Optimizing the Estimation of a Histogram-Bin Width—Application to the Multivariate Mixture-Model Estimation

• Main Authors: Branislav Panić, Jernej Klemenc, Marko Nagode
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Mathematics
• Subjects: histogram; integer optimization; parameter estimation; EM; REBMIX; mixture model
• Online Access: https://www.mdpi.com/2227-7390/8/7/1090

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 <b>rebmix</b> R package.