npMSL: Nonparametric EM-like Algorithm for Mixtures of Independent...

In mixtools: Tools for Analyzing Finite Mixture Models

Nonparametric EM-like Algorithm for Mixtures of Independent Repeated Measurements - Maximum Smoothed Likelihood version

Description

Returns nonparametric Smoothed Likelihood algorithm output (Levine et al, 2011) for mixtures of multivariate (repeated measures) data where the coordinates of a row (case) in the data matrix are assumed to be independent, conditional on the mixture component (subpopulation) from which they are drawn.

Usage

npMSL(x, mu0, blockid = 1:ncol(x), 
      bw = bw.nrd0(as.vector(as.matrix(x))), samebw = TRUE, 
      bwmethod = "S", h = bw, eps = 1e-8, 
      maxiter=500, bwiter = maxiter, nbfold = NULL,
      ngrid=200, post=NULL, verb = TRUE)

Arguments

x	An n x r matrix of data. Each of the n rows is a case, and each case has r repeated measurements. These measurements are assumed to be conditionally independent, conditional on the mixture component (subpopulation) from which the case is drawn.
mu0	Either an m x r matrix specifying the initial centers for the kmeans function, or an integer m specifying the number of initial centers, which are then choosen randomly in kmeans
blockid	A vector of length r identifying coordinates (columns of x) that are assumed to be identically distributed (i.e., in the same block). For instance, the default has all distinct elements, indicating that no two coordinates are assumed identically distributed and thus a separate set of m density estimates is produced for each column of x. On the other hand, if blockid=rep(1,ncol(x)), then the coordinates in each row are assumed conditionally i.i.d.
bw	Bandwidth for density estimation, equal to the standard deviation of the kernel density. By default, a simplistic application of the default bw.nrd0 bandwidth used by density to the entire dataset.
samebw	Logical: If TRUE, use the same bandwidth for each iteration and for each component and block. If FALSE, use a separate bandwidth for each component and block, and update this bandwidth at each iteration of the algorithm until bwiter is reached (see below). Two adaptation methods are provided, see bwmethod below.
bwmethod	Define the adaptive bandwidth strategy when samebw = FALSE, in which case the bandwidth depends on each component, block, and iteration of the algorithm. If set to "S" (the default), adaptation is done using a suitably modified bw.nrd0 method as described in Benaglia et al (2011). If set to "CV", an adaptive k-fold Cross Validation method is applied, as described in Chauveau et al (2014), where nbfold is the number of subsamples. This corresponds to a Leave-[n/nbfold]-Out CV.
h	Alternative way to specify the bandwidth, to provide backward compatibility.
eps	Tolerance limit for declaring algorithm convergence. Convergence is declared whenever the maximum change in any coordinate of the lambda vector (of mixing proportion estimates) does not exceed eps.
maxiter	The maximum number of iterations allowed, convergence may be declared before maxiter iterations (see eps above).
bwiter	The maximum number of iterations allowed for adaptive bandwidth stage, when samebw = FALSE. If set to 0, then the initial bandwidth matrix is used without adaptation.
nbfold	A parameter passed to the internal function wbs.kCV, which controls the weighted bandwidth selection by k-fold cross-validation.
ngrid	Number of points in the discretization of the intervals over which are approximated the (univariate) integrals for non linear smoothing of the log-densities, as required in the E step of the npMSL algorithm, see Levine et al (2011).
post	If non-NULL, an n x m matrix specifying the initial posterior probability vectors for each of the observations, i.e., the initial values to start the EM-like algorithm.
verb	If TRUE, print updates for every iteration of the algorithm as it runs

Value

npMSL returns a list of class npEM with the following items:

data	The raw data (an n x r matrix).
posteriors	An n x m matrix of posterior probabilities for observation.
bandwidth	If samebw==TRUE, same as the bw input argument; otherwise, value of bw matrix at final iteration. This information is needed by any method that produces density estimates from the output.
blockid	Same as the blockid input argument, but recoded to have positive integer values. Also needed by any method that produces density estimates from the output.
lambda	The sequence of mixing proportions over iterations.
lambdahat	The final mixing proportions.
loglik	The sequence of log-likelihoods over iterations.
f	An array of size ngrid x m x l, returning last values of density for component j and block k over grid points.
meanNaN	Average number of NaN that occured over iterations (for internal testing and control purpose).
meanUdfl	Average number of "underflow" that occured over iterations (for internal testing and control purpose).

References

• Benaglia, T., Chauveau, D., and Hunter, D. R. (2009), An EM-like algorithm for semi- and non-parametric estimation in multivariate mixtures, Journal of Computational and Graphical Statistics, 18, 505-526.

• Benaglia, T., Chauveau, D. and Hunter, D.R. (2011), Bandwidth Selection in an EM-like algorithm for nonparametric multivariate mixtures. Nonparametric Statistics and Mixture Models: A Festschrift in Honor of Thomas P. Hettmansperger. World Scientific Publishing Co., pages 15-27.

• Chauveau D., Hunter D. R. and Levine M. (2014), Semi-Parametric Estimation for Conditional Independence Multivariate Finite Mixture Models. Preprint (under revision).

• Levine, M., Hunter, D. and Chauveau, D. (2011), Maximum Smoothed Likelihood for Multivariate Mixtures, Biometrika 98(2): 403-416.

See Also

npEM, plot.npEM, normmixrm.sim, spEMsymloc, spEM, plotseq.npEM

Examples

## Examine and plot water-level task data set.
## Block structure pairing clock angles that are directly opposite one
## another (1:00 with 7:00, 2:00 with 8:00, etc.)
set.seed(111) # Ensure that results are exactly reproducible
data(Waterdata)
blockid <- c(4,3,2,1,3,4,1,2) # see Benaglia et al (2009a)

## Not run: 
a <- npEM(Waterdata[,3:10], mu0=3, blockid=blockid, bw=4)  # npEM solution
b <- npMSL(Waterdata[,3:10], mu0=3, blockid=blockid, bw=4) # smoothed version

# Comparisons on the 4 default plots, one for each block
par(mfrow=c(2,2))
for (l in 1:4){
plot(a, blocks=l, breaks=5*(0:37)-92.5,
	xlim=c(-90,90), xaxt="n",ylim=c(0,.035), xlab="")
plot(b, blocks=l, hist=FALSE, newplot=FALSE, addlegend=FALSE, lty=2,
	dens.col=1)
axis(1, at=30*(1:7)-120, cex.axis=1)
legend("topleft",c("npMSL"),lty=2, lwd=2)}

## End(Not run)

mixtools documentation built on Dec. 5, 2022, 5:23 p.m.