Bootstrap Likelihood Ratio Test for the Number of Mixture Components

Perform the likelihood ratio test (LRT) for assessing the number of mixture components in a specific finite mixture model parameterisation. The observed significance is approximated by using the (parametric) bootstrap for the likelihood ratio test statistic (LRTS).

Usage

mclustBootstrapLRT(data, modelName = NULL, nboot = 999, level = 0.05, maxG = NULL, 
                   verbose = interactive(), ...)
                   
# S3 method for mclustBootstrapLRT
print(x, ...)

# S3 method for mclustBootstrapLRT
plot(x, G = 1, hist.col = "grey", hist.border = "lightgrey", breaks = "Scott", 
    col = "forestgreen", lwd = 2, lty = 3, main = NULL, ...)

Arguments

data: A numeric vector, matrix, or data frame of observations. Categorical variables are not allowed. If a matrix or data frame, rows correspond to observations and columns correspond to variables.
modelName: A character string indicating the mixture model to be fitted. The help file for mclustModelNames describes the available models.
nboot: The number of bootstrap replications to use (by default 999).
level: The significance level to be used to terminate the sequential bootstrap procedure.
maxG: The maximum number of mixture components $G$ to test. If not provided the procedure is stopped when a test is not significant at the specified level.
verbose: A logical controlling if a text progress bar is displayed during the bootstrap procedure. By default is TRUE if the session is interactive, and FALSE otherwise.
...: Further arguments passed to or from other methods. In particular, see the optional arguments in mclustBIC.
x: An 'mclustBootstrapLRT' object.
G: A value specifying the number of components for which to plot the bootstrap distribution.
hist.col: The colour to be used to fill the bars of the histogram.
hist.border: The color of the border around the bars of the histogram.
breaks: See the argument in function hist.
col, lwd, lty: The color, line width and line type to be used to represent the observed LRT statistic.
main: The title for the graph.

Details

The implemented algorithm for computing the LRT observed significance using the bootstrap is the following. Let $G_0$ be the number of mixture components under the null hypothesis versus $G_1 = G_0+1$ under the alternative. Bootstrap samples are drawn by simulating data under the null hypothesis. Then, the p-value may be approximated using eq. (13) on McLachlan and Rathnayake (2014). Equivalently, using the notation of Davison and Hinkley (1997) it may be computed as $$\textnormal{p-value} = \frac{1 + \#\{LRT^*_b \ge LRTS_{obs}\}}{B+1}$$ where
$B$ = number of bootstrap samples
$LRT_{obs}$ = LRTS computed on the observed data
$LRT^*_b$ = LRTS computed on the $b$th bootstrap sample.

Value

An object of class 'mclustBootstrapLRT' with the following components:

G: A vector of number of components tested under the null hypothesis.
modelName: A character string specifying the mixture model as provided in the function call (see above).
obs: The observed values of the LRTS.
boot: A matrix of dimension nboot x the number of components tested containing the bootstrap values of LRTS.
p.value: A vector of p-values.

References

Davison, A. and Hinkley, D. (1997) Bootstrap Methods and Their Applications. Cambridge University Press.

McLachlan G.J. (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied Statistics, 36, 318-324.

McLachlan, G.J. and Peel, D. (2000) Finite Mixture Models. Wiley.

McLachlan, G.J. and Rathnayake, S. (2014) On the number of components in a Gaussian mixture model. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 4(5), pp. 341-355.

Examples

# \donttest{
data(faithful)
faithful.boot = mclustBootstrapLRT(faithful, model = "VVV")
faithful.boot
#> ------------------------------------------------------------- 
#> Bootstrap sequential LRT for the number of mixture components 
#> ------------------------------------------------------------- 
#> Model        = VVV 
#> Replications = 999 
#>                LRTS bootstrap p-value
#> 1 vs 2   319.065354             0.001
#> 2 vs 3     6.130516             0.560
plot(faithful.boot, G = 1)

plot(faithful.boot, G = 2)

# }