168 lines
4.2 KiB
TeX
168 lines
4.2 KiB
TeX
\chapter{Unsupervised Learning}
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\begin{definition}[Precision Medicine]
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Design of treatment for a given patient, based on genomic data.
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\end{definition}
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\begin{definition}[Hierarchical clustering]
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\end{definition}
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Gene expression time series: look for genes with similar expression footprint.
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\paragraph{Representation of data}
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\begin{itemize}
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\item Tables;
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\item Trees / Graphs;
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\item Time series...
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\end{itemize}
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\begin{figure}
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\includestandalone{figures/plots/genes_expression_timeseries}
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\caption{Example of gene expression time series}
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\end{figure}
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\section{Distances and Similarities}
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\begin{property}[Distance]
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\begin{description}
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\item[non-negativity] $d(i, j) \geq 0$
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\item[isolation] $d(i, i) = 0$
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\item[symmetry] $d(i, j) = d(j, i)$
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\item[triangular inequality] $d(i, j) \leq d(i, h) + d(h, j)$
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\end{description}
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\end{property}
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\begin{definition}[Dissimilarity]
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Distance without triangular inequality.
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\end{definition}
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\begin{definition}[Similarity]
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Function $s$ from $X \times X$ to $\RR_+$ such that:
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\begin{enumerate}
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\item $s$ is symmetric: $(x, y) \in X \times X; s(x, y) = s(y, x)$
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\item $(x, y) \in X \times X; s(x, x) = s(y, y) > s(x, y)$.
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\end{enumerate}
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\end{definition}
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\begin{exercise}
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Let $d(x, y)$ be the distance, $d(x, y) \in [0, +\infty[$.
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What should be the similarity measure $S(x, y) = f(d(x, y))$ that satisfies the following property:
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\[
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(x, y) \in X \times X \: | \: S(x, y) > S(x, y)
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\]
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having $S(x, y) \leq M$, $S(x, y) \in ]0, M]$.
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\end{exercise}
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$d(x, y) \geq 0 \: \forall (x, y)$
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\begin{equation}
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S(x, y) = \frac{M}{d(x, y) + 1}
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\label{eq:similarity-first}
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\end{equation}
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In \cref{eq:similarity-first}, $S(x, y)$ ranges from 0 to M.
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\begin{eqnarray}
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\lim_{n \to \infty} \frac{M}{n + 1} = 0 && \lim_{n \to 0} \frac{M}{n + 1} = M
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\end{eqnarray}
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\section{Data Representation}
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\paragraph{Data matrix}
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\paragraph{Distance matrix}
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\[
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\begin{bmatrix}
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0 \\
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d(2, 1) & 0 \\
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d(3, 1) & d(3, 2) & 0 \\
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\vdots & \vdots & \ddots \\
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d(n, 1) & d(n,2) & \dots & \dots & 0
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\end{bmatrix}
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\]
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\begin{table}
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\centering
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\begin{tabular}{c|cc}
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&$s_{1}$ & $s_{2}$ \\
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\hline
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$p_{1}$ & 0 & 1 \\
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$p_{2}$ & 1 & 0 \\
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$p_{3}$ & 3 & 2 \\
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\end{tabular}
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\caption{Example data matrix: 2 symptoms for 3 patients.}
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\end{table}
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\begin{definition}[Minkowski distance]
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\[
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L_p (x, y) = \left(\abs{x_1 - y_1}^p + \abs{x_2 - y_2}^p + \ldots + \abs{x_d - y_d}^p\right)^{\sfrac{1}{p}} = \left(\sum_{i=1}^d \left(x_i - y_i\right)^p\right)^{\sfrac{1}{p}}
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\]
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where $p$ is a positive integer.
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\end{definition}
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\begin{definition}[Manhattan distance]
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\[
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L_1(x, y) = \sum_{i=1}^d \abs{x_i - y_i}
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\]
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\end{definition}
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\begin{definition}[Euclidian distance]
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Let $A$ and $B$ be two points, with $(x_{A}, y_{A})$ and $(x_{B}, y_{B})$ their respective coordinates,
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\end{definition}
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If $p=2$, $L_2$ is the Euclidian distance:
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\begin{definition}[Euclidian distance]
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\[
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d(x, y) = \sqrt{\abs{x_1 - y_1}^2 + \abs{x_2 - y_2} + \ldots + \abs{x_d - y_d}^2}
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\]
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\end{definition}
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We can add weights
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\subsection{K-means}
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The cost function is minimized:
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\[
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Cost(C) \sum_{i=1}^{k}...
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\]
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\begin{algorithm}[H]
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Choose the number of clusters $k$.
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Choose randomly $k$ means.
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For each point, compute the distance between the point and each means.
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We allocate the point to the cluster represented by the clostest center.
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We set each means to the center of the cluster, and reiterate.
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\caption{$K$-means algorithm}
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\end{algorithm}
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\begin{exercise}
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We have six genes:
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\begin{table}[H]
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\centering
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\begin{tabular}{ccccccc}
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\toprule
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& $g_{1}$ & $g_{2}$ & $g_{3}$ & $g_{4}$ & $g_{5}$ & $g_{6}$ \\
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\midrule
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$\times 10^{-2}$ & 10 & 12 & 9 & 15 & 17 & 18 \\
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\bottomrule
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\end{tabular}
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\caption{Sample values for six gene expressions.}
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\end{table}
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With $k=2$ and $m_{1} = 10 \cdot 10^{-2}$ and $m_{2} = 9 \cdot 10^{-2}$ the two initial randomly chosen means, run the $k$-means algorithm.
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\end{exercise}
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\begin{figure}
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\centering
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\includegraphics[scale=1]{figures/plots/kmeans.pdf}
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\caption{$k$-means states at each of the 3 steps}
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\end{figure} |