 75); // grey outline
if (has_cursor) {
if (is_cursor_active) {
fill(255, 255, 0); // yellow
} else {
fill(175, 175, 0); // fader yellow
}
} else if (has_trace) {
fill(0, 0, 255); // blue
} else if (isAnchor()) {
fill(127); // grey
} else if (isWall()) {
if (wall_state == -1) {
fill(225); // wall default: white
} else if (wall_state == 0) {
fill(255, 0, 0); // wall bad: red
} else if (wall_state == 1) {
fill(0, 255, 0); // wall ok: green
}
} else if (isButton() && button_color != null) {
if (is_button_pushed) {
fill(button_push_color.x, button_push_color.y, button_push_color.z);
} else {
fill(button_color.x, button_color.y, button_color.z);
}
} else {
fill(0); // black
}
rect(pos.x, pos.y, cell_size, cell_size);
}
void setZoneColor() {
if (zone_id >= 0) {
stroke(127);
fill(0, (zone_id + 3) * 25, 0);
rect(pos.x, pos.y, cell_size, cell_size);
}
}
}
class Cursor {
PVector curr_gc;
boolean is_picked, is_active;
int prev_zone_id;
int curr_wall_id;
ArrayList<PVector> trace_history;
Cursor(PVector new_gc) {
curr_gc = new_gc.get();
is_picked = false;
is_active = false;
prev_zone_id = -1;
curr_wall_id = -1;
trace_history = new ArrayList();
set(true);
//println("cursor (" + int(curr_gc.x) + "," + int(curr_gc.y) + ")");
}
void set(boolean b) {
int i = int(curr_gc.x);
int j = int(curr_gc.y);
grid.get(i-1, j).setCursor(b, is_active);
grid.get(i+1, j).setCursor(b, is_active);
grid.get(i, j).setCursor(b, is_active);
grid.get(i, j-1).setCursor(b, is_active);
grid.get(i, j+1).setCursor(b, is_active);
}
boolean pick(PVector mouse_gc) {
is_picked = (curr_gc.x-1 <= mouse_gc.x && mouse_gc.x <= curr_gc.x+1 &&
curr_gc.y-1 <= mouse_gc.y && mouse_gc.y <= curr_gc.y+1);
return is_picked;
}
void move(PVector new_gc) {
int new_i = int(new_gc.x);
int new_j = int(new_gc.y);
if (eq(curr_gc, new_gc)) { // below cell size threshold
return;
}
// detect boundaries and anchor cells
if (new_i < 1 || new_j < 1 || new_i > grid.n_rows ||
new_j > grid.n_cols || grid.get(new_gc).isAnchor() ||
grid.get(new_gc).isButton()) {
is_picked = false;
return;
}
// dist with curr cell; n=1 -> adjacent move (no need to interpolate)
int n = int(max(abs(new_gc.x - curr_gc.x), abs(new_gc.y - curr_gc.y)));
PVector mid_gc = curr_gc.get(); // mid_gc is a vector that we'll move stepwise in the direction of new_gc
ArrayList<PVector> mid_gcs = new ArrayList();
mid_gcs.add(mid_gc.get());
for (int i = 0; i < n-1; i++) { // interpolation
PVector d = new_gc.get();
d.sub(mid_gc);
d.normalize();
mid_gc.add(d);
mid_gc.x = round(mid_gc.x);
mid_gc.y = round(mid_gc.y);
if (grid.get(mid_gc).isAnchor()) {
is_picked = false;
return;
}
mid_gcs.add(mid_gc.get());
}
for (int i = 0; i < mid_gcs.size(); i++) {
mid_gc = mid_gcs.get(i);
grid.get(mid_gc).setTrace(true);
detectWallCrossing(mid_gc);
trace_history.add(mid_gc.get());
}
detectWallCrossing(new_gc);
set(false);
curr_gc = new_gc.get();
cursors[0].set(true);
cursors[1].set(true);
}
void detectWallCrossing(PVector new_gc) {
if (grid.get(new_gc).isWall()) {
if (curr_wall_id < 0) {
//println("entering wall " + grid.get(i, j).wall_id + " at (" + i + "," + j + ")");
}
curr_wall_id = grid.get(new_gc).wall_id;
if (grid.getWallState(curr_wall_id) != -1) { // if wall state is determined, we know
grid.setWallState(curr_wall_id, 0); // right away that it cannot be good anymore
}
} else {
if (curr_wall_id >= 0) { // was in wall, got out
//println("exiting wall " + curr_wall_id + " at (" + i + "," + j + ")");
//println("prev zone = " + prev_zone_id + " , curr zone = " + grid.get(new_gc).zone_id);
if (grid.get(new_gc).zone_id != prev_zone_id &&
grid.getWallState(curr_wall_id) == -1) {
grid.setWallState(curr_wall_id, 1);
} else {
grid.setWallState(curr_wall_id, 0);
}
curr_wall_id = -1;
}
prev_zone_id = grid.get(new_gc).zone_id;
}
}
void back(int n) {
int m = min(n, trace_history.size());
for (int i = 0; i < m; i++) {
grid.get(curr_gc).setTrace(false);
set(false);
PVector prev_gc = trace_history.get(trace_history.size()-1);
curr_gc = prev_gc;
set(true);
trace_history.remove(trace_history.size()-1);
}
grid.resetWallStates();
cursors[0].replayHistory();
cursors[1].replayHistory();
}
void replayHistory() {
curr_wall_id = -1;
prev_zone_id = -1;
for (int i = 0; i < trace_history.size(); i++) {
detectWallCrossing(trace_history.get(i));
grid.get(trace_history.get(i)).setTrace(true);
}
detectWallCrossing(curr_gc);
grid.updateWallStates();
}
void setActive(boolean b) {
is_active = b;
set(true);
}
}
</script>
<div class="separator" style="clear: both; text-align: center;" onselectstart="return false;" ondragstart="return false;">
<canvas id="mycanvas1"></canvas>
</div>
<p>This was of course trivially easy.. but what about this slight modification, for which I once filled entire notebooks of trials? (I'm not kidding!):</p>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-_nK7doDrKRw/TzlDgoKE7PI/AAAAAAAACZs/s-EnsKhMwR0/s1600/bla-1.jpg" imageanchor="1" style=""><img border="0" height="222" width="320" src="http://2.bp.blogspot.com/-_nK7doDrKRw/TzlDgoKE7PI/AAAAAAAACZs/s-EnsKhMwR0/s320/bla-1.jpg" /></a></div>
<p>You can try it for yourself:</p>
<script type="text/processing" data-processing-target="mycanvas2">
Grid grid = null;
int width = 600;
int height = 500;
int cell_size = 5;
Cursor[] cursors = null;
int last_picked_cursor = -1;
int n_steps_back = 10;
void setup() {
size(width, height);
grid = new Grid(height / cell_size, width / cell_size);
int nr = grid.n_rows;
int nc = grid.n_cols;
grid.addAnchor(1 + nr/4, 1 + nc/8);
grid.addAnchor(1 + nr/4, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + int(nc*3/16) + 1);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + nc/8*3);
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + nc/8*3 + int(nc*3/16));
grid.addAnchor(1 + 2 * nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8 + int(nc*3/16) + 1);
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8 + nc/8*3 + int(nc*3/16));
grid.addAnchor(1 + 3 * nr/4, 1 + nc/8 + 2 * nc/8*3);
grid.addWall(0, 1);
grid.addWall(1, 2);
grid.addWall(0, 3);
grid.addWall(1, 5);
grid.addWall(2, 7);
grid.addWall(3, 4);
grid.addWall(4, 5);
grid.addWall(5, 6);
grid.addWall(6, 7);
grid.addWall(3, 8);
grid.addWall(4, 9);
grid.addWall(6, 10);
grid.addWall(7, 11);
grid.addWall(8, 9);
grid.addWall(9, 10);
grid.addWall(10, 11);
grid.finalizeWallAnchors();
grid.findZones();
//grid.showZones();
// button border
ArrayList<PVector> reset_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
reset_btn_data.add(new PVector(i, j));
}
}
}
// button cell data
reset_btn_data.add(new PVector(2, 4));
reset_btn_data.add(new PVector(2, 5));
reset_btn_data.add(new PVector(2, 7));
reset_btn_data.add(new PVector(3, 3));
reset_btn_data.add(new PVector(3, 6));
reset_btn_data.add(new PVector(3, 7));
reset_btn_data.add(new PVector(4, 2));
reset_btn_data.add(new PVector(4, 5));
reset_btn_data.add(new PVector(4, 6));
reset_btn_data.add(new PVector(4, 7));
reset_btn_data.add(new PVector(5, 2));
reset_btn_data.add(new PVector(6, 3));
reset_btn_data.add(new PVector(7, 4));
reset_btn_data.add(new PVector(7, 5));
reset_btn_data.add(new PVector(7, 6));
grid.addButton(reset_btn_data, new PVector(6, 6),
new PVector(200, 200, 0), new PVector(255, 255, 0));
ArrayList<PVector> back_btn_data = new ArrayList();
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
if (i == 0 || i == 9 || j == 0 || j == 9) {
back_btn_data.add(new PVector(i, j));
}
}
}
back_btn_data.add(new PVector(2, 4));
back_btn_data.add(new PVector(3, 3));
back_btn_data.add(new PVector(4, 2));
back_btn_data.add(new PVector(4, 3));
back_btn_data.add(new PVector(4, 4));
back_btn_data.add(new PVector(4, 5));
back_btn_data.add(new PVector(4, 6));
back_btn_data.add(new PVector(4, 7));
back_btn_data.add(new PVector(5, 3));
back_btn_data.add(new PVector(6, 4));
grid.addButton(back_btn_data, new PVector(6, grid.n_cols - 14),
new PVector(0, 0, 200), new PVector(0, 0, 255));
}
void draw() {
grid.updateButtonTimer();
}
void mousePressed() {
if (mouseButton == LEFT) {
PVector mouse_gc = grid.getCell(mouseX, mouseY);
if (cursors == null && grid.get(mouse_gc).isStartable()) {
cursors = new Cursor[2];
cursors[0] = new Cursor(mouse_gc);
cursors[1] = new Cursor(mouse_gc);
}
if (cursors != null) {
if (cursors[0].pick(mouse_gc)) {
last_picked_cursor = 0;
cursors[0].setActive(true);
cursors[1].setActive(false);
} else if (cursors[1].pick(mouse_gc)) {
last_picked_cursor = 1;
cursors[0].setActive(false);
cursors[1].setActive(true);
} else {
grid.buttonClicked(mouse_gc);
}
}
} else if (mouseButton == RIGHT) {
if (cursors != null) {
grid.pushButton(1);
}
}
}
//void mouseMoved() {
void mouseDragged() {
if (cursors != null) {
if (cursors[0].is_picked) {
cursors[0].move(grid.getCell(mouseX, mouseY));
} else if (cursors[1].is_picked) {
cursors[1].move(grid.getCell(mouseX, mouseY));
}
}
}
void mouseReleased() {
if (cursors != null) {
cursors[0].is_picked = false;
cursors[1].is_picked = false;
}
}
void keyPressed() {
if (key == 'r') {
grid.pushButton(0);
} else if (key == 'b') {
if (cursors != null) {
grid.pushButton(1);
}
}
}
boolean eq(PVector v1, PVector v2) {
return (v1.x == v2.x && v1.y == v2.y && v1.z == v2.z);
}
boolean adj(PVector v1, PVector v2) {
return abs(v1.x - v2.x) <= 1 && abs(v1.y - v2.y) <= 1;
}
class Grid {
Cell[][] cells;
int n_rows, n_cols;
ArrayList<ArrayList> wall_gcs;
ArrayList<Integer> wall_states;
ArrayList<PVector> anchor_gcs;
ArrayList<ArrayList> button_data;
int btn_timer_started_at, btn_timer_delay, btn_timer_id;
Grid(int nr, int nc) {
n_rows = nr;
n_cols = nc;
wall_gcs = new ArrayList();
wall_states = new ArrayList();
anchor_gcs = new ArrayList();
cells = new Cell[n_rows + 2][n_cols + 2]; // add 6-cell padding to each side
button_data = new ArrayList();
btn_timer_started_at = -1;
btn_timer_delay = 50;
btn_timer_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
cells[i][j] = new Cell((j-1)*cell_size, (i-1)*cell_size);
}
}
}
Cell get(PVector gc) {
return cells[int(gc.x)][int(gc.y)];
}
Cell get(int i, int j) {
return cells[i][j];
}
PVector getCell(int x, int y) {
return new PVector(int(y / cell_size) + 1, int(x / cell_size) + 1);
}
// 8x8 button with 10x10 frame
void addButton(ArrayList<PVector> data, PVector topleft_gc, PVector col, PVector pcol) {
int btn_id = button_data.size();
int top = int(topleft_gc.x);
int left = int(topleft_gc.y);
// button background must also be set
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 10; j++) {
get(top + i, left + j).setButton(btn_id, null, null);
}
}
ArrayList<PVector> data_tl = new ArrayList(); // data repositioned at topleft
for (int i = 0; i < data.size(); i++) {
PVector gc = data.get(i);
gc.add(topleft_gc);
get(gc).setButton(btn_id, col, pcol);
data_tl.add(gc);
}
button_data.add(data_tl);
}
void updateButton(int btn_id, boolean is_pushed) {
ArrayList<PVector> data = button_data.get(btn_id);
for (int i = 0; i < data.size(); i++) {
grid.get(data.get(i)).pushButton(is_pushed);
}
}
int buttonClicked(PVector mouse_gc) {
int btn_id = get(mouse_gc).button_id;
if (btn_id >= 0) {
pushButton(btn_id);
}
return btn_id;
}
void pushButton(int btn_id) {
updateButton(btn_id, true);
startButtonTimer(btn_id);
}
void startButtonTimer(int btn_id) {
btn_timer_started_at = millis();
btn_timer_id = btn_id;
}
void updateButtonTimer() {
if (btn_timer_started_at > 0) {
if (millis() - btn_timer_started_at >= btn_timer_delay) {
updateButton(btn_timer_id, false);
if (btn_timer_id == 0) {
cursors = null;
reset();
} else if (btn_timer_id == 1) {
cursors[last_picked_cursor].back(n_steps_back);
}
btn_timer_started_at = -1;
btn_timer_id = -1;
}
}
}
// a wall links an anchor to an other (they must be on same row or col)
void addWall(int a1, int a2) {
ArrayList wall_cells = new ArrayList();
PVector a1_gc = anchor_gcs.get(a1);
PVector a2_gc = anchor_gcs.get(a2);
//assert(a1_gc.x == a2_gc.x || a1_gc.y == a2_gc.y);
if (abs(a1_gc.x - a2_gc.x) > abs(a1_gc.y - a2_gc.y)) {
if (a1_gc.x > a2_gc.x) { // swap
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int j = int(a1_gc.y);
for (int i = int(a1_gc.x)+1; i < int(a2_gc.x); i++) {
get(i, j).setWall(wall_gcs.size());
wall_cells.add(new PVector(i, j));
}
} else {
if (a1_gc.y > a2_gc.y) { // swap
PVector tmp = a1_gc;
a1_gc = a2_gc;
a2_gc = tmp;
}
int i = int(a1_gc.x);
for (int j = int(a1_gc.y)+1; j < int(a2_gc.y); j++) {
get(i, j).setWall(wall_gcs.size());
wall_cells.add(new PVector(i, j));
}
}
wall_gcs.add(wall_cells);
wall_states.add(-1);
}
void addAnchor(int i, int j) {
get(i, j).setAnchor(anchor_gcs.size());
anchor_gcs.add(new PVector(i, j));
}
void findZones() {
int zone_id = 0;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
if (get(i, j).isZoneFree()) {
zoneFill(new PVector(i, j), zone_id);
zone_id += 1;
}
}
}
//println("found " + zone_id + " zones");
}
// find distinct zones by flood filling
void zoneFill(PVector start_gc, int zone_id) {
ArrayList<PVector> frontier = new ArrayList();
frontier.add(start_gc);
while (frontier.size() > 0) {
PVector gc = frontier.remove(0);
if (!get(gc).isZoneFree()) continue;
int i = int(gc.x);
int j = int(gc.y);
get(i, j).zone_id = zone_id;
if (i > 0 && get(i-1, j).isZoneFree()) {
frontier.add(new PVector(i-1, j));
}
if (i < n_rows+1 && get(i+1, j).isZoneFree()) {
frontier.add(new PVector(i+1, j));
}
if (j > 0 && get(i, j-1).isZoneFree()) {
frontier.add(new PVector(i, j-1));
}
if (j < n_cols+1 && get(i, j+1).isZoneFree()) {
frontier.add(new PVector(i, j+1));
}
}
}
void showZones() {
int max_zone_id = -1;
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).setZoneColor();
max_zone_id = max(max_zone_id, get(i, j).zone_id);
}
}
println("found " + (max_zone_id+1) + " zones");
}
void setWallState(int wid, int state) {
ArrayList cells = wall_gcs.get(wid);
for (int i = 0; i < cells.size(); i++) {
get((PVector)cells.get(i)).setWallState(state);
}
wall_states.set(wid, state);
}
int getWallState(int wid) {
return wall_states.get(wid);
}
void updateWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
setWallState(i, wall_states.get(i));
}
}
void resetWallStates() {
for (int i = 0; i < wall_states.size(); i++) {
wall_states.set(i, -1);
}
}
void reset() {
for (int i = 0; i < n_rows + 2; i++) {
for (int j = 0; j < n_cols + 2; j++) {
get(i, j).reset();
}
}
resetWallStates();
}
// for each wall, set first and last cell as anchors
void finalizeWallAnchors() {
int anchor_id = anchor_gcs.size();
for (int i = 0; i < wall_gcs.size(); i++) {
ArrayList<PVector> gcs = wall_gcs.get(i);
get(gcs.get(0)).setAnchor(anchor_id++);
get(gcs.get(gcs.size()-1)).setAnchor(anchor_id++);
}
}
}
class Cell {
PVector pos;
int wall_id, wall_state, zone_id, anchor_id, button_id;
boolean has_trace, has_cursor, is_button_pushed, is_cursor_active;
PVector button_color, button_push_color;
Cell(float x, float y) {
pos = new PVector(x, y);
wall_id = -1; // -1: no wall, 0--n: walls
wall_state = -1; // -1: not set, 0:bad, 1:good
zone_id = -1; // -1: no zone, 0--n: zones
anchor_id = -1; // -1: no anchor
button_id = -1; // -1: no button
has_trace = false;
has_cursor = false;
button_color = null;
button_push_color = null;
is_button_pushed = false;
is_cursor_active = false;
display();
}
boolean isStartable() {
return !isWall() && !isAnchor() && !isButton();
}
boolean isWall() {
return wall_id >= 0;
}
boolean isAnchor() {
return anchor_id >= 0;
}
void setAnchor(int id) {
anchor_id = id;
display();
}
void setButton(int id, PVector col, PVector push_col) {
button_id = id;
button_color = col;
button_push_color = push_col;
display();
}
boolean isButton() {
return button_id >= 0;
}
void pushButton(boolean b) {
is_button_pushed = b;
display();
}
boolean isZoneFree() {
return !isWall() && zone_id < 0 && !isAnchor();
}
void setTrace(boolean b) {
has_trace = b;
display();
}
void setWall(int wid) {
wall_id = wid;
display();
}
void setWallState(int state) {
wall_state = state;
display();
}
void setCursor(boolean c, boolean a) {
has_cursor = c;
is_cursor_active = a;
display();
}
void reset() {
has_trace = false;
has_cursor = false;
wall_state = -1;
display();
}
void display() {
stroke(75, 75, 75); // grey outline
if (has_cursor) {
if (is_cursor_active) {
fill(255, 255, 0); // yellow
} else {
fill(175, 175, 0); // fader yellow
}
} else if (has_trace) {
fill(0, 0, 255); // blue
} else if (isAnchor()) {
fill(127); // grey
} else if (isWall()) {
if (wall_state == -1) {
fill(225); // wall default: white
} else if (wall_state == 0) {
fill(255, 0, 0); // wall bad: red
} else if (wall_state == 1) {
fill(0, 255, 0); // wall ok: green
}
} else if (isButton() && button_color != null) {
if (is_button_pushed) {
fill(button_push_color.x, button_push_color.y, button_push_color.z);
} else {
fill(button_color.x, button_color.y, button_color.z);
}
} else {
fill(0); // black
}
rect(pos.x, pos.y, cell_size, cell_size);
}
void setZoneColor() {
if (zone_id >= 0) {
stroke(127);
fill(0, (zone_id + 3) * 25, 0);
rect(pos.x, pos.y, cell_size, cell_size);
}
}
}
class Cursor {
PVector curr_gc;
boolean is_picked, is_active;
int prev_zone_id;
int curr_wall_id;
ArrayList<PVector> trace_history;
Cursor(PVector new_gc) {
curr_gc = new_gc.get();
is_picked = false;
is_active = false;
prev_zone_id = -1;
curr_wall_id = -1;
trace_history = new ArrayList();
set(true);
//println("cursor (" + int(curr_gc.x) + "," + int(curr_gc.y) + ")");
}
void set(boolean b) {
int i = int(curr_gc.x);
int j = int(curr_gc.y);
grid.get(i-1, j).setCursor(b, is_active);
grid.get(i+1, j).setCursor(b, is_active);
grid.get(i, j).setCursor(b, is_active);
grid.get(i, j-1).setCursor(b, is_active);
grid.get(i, j+1).setCursor(b, is_active);
}
boolean pick(PVector mouse_gc) {
is_picked = (curr_gc.x-1 <= mouse_gc.x && mouse_gc.x <= curr_gc.x+1 &&
curr_gc.y-1 <= mouse_gc.y && mouse_gc.y <= curr_gc.y+1);
return is_picked;
}
void move(PVector new_gc) {
int new_i = int(new_gc.x);
int new_j = int(new_gc.y);
if (eq(curr_gc, new_gc)) { // below cell size threshold
return;
}
// detect boundaries and anchor cells
if (new_i < 1 || new_j < 1 || new_i > grid.n_rows ||
new_j > grid.n_cols || grid.get(new_gc).isAnchor() ||
grid.get(new_gc).isButton()) {
is_picked = false;
return;
}
// dist with curr cell; n=1 -> adjacent move (no need to interpolate)
int n = int(max(abs(new_gc.x - curr_gc.x), abs(new_gc.y - curr_gc.y)));
PVector mid_gc = curr_gc.get(); // mid_gc is a vector that we'll move stepwise in the direction of new_gc
ArrayList<PVector> mid_gcs = new ArrayList();
mid_gcs.add(mid_gc.get());
for (int i = 0; i < n-1; i++) { // interpolation
PVector d = new_gc.get();
d.sub(mid_gc);
d.normalize();
mid_gc.add(d);
mid_gc.x = round(mid_gc.x);
mid_gc.y = round(mid_gc.y);
if (grid.get(mid_gc).isAnchor()) {
is_picked = false;
return;
}
mid_gcs.add(mid_gc.get());
}
for (int i = 0; i < mid_gcs.size(); i++) {
mid_gc = mid_gcs.get(i);
grid.get(mid_gc).setTrace(true);
detectWallCrossing(mid_gc);
trace_history.add(mid_gc.get());
}
detectWallCrossing(new_gc);
set(false);
curr_gc = new_gc.get();
cursors[0].set(true);
cursors[1].set(true);
}
void detectWallCrossing(PVector new_gc) {
if (grid.get(new_gc).isWall()) {
if (curr_wall_id < 0) {
//println("entering wall " + grid.get(i, j).wall_id + " at (" + i + "," + j + ")");
}
curr_wall_id = grid.get(new_gc).wall_id;
if (grid.getWallState(curr_wall_id) != -1) { // if wall state is determined, we know
grid.setWallState(curr_wall_id, 0); // right away that it cannot be good anymore
}
} else {
if (curr_wall_id >= 0) { // was in wall, got out
//println("exiting wall " + curr_wall_id + " at (" + i + "," + j + ")");
//println("prev zone = " + prev_zone_id + " , curr zone = " + grid.get(new_gc).zone_id);
if (grid.get(new_gc).zone_id != prev_zone_id &&
grid.getWallState(curr_wall_id) == -1) {
grid.setWallState(curr_wall_id, 1);
} else {
grid.setWallState(curr_wall_id, 0);
}
curr_wall_id = -1;
}
prev_zone_id = grid.get(new_gc).zone_id;
}
}
void back(int n) {
int m = min(n, trace_history.size());
for (int i = 0; i < m; i++) {
grid.get(curr_gc).setTrace(false);
set(false);
PVector prev_gc = trace_history.get(trace_history.size()-1);
curr_gc = prev_gc;
set(true);
trace_history.remove(trace_history.size()-1);
}
grid.resetWallStates();
cursors[0].replayHistory();
cursors[1].replayHistory();
}
void replayHistory() {
curr_wall_id = -1;
prev_zone_id = -1;
for (int i = 0; i < trace_history.size(); i++) {
detectWallCrossing(trace_history.get(i));
grid.get(trace_history.get(i)).setTrace(true);
}
detectWallCrossing(curr_gc);
grid.updateWallStates();
}
void setActive(boolean b) {
is_active = b;
set(true);
}
}
</script>
<div class="separator" style="clear: both; text-align: center;" onselectstart="return false;" ondragstart="return false;">
<canvas id="mycanvas2"></canvas>
</div>
<p>A few attempts will probably convince you that it's much harder.. but can you see why? In a next post, I'd like to say more about it. I also plan to describe the code and algorithms I have devised, as I greatly enjoyed my first contact with <a href="http://processing.org/">Processing</a>, and I have many good things to say about it. I'm also thinking that this simple pixel grid world engine could be reused to build some other childhood-inspired games..</p>
<p>Don't miss the <a href="http://cjauvin.blogspot.ca/2012/02/eulerian-hoax.html">following conclusion</a>!</p>Christian Jauvinhttp://www.blogger.com/profile/11889328936089877334noreply@blogger.com4