KMeansSparseCluster.permute: Choose tuning parameter for sparse k-means clustering

Description

The tuning parameter controls the L1 bound on w, the feature weights. A permutation approach is used to select the tuning parameter.

Usage

KMeansSparseCluster.permute(x, K=NULL, nperms = 25, wbounds = NULL,
silent = FALSE, nvals = 10, centers=NULL)
"print"(x,...)
"plot"(x,...)

Arguments

The nxp data matrix, n is the number of observations and p the number of features.

The number of clusters desired - that is, the "K" in K-means clustering. Must specify K or centers.

nperms

Number of permutations.

wbounds

The range of tuning parameters to consider. This is the L1 bound on w, the feature weights. If NULL, then a range of values will be chosen automatically. Should be greater than 1 if non-null.

silent

Print out progress?

nvals

If wbounds is NULL, then the number of candidate tuning parameter values to consider.

centers

Optional argument. If you want to run the k-means algorithm starting from a particular set of clusters, then you can enter the Kxp matrix of cluster centers here. Default use case involves taking centers=NULL and instead specifying K.

...

not used.

Value

gaps: The gap statistics obtained (one for each of the tuning parameters tried). If O(s) is the objective function evaluated at the tuning parameter s, and O*(s) is the same quantity but for the permuted data, then Gap(s)=log(O(s))-mean(log(O*(s))).
sdgaps: The standard deviation of log(O*(s)), for each value of the tuning parameter s.
nnonzerows: The number of features with non-zero weights, for each value of the tuning parameter.
wbounds: The tuning parameters considered.
bestw: The value of the tuning parameter corresponding to the highest gap statistic.

Details

Sparse k-means clustering seeks a p-vector of weights w (one per feature) and a set of clusters C1,...,CK that optimize $maximize_C1,...,CK,w sum_j w_j BCSS_j$ subject to $||w||_2 <= 1,="" ||w||_1="" <="s," w_j="">= 0$, where $BCSS_j$ is the between cluster sum of squares for feature j, and s is a value for the L1 bound on w. Let O(s) denote the objective function with tuning parameter s: i.e. $O(s)=sum_j w_j BCSS_j$.

We permute the data as follows: within each feature, we permute the observations. Using the permuted data, we can run sparse K-means with tuning parameter s, yielding the objective function O*(s). If we do this repeatedly we can get a number of O*(s) values.

Then, the Gap statistic is given by $Gap(s)=log(O(s))-mean(log(O*(s)))$. The optimal s is that which results in the highest Gap statistic. Or, we can choose the smallest s such that its Gap statistic is within $sd(log(O*(s)))$ of the largest Gap statistic.

References

Witten and Tibshirani (2009) A framework for feature selection in clustering.

Examples

Run this code

# generate data
set.seed(11)
x <- matrix(rnorm(50*70),ncol=70)
x[1:25,1:10] <- x[1:25,1:10]+1.5
x <- scale(x, TRUE, TRUE)
# choose tuning parameter
km.perm <-
KMeansSparseCluster.permute(x,K=2,wbounds=seq(2,5,len=8),nperms=3)
print(km.perm)
plot(km.perm)
# run sparse k-means
km.out <- KMeansSparseCluster(x,K=2,wbounds=km.perm$bestw)
print(km.out)
plot(km.out)
# run sparse k-means for a range of tuning parameter values
km.out <- KMeansSparseCluster(x,K=2,wbounds=2:7)
print(km.out)
plot(km.out)
# Repeat, but this time start with a particular choice of cluster
# centers.
# This will do 4-means clustering starting with this particular choice
# of cluster centers.
km.perm.out <- KMeansSparseCluster.permute(x,wbounds=2:6, centers=x[1:4,],nperms=3)
print(km.out)
plot(km.out)

Run the code above in your browser using DataLab

Get 50% off unlimited learning