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mhjensen 133fe9d0ea update on dim red
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<h2 id="___sec1" class="anchor">Principal Component Analysis </h2>
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
<p>
The following Python code uses NumPy&#8217;s <b>svd()</b> function to obtain all the principal components of the
training set, then extracts the first two principal components
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</span>)
U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
</pre></div>
<p>
PCA assumes that the dataset is centered around the origin. Scikit-Learn&#8217;s PCA classes take care of centering
the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don&#8217;t
forget to center the data first.
<p>
Once you have identified all the principal components, you can reduce the dimensionality of the dataset
down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
Selecting this hyperplane ensures that the projection will preserve as much variance as possible.
<p>
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
</pre></div>
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