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\title{Test Document}


\chapter{Clustering}

In the context of data preprocessing, clustering is just a single step in some larger system, with the utility of a clustering algorithm measured only in relation to the end goal.
Online retailers, for instance, cluster customers based on their purchases and recommend to you those products other ``similar customers'' have purchased.
The best clustering methodology in this case would be whichever one whose recommended products are most often purchased, regardless of the elegant mathematical foundations of the underlying algorithm or whether the resultant clusters have some obvious ``meaning''.


A metric similarity measure is one for which also
\begin{align*}
  s\!\left(\vec x, \vec y\right) = s_{0} \Leftrightarrow \vec x = \vec y
\end{align*}
and
\begin{align*}
  s\!\left(\vec x, \vec y\right)s\!\left(\vec y, \vec z\right) \le \left[s\!\left(\vec x, \vec y\right) + s\!\left(\vec y, \vec z\right)\right]s\!\left(\vec x, \vec z\right) \qquad \forall \vec x, \vec y, \vec z \in X.
\end{align*}

Since we want the affinity matrix $\mathbf A$ to emphasize the closeness of neighboring residues, we apply a \textit{kernel function}, say $\frac{1}{d^{2}}$ or $\mathrm{e}^{-d/\sigma^{2}}$, to twiddle the relationships of residue pairs.


\chapter{Another Chapter}

More text here.
For the second chapter\ldots.


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