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  "body": "\n## Definition\n\nLet $x_1, \\ldots, x_n$ be data points and let $d(x_i, x_j) \\ge 0$ be a dissimilarity with $d(x_i, x_i) = 0$ and $d(x_i, x_j) = d(x_j, x_i)$.\nA (hard) *clustering* of the data into $K$ clusters is a [[partition|08QMKTWF9T]] of $\\set{1, \\ldots, n}$ into $K$ non-empty sets $C_1, \\ldots, C_K$.\n*Cluster analysis* is the task of choosing a clustering in which dissimilarities within clusters are small and dissimilarities between clusters are large.\nA method of cluster analysis makes this precise.\nIt may minimise an objective function over all clusterings, or follow a rule that merges or splits clusters.\nA *hierarchical clustering* is a nested sequence of clusterings, from $n$ singletons to one cluster.\n\n## Remarks\n\nExample: the points $0, 1, 10, 11$ on the real line with $d(x, y) = \\abs{x - y}$ have the natural clustering $\\set{0, 1}$, $\\set{10, 11}$: within-cluster distances are $1$ and between-cluster distances are at least $9$.\nNon-example: the sets $\\set{0, 1}$ and $\\set{1, 10, 11}$ are not a clustering, because the point $1$ lies in two sets, so they do not form a partition.\nSoft or fuzzy clusterings relax the partition to membership weights in $[0, 1]$ that add up to $1$ for each point.\nSingle linkage merges at each step the two clusters with the smallest minimum distance; complete linkage uses the maximum distance instead.\n",
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