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<div class="section" id="inferring-gaussians-with-the-dirichlet-process-mixture-model">
<span id="gauss2d"></span><h1>Inferring Gaussians with the Dirichlet Process Mixture Model<a class="headerlink" href="#inferring-gaussians-with-the-dirichlet-process-mixture-model" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>Let’s set up our environment</p>
<div class="code python highlight-python"><div class="highlight"><pre>%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import time
import seaborn as sns
import pandas as pd
sns.set_style('darkgrid')
sns.set_context('talk')
sns.set_palette("Set2", 30)
</pre></div>
</div>
<p>Now let’s import our functions from datamicroscopes</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">microscopes.common.rng</span> <span class="kn">import</span> <span class="n">rng</span>
<span class="kn">from</span> <span class="nn">microscopes.common.recarray.dataview</span> <span class="kn">import</span> <span class="n">numpy_dataview</span>
<span class="kn">from</span> <span class="nn">microscopes.models</span> <span class="kn">import</span> <span class="n">niw</span> <span class="k">as</span> <span class="n">normal_inverse_wishart</span>
<span class="kn">from</span> <span class="nn">microscopes.mixture.definition</span> <span class="kn">import</span> <span class="n">model_definition</span>
<span class="kn">from</span> <span class="nn">microscopes.mixture</span> <span class="kn">import</span> <span class="n">model</span><span class="p">,</span> <span class="n">runner</span><span class="p">,</span> <span class="n">query</span>
<span class="kn">from</span> <span class="nn">microscopes.common.query</span> <span class="kn">import</span> <span class="n">zmatrix_heuristic_block_ordering</span><span class="p">,</span> <span class="n">zmatrix_reorder</span>
</pre></div>
</div>
<p>From here, we’ll generate four isotropic 2D gaussian clusters in each
quadrant, varying the scale parameter</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">nsamples_per_cluster</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">means</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</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="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</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="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float</span><span class="p">)</span>
<span class="n">scales</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.08</span><span class="p">,</span> <span class="mf">0.09</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span>
<span class="n">Y_clusters</span> <span class="o">=</span> <span class="p">[</span>
<span class="n">np</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="n">mean</span><span class="o">=</span><span class="n">mu</span><span class="p">,</span>
<span class="n">cov</span><span class="o">=</span><span class="n">var</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span>
<span class="n">size</span><span class="o">=</span><span class="n">nsamples_per_cluster</span><span class="p">)</span>
<span class="k">for</span> <span class="n">mu</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">means</span><span class="p">,</span> <span class="n">scales</span><span class="p">)]</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">Yc</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">Y_clusters</span><span class="p">):</span>
<span class="n">cl</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">Yc</span><span class="p">,</span> <span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s">'x'</span><span class="p">,</span><span class="s">'y'</span><span class="p">])</span>
<span class="n">cl</span><span class="p">[</span><span class="s">'cluster'</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cl</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">(</span><span class="n">Y_clusters</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">permutation</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">df</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div style="max-height:1000px;max-width:1500px;overflow:auto;">
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>x</th>
<th>y</th>
<th>cluster</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>1.557005</td>
<td>1.266202</td>
<td>0</td>
</tr>
<tr>
<th>1</th>
<td>1.465262</td>
<td>0.842641</td>
<td>0</td>
</tr>
<tr>
<th>2</th>
<td>0.619352</td>
<td>1.309368</td>
<td>0</td>
</tr>
<tr>
<th>3</th>
<td>1.130965</td>
<td>0.700129</td>
<td>0</td>
</tr>
<tr>
<th>4</th>
<td>1.447409</td>
<td>1.112726</td>
<td>0</td>
</tr>
</tbody>
</table>
</div><p>Let’s have a look at the generated data</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">lmplot</span><span class="p">(</span><span class="s">'x'</span><span class="p">,</span> <span class="s">'y'</span><span class="p">,</span> <span class="n">hue</span><span class="o">=</span><span class="s">"cluster"</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">,</span> <span class="n">fit_reg</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Simulated Gaussians: 4 Clusters'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x112cf7290>
</pre></div>
</div>
<img alt="_images/gauss2d_8_1.png" src="_images/gauss2d_8_1.png" />
<p>Now let’s learn this clustering non-parametrically!</p>
<p>There are 5 steps necessary to set up your model:</p>
<ol class="arabic simple">
<li>Decide on the number of chains we want – it is important to run
multiple chains from different starting points!</li>
<li>Define our DP-GMM model</li>
<li>Munge the data into numpy recarray format then wrap the data for our
model</li>
<li>Randomize start points</li>
<li>Create runners for each chain</li>
</ol>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">nchains</span> <span class="o">=</span> <span class="mi">8</span>
<span class="c"># The random state object</span>
<span class="n">prng</span> <span class="o">=</span> <span class="n">rng</span><span class="p">()</span>
<span class="c"># Define a DP-GMM where the Gaussian is 2D</span>
<span class="n">defn</span> <span class="o">=</span> <span class="n">model_definition</span><span class="p">(</span><span class="n">Y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">normal_inverse_wishart</span><span class="p">(</span><span class="mi">2</span><span class="p">)])</span>
<span class="c"># Munge the data into numpy recarray format</span>
<span class="n">Y_rec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([(</span><span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="p">),)</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="p">[(</span><span class="s">''</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
<span class="c"># Create a wrapper around the numpy recarray which</span>
<span class="c"># data-microscopes understands</span>
<span class="n">view</span> <span class="o">=</span> <span class="n">numpy_dataview</span><span class="p">(</span><span class="n">Y_rec</span><span class="p">)</span>
<span class="c"># Initialize nchains start points randomly in the state space</span>
<span class="n">latents</span> <span class="o">=</span> <span class="p">[</span><span class="n">model</span><span class="o">.</span><span class="n">initialize</span><span class="p">(</span><span class="n">defn</span><span class="p">,</span> <span class="n">view</span><span class="p">,</span> <span class="n">prng</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">nchains</span><span class="p">)]</span>
<span class="c"># Create a runner for each chain</span>
<span class="n">runners</span> <span class="o">=</span> <span class="p">[</span><span class="n">runner</span><span class="o">.</span><span class="n">runner</span><span class="p">(</span><span class="n">defn</span><span class="p">,</span> <span class="n">view</span><span class="p">,</span> <span class="n">latent</span><span class="p">,</span> <span class="n">kernel_config</span><span class="o">=</span><span class="p">[</span><span class="s">'assign'</span><span class="p">])</span> <span class="k">for</span> <span class="n">latent</span> <span class="ow">in</span> <span class="n">latents</span><span class="p">]</span>
</pre></div>
</div>
<p>We will visualize our data to examine the cluster assignment</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">plot_assignment</span><span class="p">(</span><span class="n">assignment</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">Y</span><span class="p">):</span>
<span class="n">cl</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s">'x'</span><span class="p">,</span><span class="s">'y'</span><span class="p">])</span>
<span class="n">cl</span><span class="p">[</span><span class="s">'cluster'</span><span class="p">]</span> <span class="o">=</span> <span class="n">assignment</span>
<span class="n">n_clusters</span> <span class="o">=</span> <span class="n">cl</span><span class="p">[</span><span class="s">'cluster'</span><span class="p">]</span><span class="o">.</span><span class="n">nunique</span><span class="p">()</span>
<span class="n">sns</span><span class="o">.</span><span class="n">lmplot</span><span class="p">(</span><span class="s">'x'</span><span class="p">,</span> <span class="s">'y'</span><span class="p">,</span> <span class="n">hue</span><span class="o">=</span><span class="s">"cluster"</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">cl</span><span class="p">,</span> <span class="n">fit_reg</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="p">(</span><span class="n">n_clusters</span><span class="o"><</span><span class="mi">10</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Simulated Gaussians: </span><span class="si">%d</span><span class="s"> Learned Clusters'</span> <span class="o">%</span> <span class="n">n_clusters</span><span class="p">)</span>
</pre></div>
</div>
<p>Let’s peek at the starting state for one of our chains</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">plot_assignment</span><span class="p">(</span><span class="n">latents</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assignments</span><span class="p">())</span>
</pre></div>
</div>
<img alt="_images/gauss2d_14_0.png" src="_images/gauss2d_14_0.png" />
<p>Let’s watch one of the chains evolve for a few steps</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">first_runner</span> <span class="o">=</span> <span class="n">runners</span><span class="p">[</span><span class="mi">0</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="mi">5</span><span class="p">):</span>
<span class="n">first_runner</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">r</span><span class="o">=</span><span class="n">prng</span><span class="p">,</span> <span class="n">niters</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plot_assignment</span><span class="p">(</span><span class="n">first_runner</span><span class="o">.</span><span class="n">get_latent</span><span class="p">()</span><span class="o">.</span><span class="n">assignments</span><span class="p">())</span>
</pre></div>
</div>
<img alt="_images/gauss2d_16_0.png" src="_images/gauss2d_16_0.png" />
<img alt="_images/gauss2d_16_1.png" src="_images/gauss2d_16_1.png" />
<img alt="_images/gauss2d_16_2.png" src="_images/gauss2d_16_2.png" />
<img alt="_images/gauss2d_16_3.png" src="_images/gauss2d_16_3.png" />
<img alt="_images/gauss2d_16_4.png" src="_images/gauss2d_16_4.png" />
<p>Now let’s burn all our runners in for 100 iterations.</p>
<p>We’ll do this sequentially since the model is simple, but check
microscopes.parallel.runner for parallel implementions (with support for
either multiprocessing or multyvac)</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="k">for</span> <span class="n">runner</span> <span class="ow">in</span> <span class="n">runners</span><span class="p">:</span>
<span class="n">runner</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">r</span><span class="o">=</span><span class="n">prng</span><span class="p">,</span> <span class="n">niters</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
</pre></div>
</div>
<p>Let’s now peek again at the first state</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">plot_assignment</span><span class="p">(</span><span class="n">first_runner</span><span class="o">.</span><span class="n">get_latent</span><span class="p">()</span><span class="o">.</span><span class="n">assignments</span><span class="p">())</span>
</pre></div>
</div>
<img alt="_images/gauss2d_20_0.png" src="_images/gauss2d_20_0.png" />
<p>Let’s build a z-matrix to compare our result with the rest of the chains</p>
<p>We’ll be sure to sort our z-matrix before plotting. Sorting the
datapoints allows us to organize the clusters into a block matrix.</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">infers</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="o">.</span><span class="n">get_latent</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">runners</span><span class="p">]</span>
<span class="n">zmat</span> <span class="o">=</span> <span class="n">query</span><span class="o">.</span><span class="n">zmatrix</span><span class="p">(</span><span class="n">infers</span><span class="p">)</span>
<span class="n">ordering</span> <span class="o">=</span> <span class="n">zmatrix_heuristic_block_ordering</span><span class="p">(</span><span class="n">zmat</span><span class="p">)</span>
<span class="n">zmat</span> <span class="o">=</span> <span class="n">zmatrix_reorder</span><span class="p">(</span><span class="n">zmat</span><span class="p">,</span> <span class="n">ordering</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">zmat</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">xticklabels</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'entities (sorted)'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'entities (sorted)'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Z-matrix of Cluster Assignments'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x116f73510>
</pre></div>
</div>
<img alt="_images/gauss2d_23_1.png" src="_images/gauss2d_23_1.png" />
</div>
</div>
</div>
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<center> Datamicroscopes is developed by <a href="http://www.qadium.com">Qadium</a>, with funding from the <a href="http://www.darpa.mil">DARPA</a> <a href="http://www.darpa.mil/program/xdata">XDATA</a> program. Copyright Qadium 2015. </center>
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