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<div class="section" id="univariate-data-with-the-normal-inverse-chi-square-distribution">
<h1>Univariate Data with the Normal Inverse Chi-Square Distribution<a class="headerlink" href="#univariate-data-with-the-normal-inverse-chi-square-distribution" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>One of the simplest examples of data is univariate data</p>
<p>Let’s consider a timeseries example:</p>
<p><a class="reference external" href="https://vincentarelbundock.github.io/Rdatasets/doc/datasets/lynx.html">The Annual Canadian Lynx Trappings
Dataset</a>
as described by <a class="reference external" href="http://www.jstor.org/stable/2345277">Campbel and Walker
1977</a> contains the number of
Lynx trapped near the McKenzie River in the Northwest Territories in
Canada between 1821 and 1934.</p>
<div class="code python highlight-python"><div class="highlight"><pre>import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
%matplotlib inline
sns.set_context('talk')
sns.set_style('darkgrid')
</pre></div>
</div>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">lynx</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/lynx.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>
</pre></div>
</div>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">lynx</span> <span class="o">=</span> <span class="n">lynx</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s">'time'</span><span class="p">)</span>
<span class="n">lynx</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>lynx</th>
</tr>
<tr>
<th>time</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<th>1821</th>
<td>269</td>
</tr>
<tr>
<th>1822</th>
<td>321</td>
</tr>
<tr>
<th>1823</th>
<td>585</td>
</tr>
<tr>
<th>1824</th>
<td>871</td>
</tr>
<tr>
<th>1825</th>
<td>1475</td>
</tr>
</tbody>
</table>
</div><div class="code python highlight-python"><div class="highlight"><pre><span class="n">lynx</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">legend</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">'Year'</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">'Annual Canadian Lynx Trappings 1821-1934'</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">'Lynx'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x1129d0090>
</pre></div>
</div>
<img alt="_images/normal-inverse-chisquare_4_1.png" src="_images/normal-inverse-chisquare_4_1.png" />
<p>Let’s plot the kernel density estimate of annual lynx trapping</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">lynx</span><span class="p">[</span><span class="s">'lynx'</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">'Kernel Density Estimate of Annual Lynx Trapping'</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">'Probability'</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">'Number of Lynx'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x1129d0410>
</pre></div>
</div>
<img alt="_images/normal-inverse-chisquare_6_1.png" src="_images/normal-inverse-chisquare_6_1.png" />
<p>Our plot suggests there could be three modes in the Lynx data.</p>
<p>In modeling this timeseries, we could assume that the number of lynx
trapped in a given year is falls into one of <span class="math">\(k\)</span> states, which are
normally distributed with some unknown mean <span class="math">\(\mu_i\)</span> and variance
<span class="math">\(\sigma^2_i\)</span> for each state</p>
<p>In the case of our Lynx data</p>
<div class="math">
\[\forall i \in [1,...,k] \hspace{2mm} p(\text{lynx trapped}| \text{state} = i) \sim \mathcal{N}(\mu_i, \sigma^2_i)\]</div>
<hr class="docutils" />
<p>Now let’s consider demographics data from the Titanic Dataset</p>
<p>The Titanic Dataset contains information about passengers of the
Titanic.</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">ti</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">'titanic'</span><span class="p">)</span>
<span class="n">ti</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>survived</th>
<th>pclass</th>
<th>sex</th>
<th>age</th>
<th>sibsp</th>
<th>parch</th>
<th>fare</th>
<th>embarked</th>
<th>class</th>
<th>who</th>
<th>adult_male</th>
<th>deck</th>
<th>embark_town</th>
<th>alive</th>
<th>alone</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0</td>
<td>3</td>
<td>male</td>
<td>22</td>
<td>1</td>
<td>0</td>
<td>7.2500</td>
<td>S</td>
<td>Third</td>
<td>man</td>
<td>True</td>
<td>NaN</td>
<td>Southampton</td>
<td>no</td>
<td>False</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>1</td>
<td>female</td>
<td>38</td>
<td>1</td>
<td>0</td>
<td>71.2833</td>
<td>C</td>
<td>First</td>
<td>woman</td>
<td>False</td>
<td>C</td>
<td>Cherbourg</td>
<td>yes</td>
<td>False</td>
</tr>
<tr>
<th>2</th>
<td>1</td>
<td>3</td>
<td>female</td>
<td>26</td>
<td>0</td>
<td>0</td>
<td>7.9250</td>
<td>S</td>
<td>Third</td>
<td>woman</td>
<td>False</td>
<td>NaN</td>
<td>Southampton</td>
<td>yes</td>
<td>True</td>
</tr>
<tr>
<th>3</th>
<td>1</td>
<td>1</td>
<td>female</td>
<td>35</td>
<td>1</td>
<td>0</td>
<td>53.1000</td>
<td>S</td>
<td>First</td>
<td>woman</td>
<td>False</td>
<td>C</td>
<td>Southampton</td>
<td>yes</td>
<td>False</td>
</tr>
<tr>
<th>4</th>
<td>0</td>
<td>3</td>
<td>male</td>
<td>35</td>
<td>0</td>
<td>0</td>
<td>8.0500</td>
<td>S</td>
<td>Third</td>
<td>man</td>
<td>True</td>
<td>NaN</td>
<td>Southampton</td>
<td>no</td>
<td>True</td>
</tr>
</tbody>
</table>
</div><p>Passenger age and fare are both real valued. Are they related? Let’s
examine the correlation matrix</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">ti</span><span class="p">[[</span><span class="s">'age'</span><span class="p">,</span><span class="s">'fare'</span><span class="p">]]</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>age</th>
<th>fare</th>
</tr>
</thead>
<tbody>
<tr>
<th>age</th>
<td>1.000000</td>
<td>0.096067</td>
</tr>
<tr>
<th>fare</th>
<td>0.096067</td>
<td>1.000000</td>
</tr>
</tbody>
</table>
</div><p>Since the correlation is between the two variables is zero, we can model
these two real valued columns independently.</p>
<p>Let’s plot the kernel density estimate of each variable</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">ti</span><span class="p">[</span><span class="s">'age'</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">'Kernel Density Estimate of Passenger Age in the Titanic Datset'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x117426c50>
</pre></div>
</div>
<img alt="_images/normal-inverse-chisquare_12_1.png" src="_images/normal-inverse-chisquare_12_1.png" />
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">ti</span><span class="p">[</span><span class="s">'fare'</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">'Kernel Density Estimate of Passenger Fare in the Titanic Datset'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x117475310>
</pre></div>
</div>
<img alt="_images/normal-inverse-chisquare_13_1.png" src="_images/normal-inverse-chisquare_13_1.png" />
<p>Given the long tail in the fare price, we might want to model this
variable on a log scale:</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">ti</span><span class="p">[</span><span class="s">'logfare'</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">ti</span><span class="p">[</span><span class="s">'fare'</span><span class="p">])</span>
<span class="n">ti</span><span class="p">[[</span><span class="s">'age'</span><span class="p">,</span><span class="s">'logfare'</span><span class="p">]]</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>age</th>
<th>logfare</th>
</tr>
</thead>
<tbody>
<tr>
<th>age</th>
<td>1.000000</td>
<td>0.135352</td>
</tr>
<tr>
<th>logfare</th>
<td>0.135352</td>
<td>1.000000</td>
</tr>
</tbody>
</table>
</div><p>Again, <code class="docutils literal"><span class="pre">logfare</span></code> and <code class="docutils literal"><span class="pre">age</span></code> have near zero correlation, so we can
again model these two variables independently</p>
<p>Let’s see what a kernel density estimate of log fare would look like</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">ti</span><span class="p">[</span><span class="s">'logfare'</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">'Kernel Density Estimate of Log Passenger Fare in the Titanic Datset'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x1175a5110>
</pre></div>
</div>
<img alt="_images/normal-inverse-chisquare_17_1.png" src="_images/normal-inverse-chisquare_17_1.png" />
<p>In logspace, passenger fare is multimodal, suggesting that we could
model this variable with a normal distirbution</p>
<p>If we were to model the passenger list using our Mixture Model, we would
have separate likelihoods for <code class="docutils literal"><span class="pre">logfare</span></code> and <code class="docutils literal"><span class="pre">age</span></code></p>
<div class="math">
\[\forall i \in [1,...,k] \hspace{2mm} p(\text{logfare}|\text{cluster}=i)=\mathcal{N}(\mu_{i,l}, \sigma^2_{i,l})\]</div>
<div class="math">
\[\forall i \in [1,...,k] \hspace{2mm} p(\text{age}|\text{cluster}=c)=\mathcal{N}(\mu_{i,a}, \sigma^2_{i,a})\]</div>
<hr class="docutils" />
<p>Often, real value data is assumed to be normally distributed.</p>
<p>To learn the latent variables, <span class="math">\(\mu_i\)</span> <span class="math">\(\sigma^2_i\)</span>, we
would use a normal inverse-chi-square likelihood</p>
<p>The normal inverse-chi-square likelihood is the conjugate univariate
normal likelihood in data microscopes. We also have normal likelihood,
the normal inverse-wishart likelihood, optimized for multivariate
datasets.</p>
<p>It is important to model univariate normal data with this likelihood as
it acheives superior performance on univariate data.</p>
<p>In both these examples, we found variables that were amenable to being
modeled as univariate normal:</p>
<ol class="arabic simple">
<li>Univariate datasets</li>
<li>Datasets containing real valued variables with near zero correlation</li>
</ol>
<p>To import our univariate normal inverse-chi-squared likelihood, call:</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">microscopes.models</span> <span class="kn">import</span> <span class="n">nich</span> <span class="k">as</span> <span class="n">normal_inverse_chisquared</span>
</pre></div>
</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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