fpc (version 2.1-6)

cluster.stats: Cluster validation statistics

Description

Computes a number of distance based statistics, which can be used for cluster validation, comparison between clusterings and decision about the number of clusters: cluster sizes, cluster diameters, average distances within and between clusters, cluster separation, biggest within cluster gap, average silhouette widths, the Calinski and Harabasz index, a Pearson version of Hubert's gamma coefficient, the Dunn index and two indexes to assess the similarity of two clusterings, namely the corrected Rand index and Meila's VI.

Usage

cluster.stats(d = NULL, clustering, alt.clustering = NULL,
                           noisecluster=FALSE,
                              silhouette = TRUE, G2 = FALSE, G3 = FALSE,
                              wgap=TRUE, sepindex=TRUE, sepprob=0.1,
                              sepwithnoise=TRUE,
                              compareonly = FALSE,
                              aggregateonly = FALSE)

Arguments

d

a distance object (as generated by dist) or a distance matrix between cases.

clustering

an integer vector of length of the number of cases, which indicates a clustering. The clusters have to be numbered from 1 to the number of clusters.

alt.clustering

an integer vector such as for clustering, indicating an alternative clustering. If provided, the corrected Rand index and Meila's VI for clustering vs. alt.clustering are computed.

noisecluster

logical. If TRUE, it is assumed that the largest cluster number in clustering denotes a 'noise class', i.e. points that do not belong to any cluster. These points are not taken into account for the computation of all

silhouette

logical. If TRUE, the silhouette statistics are computed, which requires package cluster.

G2

logical. If TRUE, Goodman and Kruskal's index G2 (cf. Gordon (1999), p. 62) is computed. This executes lots of sorting algorithms and can be very slow (it has been improved by R. Francois - thanks!)

G3

logical. If TRUE, the index G3 (cf. Gordon (1999), p. 62) is computed. This executes sort on all distances and can be extremely slow.

wgap

logical. If TRUE, the widest within-cluster gaps (largest link in within-cluster minimum spanning tree) are computed. This is used for finding a good number of clusters in Hennig (2013).

sepindex

logical. If TRUE, a separation index is computed, defined based on the distances for every point to the closest point not in the same cluster. The separation index is then the mean of the smallest proportion sepprob o

sepprob

numerical between 0 and 1, see sepindex.

sepwithnoise

logical. If TRUE and sepindex and noisecluster are both TRUE, the noise points are incorporated as cluster in the separation index computation. Also they are taken into account for the comput

compareonly

logical. If TRUE, only the corrected Rand index and Meila's VI are computed and given out (this requires alt.clustering to be specified).

aggregateonly

logical. If TRUE (and not compareonly), no clusterwise but only aggregated information is given out (this cuts the size of the output down a bit).

Value

• cluster.stats returns a list containing the components n, cluster.number, cluster.size, min.cluster.size, noisen, diameter, average.distance, median.distance, separation, average.toother, separation.matrix, average.between, average.within, n.between, n.within, within.cluster.ss, clus.avg.silwidths, avg.silwidth, g2, g3, pearsongamma, dunn, entropy, wb.ratio, ch, corrected.rand, vi except if compareonly=TRUE, in which case only the last two components are computed.
• n
  number of cases.
• cluster.number
  number of clusters.
• cluster.size
  vector of cluster sizes (number of points).
• min.cluster.size
  size of smallest cluster.
• noisen
  number of noise points, see argument noisecluster (noisen=0 if noisecluster=FALSE).
• diameter
  vector of cluster diameters (maximum within cluster distances).
• average.distance
  vector of clusterwise within cluster average distances.
• median.distance
  vector of clusterwise within cluster distance medians.
• separation
  vector of clusterwise minimum distances of a point in the cluster to a point of another cluster.
• average.toother
  vector of clusterwise average distances of a point in the cluster to the points of other clusters.
• separation.matrix
  matrix of separation values between all pairs of clusters.
• ave.between.matrix
  matrix of mean dissimilarities between points of every pair of clusters.
• average.between
  average distance between clusters.
• average.within
  average distance within clusters.
• n.between
  number of distances between clusters.
• n.within
  number of distances within clusters.
• max.diameter
  maximum cluster diameter.
• min.separation
  minimum cluster separation.
• within.cluster.ss
  a generalisation of the within clusters sum of squares (k-means objective function), which is obtained if d is a Euclidean distance matrix. For general distance measures, this is half the sum of the within cluster squared dissimilarities divided by the cluster size.
• clus.avg.silwidths
  vector of cluster average silhouette widths. See silhouette.
• avg.silwidth
  average silhouette width. See silhouette.
• g2
  Goodman and Kruskal's Gamma coefficient. See Milligan and Cooper (1985), Gordon (1999, p. 62).
• g3
  G3 coefficient. See Gordon (1999, p. 62).
• pearsongamma
  correlation between distances and a 0-1-vector where 0 means same cluster, 1 means different clusters. "Normalized gamma" in Halkidi et al. (2001).
• dunn
  minimum separation / maximum diameter. Dunn index, see Halkidi et al. (2002).
• dunn2
  minimum average dissimilarity between two cluster / maximum average within cluster dissimilarity, another version of the family of Dunn indexes.
• entropy
  entropy of the distribution of cluster memberships, see Meila(2007).
• wb.ratio
  average.within/average.between.
• ch
  Calinski and Harabasz index (Calinski and Harabasz 1974, optimal in Milligan and Cooper 1985; generalised for dissimilarites in Hennig and Liao 2010).
• cwidegap
  vector of widest within-cluster gaps.
• widestgap
  widest within-cluster gap.
• sindex
  separation index, see argument sepindex.
• corrected.rand
  corrected Rand index (if alt.clustering has been specified), see Gordon (1999, p. 198).
• vi
  variation of information (VI) index (if alt.clustering has been specified), see Meila (2007).

References

Calinski, T., and Harabasz, J. (1974) A Dendrite Method for Cluster Analysis, Communications in Statistics, 3, 1-27.

Gordon, A. D. (1999) Classification, 2nd ed. Chapman and Hall.

Halkidi, M., Batistakis, Y., Vazirgiannis, M. (2001) On Clustering Validation Techniques, Journal of Intelligent Information Systems, 17, 107-145.

Hennig, C. and Liao, T. (2010) Comparing latent class and dissimilarity based clustering for mixed type variables with application to social stratification. Research report no. 308, Department of Statistical Science, UCL. http://www.ucl.ac.uk/Stats/research/reports/psfiles/rr308.pdf. Revised version accepted for publication by Journal of the Royal Statistical Society Series C.

Hennig, C. (2013) How many bee species? A case study in determining the number of clusters. To appear in Proceedings of GfKl-2012, Hildesheim. Kaufman, L. and Rousseeuw, P.J. (1990). "Finding Groups in Data: An Introduction to Cluster Analysis". Wiley, New York. Meila, M. (2007) Comparing clusterings?an information based distance, Journal of Multivariate Analysis, 98, 873-895. Milligan, G. W. and Cooper, M. C. (1985) An examination of procedures for determining the number of clusters. Psychometrika, 50, 159-179.

See Also

silhouette, dist, calinhara, distcritmulti. clusterboot computes clusterwise stability statistics by resampling.

Examples

set.seed(20000)
  face <- rFace(200,dMoNo=2,dNoEy=0,p=2)
  dface <- dist(face)
  complete3 <- cutree(hclust(dface),3)
  cluster.stats(dface,complete3,
                alt.clustering=as.integer(attr(face,"grouping")))