Abstract
A maximum-likelihood estimation of a multivariate mixture model’s parameters is a difficult problem. One approach is to combine the REBMIX and EM algorithms. However, the REBMIX algorithm requires the use of histogram estimation, which is the most rudimentary approach to an empirical density estimation and has many drawbacks. Nevertheless, because of its simplicity, it is still one of the most commonly used techniques. The main problem is to estimate the optimum histogram-bin width, which is usually set by the number of non-overlapping, regularly spaced bins. For univariate problems it is usually denoted by an integer value; i.e., the number of bins. However, for multivariate problems, in order to obtain a histogram estimation, a regular grid must be formed. Thus, to obtain the optimum histogram estimation, an integer-optimization problem must be solved. The aim is therefore the estimation of optimum histogram binning, alone and in application to the mixture model parameter estimation with the REBMIX&EM strategy. As an estimator, the Knuth rule was used. For the optimization algorithm, a derivative based on the coordinate-descent optimization was composed. These proposals yielded promising results. The optimization algorithm was efficient and the results were accurate. When applied to the multivariate, Gaussian-mixture-model parameter estimation, the results were competitive. All the improvements were implemented in the rebmix R package.
Highlights
Let the random variable y ∈ Rd follow the multivariate, finite-mixture model with the probability density function c f (y|c, w, Θ) =∑ wl f l (y|Θl ). (1) l =1The main task is to estimate the c, w, Θ parameters
As the number of components c can be seen as a complexity parameter, the value of the likelihood function increases as this value increases, Mathematics 2020, 8, 1090; doi:10.3390/math8071090
For the first useful insight, we evaluated on how many datasets for each dimension d, each algorithm used was capable of finding the true binning v from which the dataset was simulated
Summary
The main task is to estimate the c, w, Θ parameters. The parameters c, w, Θ are, respectively, the number of components in the mixture model, the component weights and the component parameters. The maximum-likelihood estimation of such parameters is a difficult task and mostly requires a combination of multiple procedures for its estimation [1]. Obtaining a direct analytical derivation of the likelihood function for mixture models is difficult and inconvenient; instead, the complete-data-likelihood function can be used and the maximum-likelihood parameter estimates can be obtained using the expectation-maximization (EM) algorithm [2]. As the number of components c can be seen as a complexity parameter, the value of the likelihood function increases as this value increases, Mathematics 2020, 8, 1090; doi:10.3390/math8071090 www.mdpi.com/journal/mathematics
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
