fpc (version 2.2-9)

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 functions of within and between cluster distances including the validation indexes.

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 of these. This allows to formalise separation less sensitive to a single or a few ambiguous points. The output component corresponding to this is sindex, not separation! This is used for finding a good number of clusters in Hennig (2013).

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 (sepindex) computation. Also they are taken into account for the computation for the minimum cluster separation.

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, cwidegap, widestgap, sindex, 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 (reweighted so that every observation, rather than every distance, has the same weight).

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 2013).

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. (2013) How to find an appropriate clustering for mixed-type variables with application to socio-economic stratification, Journal of the Royal Statistical Society, Series C Applied Statistics, 62, 309-369.

Hennig, C. (2013) How many bee species? A case study in determining the number of clusters. In: Spiliopoulou, L. Schmidt-Thieme, R. Janning (eds.): "Data Analysis, Machine Learning and Knowledge Discovery", Springer, Berlin, 41-49.

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

cqcluster.stats is a more sophisticated version of cluster.stats with more options. silhouette, dist, calinhara, distcritmulti. clusterboot computes clusterwise stability statistics by resampling.

Examples

# NOT RUN {
  set.seed(20000)
  options(digits=3)
  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")))
  
# }