Hierarchical clustering: cluster

### 9.1.1 Hierarchical clustering: cluster

The cluster command is used for hierarchical agglomerative clustering of custom data by using a custom distance function.

- cluster takes one or more arguments:
  - data, a list of data points.
  - Optionally, opt, a sequence of options which may contain the following:
    - k or count=k, where k is a positive integer specifying the number of clusters (unset by default).
    - count_inf=kl, where kl is a positive integer specifying the minimal number of clusters (unset by default).
    - count_sup=ku, where ku is a positive integer specifying the maximal number of clusters (by default, ku=16).
    - type=linkage, where linkage is a string specifying the linkage method. Avaliable methods are:
      - single (the default),
      - complete,
      - average,
      - weighted,
      - ward.
    - distance=dist, where dist is a distance function. By default, the squared Euclidean distance function distance2 is used unless data is a list of strings, in which case Levenshtein distance is used. For example, the taxicab distance function can be entered as follows:
      taxicab:=(p1,p2)->l1norm(p1-p2)
    - index=ind, where ind is a string or a list of strings specifying the index function(s) used for selecting the optimal number of clusters (unset by default). Available index functions are: silhouette, mcclain-rao, dunn, and all. ind may also take a boolean value, in which case no index is used (ind=false) or the silhouette index is used (ind=true).
    - output=out, where out is one of the following:
      - part, for outputting the partition of data into clusters.
      - list, for outputting the list of cluster indices for data points (the default).
      - plot, for outputting a colored visualization of data points with additional specifications given in display option (see below). Note that this is possible only with two- and three-dimensional numerical data.
      - count, for outputting only the number of clusters.
      - index, for outputting the list of values of the (first) used index.
      - tree, for outputting a dendrogram drawing which visualizes the linking process.
    - display=disp, where disp is a configuration for plot output (by default, points are drawn as dots of width 2).
    - labels=lab, where lab is a boolean value which specifies whether the data points are shown in the dendrogram.
- Hierarchical clustering is slower than the k-means algorithm (see Section 9.1.2) but may produce a better classification of the data. Hierarchical clustering is also a method of choice for custom data types and distance functions.
- If more than one index is computed in a clustering process, then the optimal number of clusters is decided by voting: the number which was selected by most indices is used.

#### Examples.

- We apply cluster command to the “aggregation” shape dataset obtained from http://cs.joensuu.fi/sipu/datasets/. The dataset is loaded from file in a table cell in Xcas and associated with the variable data. We use the average linkage method and silhouette index (which is used by default if index=true). By setting output parameter to plot, we obtain a visualization of colored clusters as shown in Figure 9.1.

  > Figure 9.1: Clustering in Xcas

- For string data, Levenshtein distance is used by default (see Section 5.3.15). Input:
  cluster(["cat","mouse","rat","spouse","house","cut"],output=part)
  Output:

  ⎡ ⎢ ⎣ |
  "cat" | "rat" | "cut"
  "mouse" | "spouse" | "house"
  | ⎤ ⎥ ⎦

- In this example we cluster genomic sequences into five clusters by using average linkage and Hamming distance function.
  Input:

  data:=["GTCTT","AAGCT","GGTAA","AGGCT","GTCAT","CGGCC",
  "GGGAG","GTTAT","GTCAT","AGGCT","GTCAG","AGGAT"]:;
  cluster(data,type="average",count=5,distance=hamdist,output=part)

  Output:

  [ |
  ⎡ ⎣ | "GTCTT","GTCAT","GTTAT","GTCAT","GTCAG" | ⎤ ⎦ | ,
  ⎡ ⎣ | "AAGCT","AGGCT","AGGCT","AGGAT" | ⎤ ⎦ | ,
  ⎡ ⎣ | "GGTAA" | ⎤ ⎦ | , | ⎡ ⎣ | "CGGCC" | ⎤ ⎦ | , | ⎡ ⎣ | "GGGAG" | ⎤ ⎦ | ]