prediction.strength {fpc}

Prediction strength for estimating number of clusters

Description

Computes the prediction strength of a clustering of a dataset into different numbers of components. The prediction strength is defined according to Tibshirani and Walther (2005), who recommend to choose as optimal number of cluster the largest number of clusters that leads to a prediction strength above 0.8 or 0.9. See details.

Various clustering methods can be used, see argument clustermethod. In Tibshirani and Walther (2005), only classification to the nearest centroid is discussed, but more methods are offered here, see argument classification.

Usage

  prediction.strength(xdata, Gmin=2, Gmax=10, M=50,
                      clustermethod=kmeansCBI,
                                classification="centroid", centroidname = NULL,
                                cutoff=0.8,nnk=1,
                      distances=inherits(xdata,"dist"),count=FALSE,...)
  ## S3 method for class 'predstr'
print(x, ...)

Arguments

xdata

data (something that can be coerced into a matrix).

Gmin

integer. Minimum number of clusters. Note that the prediction strength for 1 cluster is trivially 1, which is automatically included if GMin>1. Therefore GMin<2 is useless.

Gmax

integer. Maximum number of clusters.

M

integer. Number of times the dataset is divided into two halves.

clustermethod

an interface function (the function name, not a string containing the name, has to be provided!). This defines the clustering method. See the "Details"-section of clusterboot and kmeansCBI for the format. Clustering methods for prediction.strength must have a k-argument for the number of clusters, must operate on n times p data matrices and must otherwise follow the specifications in clusterboot Note that prediction.strength won't work with CBI-functions that implicitly already estimate the number of clusters such as pamkCBI; use claraCBI if you want to run it for pam/clara clustering.

classification

string. This determines how non-clustered points are classified to given clusters. Options are explained in classifnp and classifdist, the latter for dissimilarity data. Certain classification methods are connected to certain clustering methods. classification="averagedist" is recommended for average linkage, classification="centroid" is recommended for k-means, clara and pam (with distances it will work with claraCBI only), classification="knn" with nnk=1 is recommended for single linkage and classification="qda" is recommended for Gaussian mixtures with flexible covariance matrices.

centroidname

string. Indicates the name of the component of CBIoutput$result that contains the cluster centroids in case of classification="centroid", where CBIoutput is the output object of clustermethod. If clustermethod is kmeansCBI or claraCBI, centroids are recognised automatically if centroidname=NULL. If centroidname=NULL and distances=FALSE, cluster means are computed as the cluster centroids.

cutoff

numeric between 0 and 1. The optimal number of clusters is the maximum one with prediction strength above cutoff.

nnk

number of nearest neighbours if classification="knn", see classifnp.

distances

logical. If TRUE, data will be interpreted as dissimilarity matrix, passed on to clustering methods as "dist"-object, and classifdist will be used for classification.

count

logical. TRUE will print current number of clusters and simulation run number on the screen.

x

object of class predstr.

...

arguments to be passed on to the clustering method.

Details

The prediction strength for a certain number of clusters k under a random partition of the dataset in halves A and B is defined as follows. Both halves are clustered with k clusters. Then the points of A are classified to the clusters of B. In the original paper this is done by assigning every observation in A to the closest cluster centroid in B (corresponding to classification="centroid"), but other methods are possible, see classifnp. A pair of points A in the same A-cluster is defined to be correctly predicted if both points are classified into the same cluster on B. The same is done with the points of B relative to the clustering on A. The prediction strength for each of the clusterings is the minimum (taken over all clusters) relative frequency of correctly predicted pairs of points of that cluster. The final mean prediction strength statistic is the mean over all 2M clusterings.

Value

prediction.strength gives out an object of class predstr, which is a list with components

predcorr

list of vectors of length M with relative frequencies of correct predictions (clusterwise minimum). Every list entry refers to a certain number of clusters.

mean.pred

means of predcorr for all numbers of clusters.

optimalk

optimal number of clusters.

cutoff

see above.

method

a string identifying the clustering method.

Gmax

see above.

M

see above.

Author(s)

Christian Hennig [e-mail] https://www.unibo.it/sitoweb/christian.hennig/en/

References

Tibshirani, R. and Walther, G. (2005) Cluster Validation by Prediction Strength, Journal of Computational and Graphical Statistics, 14, 511-528.

See Also

kmeansCBI, classifnp

Examples

  options(digits=3)
  set.seed(98765)
  iriss <- iris[sample(150,20),-5]
  prediction.strength(iriss,2,3,M=3)
  prediction.strength(iriss,2,3,M=3,clustermethod=claraCBI)
# The examples are fast, but of course M should really be larger.