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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>
<!-- 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-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-bs008.html#___sec7" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
<!-- navigation toc: --> <li><a href="#___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-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-bs017.html#___sec16" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec18" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample mean and center the data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec19" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample covariance</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec20" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Diagonalize the sample covariance matrix to obtain the principal components</a></li>
<!-- 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-bs024.html#___sec26" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec27" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
<!-- 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-bs028.html#___sec30" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<h2 id="___sec8" class="anchor">Introducing the Covariance and Correlation functions </h2>
<p>
Before we discuss the PCA theorem, we need to remind ourselves about
the definition of the covariance and the correlation function. These are quantities
<p>
Suppose we have defined two vectors
\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
\end{bmatrix},
$$
where for example
$$
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
$$
With this definition and recalling that the variance is defined as
$$
\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
$$
we can rewrite the covariance matrix as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
\end{bmatrix}.
$$
<p>
The covariance takes values between zero and infinity and may thus
lead to problems with loss of numerical precision for particularly
large values. It is common to scale the covariance matrix by
introducing instead the correlation matrix defined via the so-called
correlation function
$$
\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
$$
<p>
The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
\in [-1,1] \). This avoids eventual problems with too large values. We
can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
and \( \boldsymbol{y} \) as
$$
\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
\end{bmatrix},
$$
<p>
In the above example this is the function we constructed using <b>pandas</b>.
<p>
<p>
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