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mhjensen 20d45a6142 typos in dim red
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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-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
<!-- 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="___sec17" class="anchor">Classical PCA Theorem </h2>
<p>
We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \)
each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
<p>
We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \). We define the reconstruction error (which is similar to the mean squared error we have seen before) as
$$
J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2,
$$
with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \). When doing PCA we want to reduce this dimensionality.
<p>
The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
<p>
<p>
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