prclust (version 1.3)

GCV: Calculate the Generalized Cross-Validation Statistic (GCV)

Description

Calculate the generalized cross-validation statistic with generalized degrees of freedom.

Usage

GCV(data,lambda1,lambda2,tau,sigma,B=100,
	loss.method = c("quadratic","lasso"),
	grouping.penalty = c("gtlp","L1","SCAD","MCP"), 
	algorithm = c("ADMM","Quadratic"), epsilon =0.001)

Arguments

data

Numeric data matrix .

lambda1

Tuning parameter or step size: lambda1, typically set at 1 for quadratic penalty based algorithm; 0.4 for revised ADMM.

lambda2

Tuning parameter: lambda2, the magnitude of grouping penalty.

tau

Tuning parameter: tau, related to grouping penalty.

sigma

The perturbation size.

B

The Monte Carlo time. The defualt value is 100.

loss.method

character may be abbreviated. "lasso" stands for $L_1$ loss function, while "quadratic" stands for the quadratic loss function.

grouping.penalty

character: may be abbreviated. "gtlp" means generalized group lasso is used for grouping penalty. "lasso" means lasso is used for grouping penalty. "SCAD" and "MCP" are two other non-convex penalty.

algorithm

character: may be abbreviated. The algorithm will use for finding the solution. The default algorithm is "ADMM", which stands for the DC-ADMM.

epsilon

The stopping critetion parameter. The default is 0.001.

Value

Return value: the Generalized cross-validation statistic (GCV)

Details

A bonus with the regression approach to clustering is the potential application of many existing model selection methods for regression or supervised learning to clustering. We propose using generalized cross-validation (GCV). GCV can be regarded as an approximation to leave-one-out cross-validation (CV). Hence, GCV provides an approximately unbiased estimate of the prediction error.

We use the generalized degrees of freedom (GDF) to consider the data-adaptive nature in estimating the centroids of the observations.

The chosen tuning parameters are the one giving the smallest GCV error.

References

Pan, W., Shen, X., & Liu, B. (2013). Cluster analysis: unsupervised learning via supervised learning with a non-convex penalty. Journal of Machine Learning Research, 14(1), 1865-1889.

Examples

set.seed(1)
library("prclust")
data = matrix(NA,2,50)
data[1,1:25] = rnorm(25,0,0.33)
data[2,1:25] = rnorm(25,0,0.33)
data[1,26:50] = rnorm(25,1,0.33)
data[2,26:50] = rnorm(25,1,0.33)

#case 1
gcv1 = GCV(data,lambda1=1,lambda2=1,tau=0.5,sigma=0.25,B =10)
gcv1

#case 2
gcv2 = GCV(data,lambda1=1,lambda2=0.7,tau=0.3,sigma=0.25,B = 10)
gcv2

# Note that the combination of tuning parameters in case 1 are better than 
# the combination of tuning parameters in case 2 since the value of GCV in case 1 is
# less than the value in case 2.