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).
mclustBootstrapLRT(data, modelName = NULL, nboot = 999, level = 0.05, maxG = NULL,
verbose = interactive(), ...)
## S3 method for class 'mclustBootstrapLRT'
print(x, ...)
## S3 method for class 'mclustBootstrapLRT'
plot(x, G = 1, hist.col = "grey", hist.border = "lightgrey", breaks = "Scott",
col = "forestgreen", lwd = 2, lty = 3, main = NULL, ...)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 |
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 |
verbose |
A logical controlling if a text progress bar is displayed during the bootstrap procedure. By default is |
... |
Further arguments passed to or from other methods. In particular, see the optional arguments in |
x |
An |
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 |
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. |
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
p-value = (1 + #{LRTS*_b ≥ LRT_obs}) / (B+1)
where
B = number of bootstrap samples
LRT_obs = LRTS computed on the observed data
LRT*_b = LRTS computed on the bth bootstrap sample.
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 |
p.value |
A vector of p-values. |
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.
data(faithful) faithful.boot = mclustBootstrapLRT(faithful, model = "VVV") faithful.boot plot(faithful.boot, G = 1) plot(faithful.boot, G = 2)
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