Conditional mixture modeling and model-based clustering

Description

The utility of this package includes fitting a conditional mixture model with EM (Expectation Maximization) algorithm, model-based clustering based on conditional mixture modeling, conditional mixture modeling with parsimonious procedures, and optimal conditional order exploration by using either a full search or the proposed searching algorithm, and illustration of clustering results through pairwise plots.

Details

Function 'cmb.em' runs the parsimonious conditional mixture modeling for a user-specified conditioning order.

Function 'cmb.search' runs the 'cmb.em' procedure for all possible conditioning orders and determines the optimal order using BIC, or runs the proposed optimal order search algorithm and then the 'cmb.em' for the obtained optimal order.

Function 'cmb.plot' builds pairwise plots to present clustering results from functions 'cmb.em' and 'cmb.search".

Author(s)

Yang Wang and Volodymyr Melnykov.

Maintainer: Yang Wang <wangy4@cofc.edu>

References

Melnykov, V., and Wang, Y. (2023). Conditional mixture modeling and model-based clustering. Pattern Recognition, 133, p. 108994.

Australian Institute of Sport data

Description

The data set considers body characteristics from 102 male and 100 female athletes at the Australian Institute of Sport. It is collected for a study of how data on various features varied with sport body size and sex of athlete.

Usage

data(ais)

Format

A data frame with 202 observations on the following 13 variables.

sex

Factor with levels: female, male;

sport

Factor with levels: B_Ball, Field, Gym, Netball, Row Swim, T_400m, Tennis, T_Sprnt, W_Polo;

RCC

Red cell count;

WCC

White cell count;

Hc

Hematocrit;

Hg

Hemoglobin;

Fe

Plasma ferritin concentration;

BMI

Body Mass Index;

SSF

Sum of skin folds;

Bfat

Body fat percentage;

LBM

Lean body mass;

Ht

Height, cm;

Wt

Weight, kg

Details

The data have been made publicly available in connection with the book by Cook, R.D. and Weisberg, S. (1994, ISBN-10:0471008397).

References

Cook, R.D. and Weisberg, S. (1994). An introduction to regression graphics. John Wiley & Sons.

Conditional mixture modeling by EM algorithm

Description

Runs conditional mixture modeling and model-based clustering by EM algorithm (Expectation Maximization) for a prespecified variables conditioning order. Runs variable selection procedure (forward, backward or stepwise) to achieve a parsimonious mixture model.

Usage

cmb.em(x, order = NULL, l, K, method = "stepwise", id0 = NULL, n.em = 200, em.iter = 5,
EM.iter = 200, nk.min = NULL, max.spur=5, tol = 1e-06, silent = FALSE, Parallel = FALSE,
n.cores = 4)

Arguments

Details

In conditional mixture modeling, each component is modeled by a product of conditional distributions with the means expressed by polynomial regression functions depending on other variables. Polynomial regression function order l and the number of clusters K are prespecified by user. The model's initialization can be determined by passing a group membership vector to the argument id, or obtained by the emEM algorithm (the default setting) in the function. There are two arguments related to the emEM procedure, the number of short EM n.em and maximum number of iterations for short EM em.iter. By default, the n.em = 200 and em.iter = 5. The method of variable selection can be specified as method = "stepwise", "forward", "backward", or "none" where method = none means no parsimonious procedure conducted. During the model fitting and variable selection phases, EM algorithm will be applied multiple times, where options EM.iter and tol are stopping criteria of EM iteration. The spurious output control argument nk.min, by default nk.min = (l x (p - 1) + 1) x 2, can be set by user. When spurious output is obtained, cmb.em will be rerun. The maximum number of rerunning is max.spur.

Notation: n - sample size, l - order of polynomial regression model, K - number of mixture components.

Value

References

Biernacki C., Celeux G., Govaert G. (2003). Choosing Starting Values for the EM Algorithm for Getting the Highest Likelihood in Multivariate Gaussian Mixture Models. Computational Statistics and Data Analysis, 41(3-4), pp. 561-575.

Examples

set.seed(1)
K <- 3
l <- 2
x <- as.matrix(iris[,-5])
id.true <- iris[,5]

# Run EM algorithm for fitting a conditioning mixture model 
obj <- cmb.em(x = x, order = c(1,3,2,4), l, K, method = "stepwise", silent = FALSE,
Parallel = FALSE)
id.cmb <- obj$id
table(id.true, id.cmb)
obj$BIC

Graphic display for the results of conditional mixture modeling and model-based clustering

Description

cmb.plot demonstrates the clustering results of functions cmb.em and cmb.search. A graph with a combination of pairwise scatter plot for data points, pairwise contour plot of estimated mixture density, and pairwise regression curves is produced.

Usage

cmb.plot(obj, allcolors = NULL, allpch = NULL, lwd = 1, cex.text = 1, cex.point = 0.6, 
mar = c(0.6,0.6,0.6,0.6), oma = c(3.5,3.5,2.5,14), nlevels = 30)

Arguments

Value

This function generates a graphic.

Examples

set.seed(4)
K <- 3
l <- 2
x <- as.matrix(iris[,-5])

# Run EM algorithm for fitting a conditioning mixture model 

obj <- cmb.em(x = x, order = c(1,2,3,4), l, K, method = "stepwise",
silent = TRUE, Parallel = FALSE)
cmb.plot(obj)
	

Optimal conditioning order search

Description

Runs forward, backward, or stepwise variable selection procedure for obtaining the parsimonious conditional mixture models when all conditional orders are considered. Alternatively, runs the optimal order search algorithm, and parsimonious conditional mixture modeling for the obtained order.

Usage

cmb.search(x, l, K, method = "stepwise", all.perms  = TRUE, id0 = NULL,  n.em = 200,
em.iter = 5, EM.iter = 200, nk.min = NULL, max.spur = 5, tol = 1e-06, silent = FALSE, 
Parallel = TRUE, n.cores = 4)

Arguments

Details

Functions 'cmb.search' and 'cmb.em' have common arguments except 'all.perm'. With all.perms = TRUE, a full search is applied to data, that is running parsimonious conditional mixture modeling for all orders and recognizing the optimal order based on the BIC. Then two lists are returned: best.model stores the results for the conditional mixture model with the optimal order, and models has results for all orders. With the option all.perms = FALSE, the optimal conditional order search algorithm is applied, and then only the list best.model is returned.

Value

The list models is returned when all.perms = TRUE.

See Also

cmb.em

Examples

set.seed(1)
K = 3
l <- 2
x <- as.matrix(iris[,-5])


obj <- cmb.search(x = x, l, K, method = "stepwise", all.perms = FALSE,
Parallel = FALSE, silent = FALSE)
obj$best.model$BIC

Simulated datasets

Description

Datasets are simulated from conditional mixture models with different numbers of components.

Usage

data(smltn)

Format

Two datasets are stored in the data smltn. smltn1 is a data matrix with 200 observations on two variables and one group membership; smltn2 is a matrix with 300 observations on two variables and one group ID.

Examples

data(smltn)
# view data matrices smltn1 and smltn2
print(smltn1)
print(smltn2)