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<div class="section" id="datatypes-and-bayesian-nonparametric-models">
<span id="datatypes"></span><h1>Datatypes and Bayesian Nonparametric Models<a class="headerlink" href="#datatypes-and-bayesian-nonparametric-models" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>To understand data, we often categorize data as falling under a specific
type of datatype. Understanding our underlying dataype gives structure
to the problem of modeling the data.</p>
<p>Datamicroscopes provides tools to understand 4 particular datatypes:</p>
<ol class="arabic simple">
<li>Real valued data</li>
<li>Social network data</li>
<li>Timeseries data</li>
<li>Text data</li>
</ol>
<div class="code python highlight-python"><div class="highlight"><pre>import numpy as np
import pandas as pd
import itertools as it
import seaborn as sns
import scipy.io
import cPickle as pickle
%matplotlib inline
import pylab as plt
sns.set_style('darkgrid')
sns.set_context('talk')
</pre></div>
</div>
<p>The two most common datatypes are real valued data and discrete data</p>
<p>For example, let’s take the iris dataset</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">iris</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">load_dataset</span><span class="p">(</span><span class="s">'iris'</span><span class="p">)</span>
<span class="n">iris</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>sepal_length</th>
<th>sepal_width</th>
<th>petal_length</th>
<th>petal_width</th>
<th>species</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>5.1</td>
<td>3.5</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>1</th>
<td>4.9</td>
<td>3.0</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>2</th>
<td>4.7</td>
<td>3.2</td>
<td>1.3</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>3</th>
<td>4.6</td>
<td>3.1</td>
<td>1.5</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>4</th>
<td>5.0</td>
<td>3.6</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
</tbody>
</table>
</div><p>In this case, <code class="docutils literal"><span class="pre">species</span></code> is a discrete variable and the other variables
are real valued</p>
<p>By understanding the form of the data, we can find a model that
represents its underlying structure</p>
<p>In the case of the <code class="docutils literal"><span class="pre">iris</span></code> dataset, plotting the data shows that
indiviudal species exhibit a typical range of measurements</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">irisplot</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">pairplot</span><span class="p">(</span><span class="n">iris</span><span class="p">,</span> <span class="n">hue</span><span class="o">=</span><span class="s">"species"</span><span class="p">,</span> <span class="n">palette</span><span class="o">=</span><span class="s">'Set2'</span><span class="p">,</span> <span class="n">diag_kind</span><span class="o">=</span><span class="s">"hist"</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mf">2.5</span><span class="p">)</span>
<span class="n">irisplot</span><span class="o">.</span><span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s">'Scatter Plots and KDE of Iris Data by Species'</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">18</span><span class="p">)</span>
<span class="n">irisplot</span><span class="o">.</span><span class="n">fig</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">top</span><span class="o">=.</span><span class="mi">9</span><span class="p">)</span>
</pre></div>
</div>
<img alt="_images/datatypes_5_0.png" src="_images/datatypes_5_0.png" />
<p>If we wanted to learn these underlying species’ measurements, we would
use these real valued measurements and make assumptions about the
structure of the data.</p>
<p>For example we could assume that each species had a latent range of
mesurements, and assume that these measurements were distributed
multivariate normal. In other words, the conditional probability of the
measurements given the species would be normally distributed</p>
<div class="math">
\[P(\mathbf{x}|species=s)\sim\mathcal{N}(\mu_{s},\Sigma_{s})\]</div>
<p>Bayesian Models allow us to leverage those assumptions.</p>
<p>In the case of the iris dataset, we would be able to learn both the
latent measurements of each Gaussian AND the number of species with a
Dirichlet Process Mixture Model, <code class="docutils literal"><span class="pre">microscopes.mituremodel</span></code></p>
<hr class="docutils" />
<p>Relational Data</p>
<p>While Dirichlet Process Mixture Models are the most common Bayesian
Nonparametric Model, there are other kinds of data to consider. For
example, let’s consider relational data in social networks</p>
<p>While social network data also has discrete valued varaibles, in this
case they have a different interpretation than the iris dataset</p>
<p>Let’s look at the Enron Email Corpus</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">enron_utils</span>
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s">'results.p'</span><span class="p">)</span> <span class="k">as</span> <span class="n">fp</span><span class="p">:</span>
<span class="n">communications</span> <span class="o">=</span> <span class="n">pickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">fp</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">allnames</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">names</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">it</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">(</span><span class="n">allnames</span><span class="p">(</span><span class="n">communications</span><span class="p">)))</span>
<span class="n">names</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">names</span><span class="p">))</span>
<span class="n">namemap</span> <span class="o">=</span> <span class="p">{</span> <span class="n">name</span> <span class="p">:</span> <span class="n">idx</span> <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">name</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">names</span><span class="p">)</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">names</span><span class="p">)</span>
<span class="n">communications_relation</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="n">N</span><span class="p">,</span> <span class="n">N</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">bool</span><span class="p">)</span>
<span class="k">for</span> <span class="n">sender</span><span class="p">,</span> <span class="n">receivers</span> <span class="ow">in</span> <span class="n">communications</span><span class="p">:</span>
<span class="n">sender_id</span> <span class="o">=</span> <span class="n">namemap</span><span class="p">[</span><span class="n">sender</span><span class="p">]</span>
<span class="k">for</span> <span class="n">receiver</span> <span class="ow">in</span> <span class="n">receivers</span><span class="p">:</span>
<span class="n">receiver_id</span> <span class="o">=</span> <span class="n">namemap</span><span class="p">[</span><span class="n">receiver</span><span class="p">]</span>
<span class="n">communications_relation</span><span class="p">[</span><span class="n">sender_id</span><span class="p">,</span> <span class="n">receiver_id</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
<span class="k">print</span> <span class="s">"</span><span class="si">%d</span><span class="s"> names in the corpus"</span> <span class="o">%</span> <span class="n">N</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre>115 names in the corpus
</pre></div>
</div>
<p>In this dataset, data is representated as a binary communication matrix
where</p>
<div class="math">
\[\mathbf{X}_{i,j} = 1 \Leftrightarrow \text{person}_{i} \text{ sent an email to person}_{j}\]</div>
<p>Let’s visualize the communication matrix</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">labels</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="mi">20</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="s">''</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="n">N</span><span class="p">)]</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">communications_relation</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">cbar</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">xticklabels</span><span class="o">=</span><span class="n">labels</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">labels</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">'person number'</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">'person number'</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">'Email Communication Matrix'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x11f764390>
</pre></div>
</div>
<img alt="_images/datatypes_10_1.png" src="_images/datatypes_10_1.png" />
<p>In this context, binary data represents communication between
individuals. With this interpretation of the data, we can model the
underlying social network.</p>
<p>To learn its structure we could use the Inifinite Relational Model,
<code class="docutils literal"><span class="pre">microscopes.irm</span></code></p>
<hr class="docutils" />
<p>In the case of time series data, the index of the data describes the
relationship between the observation and the rest of the data</p>
<div class="math">
\[\mathbf{x}_{t}\text{ s.t. }t \in \{0,...,T\}\]</div>
<p>For example, let’s look at the Old Faithful Data</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">old_faithful</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'https://vincentarelbundock.github.io/Rdatasets/csv/datasets/faithful.csv'</span><span class="p">,</span> <span class="n">index_col</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">old_faithful</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>eruptions</th>
<th>waiting</th>
</tr>
</thead>
<tbody>
<tr>
<th>1</th>
<td>3.600</td>
<td>79</td>
</tr>
<tr>
<th>2</th>
<td>1.800</td>
<td>54</td>
</tr>
<tr>
<th>3</th>
<td>3.333</td>
<td>74</td>
</tr>
<tr>
<th>4</th>
<td>2.283</td>
<td>62</td>
</tr>
<tr>
<th>5</th>
<td>4.533</td>
<td>85</td>
</tr>
</tbody>
</table>
</div><p>Let’s plot the erruptions as a function of time</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">f</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">tsplot</span><span class="p">(</span><span class="n">old_faithful</span><span class="p">[</span><span class="s">'eruptions'</span><span class="p">],</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</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">'erruptions'</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">'time'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x120e4fad0>
</pre></div>
</div>
<img alt="_images/datatypes_15_1.png" src="_images/datatypes_15_1.png" />
<p>The plot sugests that the number of errupitons has a particular set of
states</p>
<p>To learn both the number of underlying states and the states themselves,
we could use a Dirichlet-Process Hidden Markov Model</p>
<hr class="docutils" />
<p>Finally, let’s consider text data</p>
<p>Text data, like social network data, is discrete valued. However, the
values are each word or its id.</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s">'nyt_50.txt'</span><span class="p">,</span> <span class="s">'r'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
<span class="n">nyt</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
<span class="n">nyt</span> <span class="o">=</span> <span class="n">nyt</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">'</span><span class="se">\n</span><span class="s">'</span><span class="p">)</span>
</pre></div>
</div>
<p>In the case of the New York Times Dataset, we have 50 documents</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="k">print</span> <span class="n">nyt</span><span class="p">[</span><span class="mi">0</span><span class="p">][:</span><span class="mi">100</span><span class="p">]</span>
<span class="k">print</span> <span class="n">nyt</span><span class="p">[</span><span class="mi">1</span><span class="p">][:</span><span class="mi">100</span><span class="p">]</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre>the new york times said editorial for tuesday jan new year day has way stealing down upon coming the
the seminal russian filmmaker sergei eisenstein had physically matched the style his monumental film
</pre></div>
</div>
<p>One of the most common classification tasks within corpora is topic
modelling</p>
<p>While LDA is a popular method of topic modeling, we can also learn
topics and the number of topics with the Heierarchical Dirichlet Process</p>
<hr class="docutils" />
<p>These example illustrate the ways in which Bayesian Nonparametric Models
can learn structure within data. For more information about each model
within Datamicroscopes you can read about each of the models in detail.</p>
</div>
</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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