mixmod_em: Mixture Model Estimation using EM-Algorithm
In weibulltools: Statistical Methods for Life Data Analysis

Description Usage Arguments Details Value References Examples

This method uses the EM-Algorithm to estimate the parameters of a univariate mixture model. Until now, the mixture model can consist of k two-parametric Weibull distributions. If no mixture of k components can be estimated, the function is forced to stop and a message with instructions is given.

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mixmod_em(x, event, post = NULL, distribution = "weibull",
  conf_level = 0.95, k = 2, method = "EM", n_iter = 100L,
  conv_limit = 1e-06, diff_loglik = 0.5)

`x`	a numeric vector which consists of lifetime data. Lifetime data could be every characteristic influencing the reliability of a product, e.g. operating time (days/months in service), mileage (km, miles), load cycles.
`event`	a vector of binary data (0 or 1) indicating whether unit i is a right censored observation (= 0) or a failure (= 1).
`post`	a numeric matrix specifiying initial a-posteriori probabilities. If post is `NULL` (default) a-posteriori probabilities are assigned randomly using the Dirichlet distribution (`rdirichlet` from LearnBayes Package), which is the conjugate prior of a Multinomial distribution. This idea was taken from the blog post of Mr. Gelissen (linked under references).
`distribution`	supposed mixture model. Only `"weibull"` can be used. Other distributions have not been implemented yet.
`conf_level`	confidence level for the confidence intervals of the parameters of every component `k`. The default value is `conf_level = 0.95`.
`k`	integer of mixture components, default is 2.
`method`	default method is `"EM"`. Other methods have not been implemented yet.
`n_iter`	integer defining the maximum number of iterations.
`conv_limit`	numeric value defining the convergence limit.
`diff_loglik`	numeric value defining the maximum difference between log-likelihood values, which seems permissible. The default value is `0.5`. See Details for the usage of this argument.

In mixmod_em the function mixture_em_cpp is called. The computed posterior probabilities are then used as weights inside function ml_estimation to model a weighted log-likelihood. This strategy enables the computation of confidence intervals for the parameters of the separated sub-distributions, since ml_estimation provides a variance-covariance matrix. Using this strategy, a potential problem that can occur is, that the value of the complete log-likelihood, computed by mixture_em_cpp, differs considerably from the complete log-likelihood after re-estimating parameters with ml_estimation. If so, the estimated quantities like prior and posterior probabilities, as well as the model parameters are not reliable anymore and the function is forced to stop with the message: "Parameter estimation was not successful!" But if the log-likelihood values are close to each other, the presence of the mixture is strengthened and a reasonable fit is provided. Thus, a check of the absolute differences in the log-likelihood values is made and the critical difference has to be specified in argument diff_loglik.

Returns a list where the length of the list depends on the number of k subgroups. The first k lists have the same information as provided by ml_estimation, but the values logL, aic and bic are the results of a log-likelihood function, which is weighted by a-posteriori probabilities. The last list summarizes further results of the EM-Algorithm and is therefore called em_results. It contains the following elements:

a_priori : A vector with estimated a-priori probabilities.
a_posteriori : A matrix with estimated a-posteriori probabilities.
groups : Numeric vector specifying the group membership of every observation.
logL : The value of the complete log-likelihood.
aic : Akaike Information Criterion.
bic : Bayesian Information Criterion.

Doganaksoy, N.; Hahn, G.; Meeker, W. Q., Reliability Analysis by Failure Mode, Quality Progress, 35(6), 47-52, 2002
Blog posts by Stefan Gelissen: http://blogs2.datall-analyse.nl/2016/02/18/rcode_mixture_distribution_censored; last access on 19th January 2019

# Data is taken from given reference of Doganaksoy, Hahn and Meeker:
hours <- c(2, 28, 67, 119, 179, 236, 282, 317, 348, 387, 3, 31, 69, 135,
          191, 241, 284, 318, 348, 392, 5, 31, 76, 144, 203, 257, 286,
          320, 350, 412, 8, 52, 78, 157, 211, 261, 298, 327, 360, 446,
          13, 53, 104, 160, 221, 264, 303, 328, 369, 21, 64, 113, 168,
          226, 278, 314, 328, 377)
state <- c(1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1,
         1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0,
         1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1,
         0, 1, 1, 1, 1, 1, 1)

mix_mod_em <- mixmod_em(x = hours,
                        event = state,
                        distribution = "weibull",
                        conf_level = 0.95,
                        k = 2,
                        method = "EM",
                        n_iter = 150)