npMSL: 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)

Value

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

data: The raw data (an $n \times r$ matrix).
posteriors: An $n \times m$ 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 $n g r i d \times m \times l$ , 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).

Arguments

x: An $n \times r$ 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 $m \times r$ 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- $[n / n b f o l d]$ -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 $n \times m$ 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

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.

Examples

Run this code

## 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)

if (FALSE) {
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)}
}

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