k-means clustering

### 20.1.2 k-means clustering

The kmeans command is an efficient implementation of k-means algorithm for clustering numerical multidimensional data.

- kmeans accepts one or more input arguments:
  - data, which must be a list of 2D points or m-dimensional Euclidean points entered as vectors. For n such points, data should be a n× m matrix.
  - Optionally, opt, a sequence of options which may contain the following:
    - k or count=k or count=kl..ku, where k is a positive integer specifying the number of clusters and kl..ku is an interval specifying minimal and maximal number of clusters (by default, unset).
    - count_inf=kl, where kl is a positive integer specifying the minimal number of clusters (by default, kl=2).
    - count_sup=ku, where ku is a positive integer specifying the maximal number of clusters (by default, ku=n−1).
    - limit=N, where N is a positive integer specifying the maximal number of iterations for the k-means algorithm (by default, N=100).
    - index=str, where str is a string or a list of strings specifying the index functions used to find the optimal number of clusters automatically. Available indices are: ball-hall, banfeld-raftery, calinski-harabasz, davies-bouldin, det, ksq-detW, log-det, log-ss, pbm, ratkowsky-lance, ray-turi, scott-symons, trace-W, trace-WiB and all (note that you must surround these names by double-quotes). For details, see Desgraupes (2017)2. str may also take a boolean value, in which case either no index is used (str=false) or the Calinski-Harabasz index is used (str=true). By default, Hartigan’s criterion is used to determine the optimal number of clusters if k is not given.
    - output=out, where out is one of the following:
      - part, for outputting the partition of the 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 data.
      - count, for outputting only the number of clusters.
      - index, for outputting the list of values of the (first) used index.
    - display=disp, where disp is a configuration for plot output (by default, points are drawn as dots of width 2).
- kmeans(data ⟨,opt ⟩) returns the result as specified by the output argument (see above).
- kmeans applies the Hartigan-Wong implementation of k-means algorithm, which is fast and suitable for large datasets.
- kmeans does not support custom distance functions. If you want to cluster the data by using a distance function other than the Euclidean distance, use the cluster command (see Section 20.1.1).
- 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.

#### Example

We cluster some random gaussian data and use the Calinski-Harabasz index for choosing the optimal number of clusters.

d:=5:; n:=2500:; data:=[]:; C:=[[5,6],[14,10],[27,5],[10,21],[30,-2]]:; R:=[[2,2.5],[1.2,1],[3,3],[2,1.8],[1,0.5]]:; for k from 1 to n do j:=rand(d); x:=sample(randvar(normal,C[j][0],R[j][0])); y:=sample(randvar(normal,C[j][1],R[j][1])); data:=append(data,[x,y]); od:; kmeans(data,index="calinski-harabasz",output=plot)