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mhjensen 63659da7d8 added some examples
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<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
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<h2 id="___sec15" class="anchor">Towards the PCA theorem </h2>
<p>
We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
$$
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
$$
Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \).
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 \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
<p>
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 \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
<p>
That is we have
$$
\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
$$
since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S}^T \) from the left we have
$$
\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
$$
and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
$$
\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
$$
<p>
In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
<p>
The eigenvalues tell us then how much we need to stretch the
corresponding eigenvectors. Dimensions with large eigenvalues have
thus large variations (large variance) and define therefore useful
dimensions. The data points are more spread out in the direction of
these eigenvectors. Smaller eigenvalues mean on the other hand that
the corresponding eigenvectors are shrunk accordingly and the data
points are tightly bunched together and there is not much variation in
these specific directions. Hopefully then we could leave it out
dimensions where the eigenvalues are very small. If \( p \) is very large,
we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
features/predictors.
<p>
<p>
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