# apL1: Model-based Clustering with APL1 In PARSE: Model-Based Clustering with Regularization Methods for High-Dimensional Data

## Description

The adaptive L_1 penalty was proposed by Pan and Shen (2007). Under the framework of the model-based clustering, APL1 aims to identify the globally informative variables for clustering high-dimensional data.

## Usage

 1 2 3 4 apL1(tuning, K = NULL, lambda = NULL, y, N = 100, kms.iter = 100, kms.nstart = 100, adapt.kms = FALSE, eps.diff = 1e-5, eps.em = 1e-5, model.crit = 'gic') apL1(tuning = NULL, K, lambda, y, N = 100, kms.iter = 100, kms.nstart = 100, adapt.kms = FALSE, eps.diff = 1e-5, eps.em = 1e-5, model.crit = 'gic') 

## Arguments

 tuning A 2-dimensional vector or a matrix with 2 columns, the first column is the number of clusters K and the second column is the tuning parameter λ in the penalty term. If this is missing, then K and lambda must be provided. K The number of clusters K. lambda The tuning parameter λ in the penalty term. y A p-dimensional data matrix. Each row is an observation. N The maximum number of iterations in the EM algorithm. The default value is 100. kms.iter The maximum number of iterations in kmeans algorithm for generating the starting value for the EM algorithm. kms.nstart The number of starting values in K-means. adapt.kms A indicator of using the cluster means estimated by K-means to calculate the adaptive parameters in APFP. The default value is FALSE. eps.diff The lower bound of pairwise difference of two mean values. Any value lower than it is treated as 0. eps.em The lower bound for the stopping criterion. model.crit The criterion used to select the number of clusters K. It is either ‘bic’ for Bayesian Information Criterion or ‘gic’ for Generalized Information Criterion.

## Details

A variable is defined as globally informative if there exists at least one pair of clusters such that μ_{kj} \neq μ_{k'j}. Here we assume that each cluster has the same diagonal variance in the model-based clustering. APL1 is in the following form,

∑_{j=1}^d τ_{kj}∑_{k=1}^K |μ_{kj}|,

where d is the number of variables in the data, K is the number of clusters, τ_{kj} = \tilde{μ}_{kj} is the adaptive parameters. Here we provide two choices for τ_{kj}. If adapt.kms == TRUE, \tilde{μ}_{kj} is the estimates from the K-mean algorithm; otherwise, \tilde{μ}_{kj} is the estimates from the model-based clustering without penalty.

The EM algorithm is used for estimating parameters. Since the EM algorithm depends on the starting values. We use the estimates from K-means with multiple starting points as the starting values.

## Value

This function returns the esimated parameters and some statistics of the optimal model within the given K and λ, which is selected by BIC when model.crit = 'bic' or GIC when model.crit = 'gic'.

 mu.hat.best The estimated cluster means in the optimal model sigma.hat.best The estimated covariance in the optimal model p.hat.best The estimated cluster proportions in the optimal model s.hat.best The clustering assignments using the optimal model lambda.best The value of λ that provide the optimal model K.best The value of K that provide the optimal model llh.best The log-likelihood of the optimal model gic.best The GIC of the optimal model bic.best The BIC of the optimal model ct.mu.best The degrees of freedom in the cluster means of the optimal model

## References

Pan, W. and Shen, X. (2007). Penalized model-based clustering with application to variable selection. The Journal of Machine Learning Research 8, 1145–1164.

nopenalty apfp parse
 1 2 3 y <- rbind(matrix(rnorm(100,0,1),ncol=2), matrix(rnorm(100,4,1), ncol=2)) output <- apL1(K = c(1:2), lambda = c(0,0.1), y=y) output\$mu.hat.best