mhjensen
392 lines
20 KiB
HTML
392 lines
20 KiB
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<a class="navbar-brand" href="DimRed-bs.html">Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction</a>
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<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;"><b>Simple preprocessing examples, Franke function and regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;"><b>Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;"><b>More on Cancer Data, now with Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;"><b>Why should we think of reducing the dimensionality</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
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<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;"> Compute the sample covariance</a></li>
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<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec21" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec22" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec23" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec24" style="font-size: 80%;"><b>The final step</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec25" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec26" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec27" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec28" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec29" style="font-size: 80%;"><b>More on the PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec30" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec31" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec32" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs031.html#___sec33" style="font-size: 80%;"><b>LLE</b></a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs032.html#___sec34" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</ul>
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</li>
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</ul>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0018"></a>
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<!-- !split -->
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<h2 id="___sec17" class="anchor">Writing our own PCA code </h2>
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<p>
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We will use a simple example first with two-dimensional data
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drawn from a multivariate normal distribution with the following mean and covariance matrix:
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$$
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\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
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2 & 2
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\end{bmatrix}
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$$
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Note that the mean refers to each column of data.
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We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from
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this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \).
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<p>
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The following Python code aids in setting up the data
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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<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>
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<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
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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mean <span style="color: #666666">=</span> (<span style="color: #666666">-1</span>, <span style="color: #666666">2</span>)
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cov <span style="color: #666666">=</span> [[<span style="color: #666666">4</span>, <span style="color: #666666">2</span>], [<span style="color: #666666">2</span>, <span style="color: #666666">2</span>]]
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>multivariate_normal(mean, cov, n)
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<span style="color: #408080; font-style: italic"># Print the X-matrix</span>
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<span style="color: #008000; font-weight: bold">print</span>(X)
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</pre></div>
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<p>
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Make thereafter a small Python code which writes out the data. Note that the function <b>multivariate</b> returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
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<p>
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Now we are going to implement the PCA algorithm. We will break it down into various substeps.
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<h3 id="___sec18" class="anchor">Compute the sample mean and center the data </h3>
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<p>
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The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
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$$
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\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
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$$
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and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
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$$
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\bar{x}_i = x_i - \mu_n.
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$$
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When you are done with these steps, print out \( \mu_n \) to verify it is
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close to \( \mu \) and plot your mean centered data to verify it is
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centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
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<span style="color: #408080; font-style: italic"># Pandas does the centering for us</span>
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df <span style="color: #666666">=</span> df <span style="color: #666666">-</span>df<span style="color: #666666">.</span>mean()
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display(df)
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<span style="color: #408080; font-style: italic"># we center it ourselves</span>
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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>)
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<span style="color: #408080; font-style: italic"># test that we get the same as Pandas</span>
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<span style="color: #008000; font-weight: bold">print</span>(X_centered<span style="color: #666666">-</span>df)
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</pre></div>
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<h3 id="___sec19" class="anchor">Compute the sample covariance </h3>
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<p>
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Now we are going to use the mean centered data to compute the sample covariance of the data. Recall it is given by:
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$$
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\begin{equation*}
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\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
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\end{equation*}
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$$
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where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \).
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Compare the computed covariance with the answer given above.
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<h3 id="___sec20" class="anchor">Diagonalize the sample covariance matrix to obtain the principal components </h3>
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<p>
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Now we are ready to solve for the principal components! To do so we
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diagonalize the sample covariance matrix \( \Sigma_n \). We can use the
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function <b>np.linalg.eig</b> to do so. It will return the eigenvalues and
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eigenvectors of \( \Sigma_n \). Once you have these, carry out the
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following tasks:
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<ul>
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<li> Compute the percentage of the total variance captured by the first principal component</li>
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<li> Plot the mean centered data and lines along the first and second principal components</li>
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<li> Project the mean centered data onto the first and second principal components, and plot the projected data. What do you observe?</li>
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<li> Approximate the data as</li>
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</ul>
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$$
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\begin{equation*}
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x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
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\end{equation*}
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$$
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where \( v_0 \) is the first principal component. What do you observe?
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<p>
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Finally, collect all these steps and write your own PCA function and
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compare this with the functionality included in <b>Scikit-Learn</b>.
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Have the input be the data and have the output be the principal components and their associated eigenvalues, sorted in descending order. Can you think of a way to make it more efficient than the algorithm outlined above?
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">#Now we do an SVD</span>
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U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
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c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
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c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
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W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
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X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
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<span style="color: #008000; font-weight: bold">print</span>(X2D)
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<span style="color: #408080; font-style: italic">#thereafter we do a PCA with Scikit-learn</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
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pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
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X2Dsl <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Check that we get the same"</span>)
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<span style="color: #008000; font-weight: bold">print</span>(X2D<span style="color: #666666">-</span>X2Dsl)
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<span style="color: #008000; font-weight: bold">print</span>(pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>])
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</pre></div>
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<p>
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Finally, try out your own PCA function with other data sets.
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