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mhjensen 8467a2a9a2 updating pca
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<!-- navigation toc: --> <li><a href="._week43-bs001.html#___sec0" style="font-size: 80%;"><b>Plans for week 43</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs002.html#___sec1" style="font-size: 80%;"><b>Recurrent Neural Networks</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec2" style="font-size: 80%;"><b>Solving ODEs with Deep Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec21" style="font-size: 80%;"><b>Gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec22" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec23" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec26" style="font-size: 80%;"><b>The trial solution</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec50" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample mean and center the data</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample covariance</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec64" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Diagonalize the sample covariance matrix to obtain the principal components</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec65" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec68" style="font-size: 80%;"><b>The final step</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec69" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec70" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec71" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec72" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec75" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec76" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec77" style="font-size: 80%;"><b>LLE</b></a></li>
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<h2 id="___sec70" class="anchor">Principal Component Analysis </h2>
<p>
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. First we center the data using either <b>pandas</b> or our own code
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 vanilla matrix </span>
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
<span style="color: #408080; font-style: italic"># Pandas does the centering for us</span>
df <span style="color: #666666">=</span> df <span style="color: #666666">-</span>df<span style="color: #666666">.</span>mean()
display(df)
<span style="color: #408080; font-style: italic"># we center it ourselves</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>)
<span style="color: #408080; font-style: italic"># Then check the difference between pandas and our own set up</span>
<span style="color: #008000">print</span>(X_centered<span style="color: #666666">-</span>df)
<span style="color: #408080; font-style: italic">#Now we do an SVD</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>]
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)
<span style="color: #008000">print</span>(X2D)
</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>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<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>
<p>
<p>
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