normalmixMMlc: EC-MM Algorithm for Mixtures of Univariate Normals with linear constraints

Description

Return EC-MM (see below) algorithm output for mixtures of normal distributions with linear constraints on the means and variances parameters, as in Chauveau and Hunter (2013). The linear constraint for the means is of the form \(\mu = M \beta + C\), where \(M\) and \(C\) are matrix and vector specified as parameters. The linear constraints for the variances are actually specified on the inverse variances, by \(\pi = A \gamma\), where \(\pi\) is the vector of inverse variances, and \(A\) is a matrix specified as a parameter (see below).

Usage

normalmixMMlc(x, lambda = NULL, mu = NULL, sigma = NULL, k = 2,
              mean.constr = NULL, mean.lincstr = NULL, 
              mean.constant = NULL, var.lincstr = NULL, 
              gparam = NULL, epsilon = 1e-08, maxit = 1000, 
              maxrestarts=20, verb = FALSE)

Value

normalmixMMlc returns a list of class mixEM with items:

x: The raw data.
lambda: The final mixing proportions.
mu: The final mean parameters.
sigma: The final standard deviation(s)
scale: Scale factor for the component standard deviations, if applicable.
loglik: The final log-likelihood.
posterior: An nxk matrix of posterior probabilities for observations.
all.loglik: A vector of each iteration's log-likelihood. This vector includes both the initial and the final values; thus, the number of iterations is one less than its length.
restarts: The number of times the algorithm restarted due to unacceptable choice of initial values.
beta: The final \(\beta\) parameter estimate.
gamma: The final \(\gamma\) parameter estimate.
ft: A character vector giving the name of the function.

Arguments

x: A vector of length n consisting of the data.
lambda: Initial value of mixing proportions. Automatically repeated as necessary to produce a vector of length k, then normalized to sum to 1. If NULL, then lambda is random from a uniform Dirichlet distribution (i.e., its entries are uniform random and then it is normalized to sum to 1).
mu: Starting value of vector of component means. If non-NULL and a vector, k is set to length(mu). If NULL, then the initial value is randomly generated from a normal distribution with center(s) determined by binning the data.
sigma: Starting value of vector of component standard deviations for algorithm. Obsolete for linear constraints on the inverse variances; use gparam instead to specify a starting value.
k: Number of components. Initial value ignored unless mu and sigma are both NULL.
mean.constr: First, simplest way to define equality constraints on the mean parameters, given as a vector of length k, as in normalmixEM. Each vector entry specifies the constraints, if any, on the corresponding mean parameter: If NA, the corresponding parameter is unconstrained. If numeric, the corresponding parameter is fixed at that value. If a character string consisting of a single character preceded by a coefficient, such as "0.5a" or "-b", all parameters using the same single character in their constraints will fix these parameters equal to the coefficient times some the same free parameter. For instance, if mean.constr = c(NA, 0, "a", "-a"), then the first mean parameter is unconstrained, the second is fixed at zero, and the third and forth are constrained to be equal and opposite in sign. Note: if there are no linear constraints for the means, it is more efficient to use directly normalmixEM.
mean.lincstr: Matrix \(M\) \((k,p)\) in the linear constraint for the means equation \(\mu = M \beta + C\), with \(p \le k\).
mean.constant: Vector of \(k\) constants \(C\) in the linear constraint for the means equation \(\mu = M \beta + C\).
var.lincstr: Matrix \(A\) \((k,q)\) in the linear constraint for the inverse variances equation \(\pi = A \gamma\), with \(q \le k\).
gparam: Vector of \(q\) starting values for the \(\gamma\) parameter in the linear constraint for the inverse variances; see var.lincstr. If NULL, a vector of randomly generated standard exponential variables is used.
epsilon: The convergence criterion. Convergence is declared when the change in the observed data log-likelihood increases by less than epsilon.
maxit: The maximum allowed number of iterations.
maxrestarts: The maximum number of restarts allowed in case of a problem with the particular starting values chosen due to one of the variance estimates getting too small (each restart uses randomly chosen starting values). It is well-known that when each component of a normal mixture may have its own mean and variance, the likelihood has no maximizer; in such cases, we hope to find a "nice" local maximum with this algorithm instead, but occasionally the algorithm finds a "not nice" solution and one of the variances goes to zero, driving the likelihood to infinity.
verb: If TRUE, then various updates are printed during each iteration of the algorithm.

Author

Didier Chauveau

Details

This is a specific "EC-MM" algorithm for normal mixtures with linear constraints on the means and variances parameters. EC-MM here means that this algorithm is similar to an ECM algorithm as in Meng and Rubin (1993), except that it uses conditional MM (Minorization-Maximization)-steps instead of simple M-steps. Conditional means that it alternates between maximizing with respect to the mu and lambda while holding sigma fixed, and maximizing with respect to sigma and lambda while holding mu fixed. This ECM generalization of EM is forced in the case of linear constraints because there is no closed-form EM algorithm.

References

McLachlan, G. J. and Peel, D. (2000) Finite Mixture Models, John Wiley & Sons, Inc.
Meng, X.-L. and Rubin, D. B. (1993) Maximum Likelihood Estimation Via the ECM Algorithm: A General Framework, Biometrika 80(2): 267-278.
Chauveau, D. and Hunter, D.R. (2013) ECM and MM algorithms for mixtures with constrained parameters, preprint https://hal.archives-ouvertes.fr/hal-00625285.
Thomas, H., Lohaus, A., and Domsch, H. (2011) Stable Unstable Reliability Theory, British Journal of Mathematical and Statistical Psychology 65(2): 201-221.

Examples

Run this code

## Analyzing synthetic data as in the tau equivalent model  
## From Thomas et al (2011), see also Chauveau and Hunter (2013)
## a 3-component mixture of normals with linear constraints.
lbd <- c(0.6,0.3,0.1); m <- length(lbd)
sigma <- sig0 <- sqrt(c(1,9,9))
# means constaints mu = M beta
M <- matrix(c(1,1,1,0,-1,1), 3, 2)
beta <- c(1,5) # unknown constrained mean
mu0 <- mu <- as.vector(M %*% beta)
# linear constraint on the inverse variances pi = A.g
A <- matrix(c(1,1,1,0,1,0), m, 2, byrow=TRUE)
iv0 <- 1/(sig0^2)
g0 <- c(iv0[2],iv0[1] - iv0[2]) # gamma^0 init 

# simulation and EM fits
set.seed(50); n=100; x <- rnormmix(n,lbd,mu,sigma)
s <- normalmixEM(x,mu=mu0,sigma=sig0,maxit=2000) # plain EM
# EM with var and mean linear constraints
sc <- normalmixMMlc(x, lambda=lbd, mu=mu0, sigma=sig0,
					mean.lincstr=M, var.lincstr=A, gparam=g0)
# plot and compare both estimates
dnormmixt <- function(t, lam, mu, sig){
	m <- length(lam); f <- 0
	for (j in 1:m) f <- f + lam[j]*dnorm(t,mean=mu[j],sd=sig[j])
	f}
t <- seq(min(x)-2, max(x)+2, len=200)
hist(x, freq=FALSE, col="lightgrey", 
		ylim=c(0,0.3), ylab="density",main="")
lines(t, dnormmixt(t, lbd, mu, sigma), col="darkgrey", lwd=2) # true
lines(t, dnormmixt(t, s$lambda, s$mu, s$sigma), lty=2) 
lines(t, dnormmixt(t, sc$lambda, sc$mu, sc$sigma), col=1, lty=3)
legend("topleft", c("true","plain EM","constr EM"), 
	col=c("darkgrey",1,1), lty=c(1,2,3), lwd=c(2,1,1))

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