Density-based silhouette information methods
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.
## 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, ...)
x |
A matrix of data points partitioned by any density-based clustering method or an object of |
clusters |
Cluster labels of grouped data. This argument has not to be set when |
h.funct |
Function to estimate the smoothing parameters. Default is |
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 |
stage |
When |
... |
Further arguments to be passed to methods (see |
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.
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 |
See dbs-class for more details.
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"
.
Menardi, G. (2011) Density-based Silhouette diagnostics for clustering methods. Statistics and Computing, 21, 295-308.
#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)
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