St-EM algorithm for Reliability Mixture Models (RMM) of Weibull with right Censoring

Parametric Stochastic EM (St-EM) algorithm for univariate finite mixture of Weibull distributions with randomly right censored data.

Usage

weibullRMM_SEM(x, d = NULL, lambda = NULL, shape = NULL, scale = NULL,
        k = 2, maxit = 200, maxit.survreg = 200, epsilon = 1e-03,
        averaged = TRUE, verb = FALSE)

Arguments

x: A vector of \(n\) real positive lifetime (possibly censored) durations. If d is not NULL then a vector of random censoring times c occured, so that \(x= min(x,c)\) and \(d = I(x <= c)\).
d: The vector of censoring indication, where 1 means observed lifetime data, and 0 means censored lifetime data.
lambda: Initial value of mixing proportions. If NULL, then lambda is set to rep(1/k,k).
shape: Initial value of Weibull component shapes, all set to 1 if NULL.
scale: Initial value of Weibull component scales, all set to 1 if NULL.
k: Number of components of the mixture.
maxit: The number of iterations allowed, since for St-EM algorithms convergence is not based on stabilization, exactly maxit iterations are performed (see Bordes L. and Chauveau D. (2016) reference below).
maxit.survreg: The number of iterations allowed in the computations of the MLE for censored weibull data from the survival package (see Bordes L. and Chauveau D. (2016) reference below).
epsilon: Tolerance parameter used in the numerical computations of the MLE for censored weibull data by survreg from the survival package (see Bordes L. and Chauveau D. (2016) reference below).
averaged: The way of updating parameters at each iteration: if TRUE, current values of the parameters are obtained by averaging the sequence (see Bordes L. and Chauveau D. (2016) reference below).
verb: If TRUE, print updates for every iteration of the algorithm as it runs

Details

This St-EM algorithm calls functions from the survival package to compute parametric MLE for censored weibull data.

Value

weibullRMM_SEM returns a list of class "mixEM" with the following items:

x: The input data.
d: The input censoring indicator.
lambda: The estimates for the mixing proportions.
scale: The estimates for the Weibull component scales.
shape: The estimates for the Weibull component shapes.
loglik: The log-likelihood value at convergence of the algorithm.
posterior: An \(n\times k\) matrix of posterior probabilities for observation, after convergence of the algorithm.
all.loglik: The sequence of log-likelihoods over iterations.
all.lambda: The sequence of mixing proportions over iterations.
all.scale: The sequence of component scales over iterations.
all.shape: The sequence of component shapes over iterations.
ft: A character vector giving the name of the function called.

References

Bordes, L., and Chauveau, D. (2016), Stochastic EM algorithms for parametric and semiparametric mixture models for right-censored lifetime data, Computational Statistics, Volume 31, Issue 4, pages 1513-1538. https://link.springer.com/article/10.1007/s00180-016-0661-7

Author

Didier Chauveau

Examples

n = 500 # sample size
m = 2 # nb components
lambda=c(0.4, 0.6)
shape <- c(0.5,5); scale <- c(1,20) # model parameters
set.seed(321)
x <- rweibullmix(n, lambda, shape, scale) # iid ~ weibull mixture
cs=runif(n,0,max(x)+10) # iid censoring times
t <- apply(cbind(x,cs),1,min) # censored observations
d <- 1*(x <= cs)              # censoring indicator

## set arbitrary or "reasonable" (e.g., data-driven) initial values
l0 <- rep(1/m,m); sh0 <- c(1, 2); sc0 <- c(2,10)
# Stochastic EM algorithm 
a <- weibullRMM_SEM(t, d, lambda = l0, shape = sh0, scale = sc0, maxit = 200)
#> number of iterations = 200 

summary(a) # Parameters estimates etc
#> summary of weibullRMM_SEM object:
#>          comp 1   comp 2
#> lambda 0.396100  0.60390
#> shape  0.497675  5.18371
#> scale  0.620329 20.20621
#> loglik at estimate:  -960.1892 
#> 25 % of the data right censored
plotweibullRMM(a) # plot of St-EM sequences

plot(a, which=2) # or equivalently, S3 method for "mixEM" object