mhjensen
287 lines
14 KiB
HTML
287 lines
14 KiB
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Recurrent Neural Networks', 2, None, '___sec0'),
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('Solving ODEs with Deep Learning', 2, None, '___sec1'),
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('Why should we think of reducing the dimensionality',
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('Basic ideas of the Principal Component Analysis (PCA)',
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('Introducing the Covariance and Correlation functions',
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('Correlation Function and Design/Feature Matrix',
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('Covariance Matrix Examples', 2, None, '___sec6'),
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('Correlation Matrix with Pandas', 2, None, '___sec8'),
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('Rewriting the Covariance and/or Correlation Matrix',
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'___sec10'),
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('Towards the PCA theorem', 2, None, '___sec11'),
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('The Algorithm before the Theorem', 2, None, '___sec12'),
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('Writing our own PCA code', 2, None, '___sec13'),
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('Compute the sample mean and center the data',
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3,
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None,
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'___sec14'),
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('Compute the sample covariance', 3, None, '___sec15'),
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('Diagonalize the sample covariance matrix to obtain the '
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'principal components',
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3,
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None,
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'___sec16'),
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('Classical PCA Theorem', 2, None, '___sec17'),
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('Proof of the PCA Theorem', 2, None, '___sec18'),
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('PCA Proof continued', 2, None, '___sec19'),
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('The final step', 2, None, '___sec20'),
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('Geometric Interpretation and link with Singular Value '
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'Decomposition',
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2,
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('Principal Component Analysis', 2, None, '___sec22'),
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('PCA and scikit-learn', 2, None, '___sec23'),
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('Back to the Cancer Data', 2, None, '___sec24'),
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('More on the PCA', 2, None, '___sec25'),
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('Incremental PCA', 2, None, '___sec26'),
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('Randomized PCA', 2, None, '___sec27'),
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('Kernel PCA', 2, None, '___sec28'),
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('LLE', 2, None, '___sec29'),
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('Other techniques', 2, None, '___sec30')]}
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<a class="navbar-brand" href="week43-bs.html">Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week43-bs002.html#___sec0" style="font-size: 80%;"><b>Recurrent Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec1" style="font-size: 80%;"><b>Solving ODEs with Deep Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs004.html#___sec2" 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="._week43-bs005.html#___sec3" 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="._week43-bs006.html#___sec4" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs007.html#___sec5" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs008.html#___sec6" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs009.html#___sec7" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs010.html#___sec8" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs011.html#___sec9" 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="._week43-bs012.html#___sec10" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs014.html#___sec12" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec13" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec14" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec15" style="font-size: 80%;"> Compute the sample covariance</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec16" 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="._week43-bs016.html#___sec17" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs017.html#___sec18" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec19" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec20" style="font-size: 80%;"><b>The final step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec21" 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="._week43-bs021.html#___sec22" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec23" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec24" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec25" style="font-size: 80%;"><b>More on the PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs025.html#___sec26" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs026.html#___sec27" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec28" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec29" style="font-size: 80%;"><b>LLE</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec30" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</li>
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</ul>
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<a name="part0013"></a>
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<!-- !split -->
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<h2 id="___sec11" class="anchor">Towards the PCA theorem </h2>
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<p>
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We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
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$$
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\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
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$$
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Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \).
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These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
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<p>
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Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
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<p>
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That is we have
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$$
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\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
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$$
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since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S}^T \) from the left we have
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$$
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\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
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$$
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and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
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$$
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\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
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$$
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<p>
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In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
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\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
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<p>
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The eigenvalues tell us then how much we need to stretch the
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corresponding eigenvectors. Dimensions with large eigenvalues have
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thus large variations (large variance) and define therefore useful
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dimensions. The data points are more spread out in the direction of
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these eigenvectors. Smaller eigenvalues mean on the other hand that
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the corresponding eigenvectors are shrunk accordingly and the data
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points are tightly bunched together and there is not much variation in
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these specific directions. Hopefully then we could leave it out
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dimensions where the eigenvalues are very small. If \( p \) is very large,
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we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
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features/predictors.
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<p>
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<p>
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