# dbs: Density-based silhouette information methods In pdfCluster: Cluster Analysis via Nonparametric Density Estimation

## Description

Computes the density-based silhouette information of clustered data. Two methods are associated to this function. The first method applies to two arguments: the matrix of data and the vector of cluster labels; the second method applies to objects of pdfCluster-class.

## Usage

 1 2 3 4 5 6 7 ## S4 method for signature 'matrix' dbs(x, clusters, h.funct="h.norm", hmult=1, prior, ...) ## S4 method for signature 'pdfCluster' dbs(x, h.funct="h.norm", hmult = 1, prior = as.vector(table(x@cluster.cores)/sum(table(x@cluster.cores))), stage=NULL, ...) 

## Arguments

 x A matrix of data points partitioned by any density-based clustering method or an object of pdfCluster-class. clusters Cluster labels of grouped data. This argument has not to be set when x is a pdfCluster-class object. h.funct Function to estimate the smoothing parameters. Default is h.norm. hmult Shrink factor to be multiplied by the smoothing parameters. Default value is 1. prior Vector of prior probabilities of belonging to the groups. When x is of pdfCluster-class, default value is set proportional to the cluster cores cardinalities. Otherwise, equal prior probabilities are given to the clusters by default. stage When x is a pdfCluster-class object, this is the stage of classification of low-density data at which the dbs has to be computed. Default value is the number of stages of the procedure. Set it to 0 if the dbs has to be computed at cluster cores only. ... Further arguments to be passed to methods (see dbs-methods) or arguments to kepdf. See details below.

## Details

This function provides diagnostics for a clustering produced by any density-based clustering method. The dbs information is a suitable modification of the silhouette information aimed at evaluating the cluster quality in a density based framework. It is based on the estimation of data posterior probabilities of belonging to the clusters. It may be used to measure the quality of data allocation to the clusters. High values of the \hat{dbs} are evidence of a good quality clustering.

Define

\hat{τ}_m(x_i)=\frac{π_{m} \hat{f}(x_i|x_ \in m)}{∑_{m=1}^M π_{m}\hat{f}(x_i|x_i \in m)} \quad m=1,…,M,

where π_{m} is a prior probability of m and \hat{f}(x_i|x_i \in m) is a density estimate at x_i evaluated with function kepdf by using the only data points in m. Density estimation is performed with fixed bandwidths h, as evaluated by function h.funct, possibly multiplied by the shrink factor hmult.

Density-based silhouette information of x_i, the i^{th} row of the data matrix x, is defined as follows:

\hat{dbs}_i=\frac{\log≤ft(\frac{\hat{τ}_{m_{0}}(x_i)}{\hat{τ}_{m_{1}}(x_i)}\right)}{{\max}_{x_i }≤ft| \log≤ft(\frac{\hat{τ}_{m_{0}}(x_i)}{\hat{τ}_{m_{1}}(x_i)}\right)\right|},

where m_0 is the group where x_i has been allocated and m_1 is the group for which τ_m is maximum, m\neq m_0.

Note: when there exists x_j such that \hat{τ}_{m_{1}}(x_j) is zero, \hat{dbs}_j is forced to 1 and {\max}_{x_i }≤ft| \log≤ft(\frac{\hat{τ}_{m_{0}}(x_i)}{\hat{τ}_{m_{1}}(x_i)}\right)\right| is computed by excluding x_j from tha data matrix x.

See Menardi (2011) for a detailed treatment.

## Value

An object of class "dbs", with slots:

 call  The matched call. x  The matrix of clustered data points. prior  The vector of prior probabilities of belonging to the groups. dbs  A vector reporting the density-based silhouette information of the clustered data. clusters  Cluster labels of grouped data. noc  Number of clusters stage  If argument x of dbs is a pdfCluster-class object, this slot provides the stage of the classification at which the dbs is computed.

See dbs-class for more details.

## Methods

signature(x = "matrix", clusters = "numeric")

Computes the density based silhouette information for objects partitioned according to any density-based clustering method.

signature(x = "pdfCluster", clusters = "missing")

Computes the density based silhouette information for objects of class "pdfCluster"

.

## References

Menardi, G. (2011) Density-based Silhouette diagnostics for clustering methods. Statistics and Computing, 21, 295-308.

dbs-class, plot,dbs-method, silhouette.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 #example 1: no groups in data #random generation of group labels set.seed(54321) x <- rnorm(50) groups <- sample(1:2, 50, replace = TRUE) groups dsil <- dbs(x = as.matrix(x), clusters=groups) dsil summary(dsil) plot(dsil, labels=TRUE, lwd=6) #example 2: wines data # load data data(wine) # select a subset of variables x <- wine[, c(2,5,8)] #clustering cl <- pdfCluster(x) dsil <- dbs(cl) plot(dsil)