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Multivariate Beta Mixture Model: Probabilistic Clustering With Flexible Cluster Shapes

Yung-Peng Hsu, Hung-Hsuan Chen

TL;DR

The properties of MBMM are introduced, the parameter learning procedure is described, and the experimental results are presented, showing that MBMM fits diverse cluster shapes on synthetic and real datasets.

Abstract

This paper introduces the multivariate beta mixture model (MBMM), a new probabilistic model for soft clustering. MBMM adapts to diverse cluster shapes because of the flexible probability density function of the multivariate beta distribution. We introduce the properties of MBMM, describe the parameter learning procedure, and present the experimental results, showing that MBMM fits diverse cluster shapes on synthetic and real datasets. The code is released anonymously at https://github.com/hhchen1105/mbmm/.

Multivariate Beta Mixture Model: Probabilistic Clustering With Flexible Cluster Shapes

TL;DR

The properties of MBMM are introduced, the parameter learning procedure is described, and the experimental results are presented, showing that MBMM fits diverse cluster shapes on synthetic and real datasets.

Abstract

This paper introduces the multivariate beta mixture model (MBMM), a new probabilistic model for soft clustering. MBMM adapts to diverse cluster shapes because of the flexible probability density function of the multivariate beta distribution. We introduce the properties of MBMM, describe the parameter learning procedure, and present the experimental results, showing that MBMM fits diverse cluster shapes on synthetic and real datasets. The code is released anonymously at https://github.com/hhchen1105/mbmm/.
Paper Structure (17 sections, 9 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 9 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multivariate Beta Mixture Model
  3. Multivariate beta distribution
  4. MBMM density function and generative process
  5. Parameter Learning for the MBMM
  6. The similarity score between data points
  7. Experiments
  8. Comparisons on the synthetic datasets
  9. Baseline models
  10. Synthetic dataset generation
  11. Visualizing clustering results
  12. Comparison on the real datasets
  13. Real datasets
  14. Results
  15. Distance between data points
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Examples of the versatile shape of the bivariate beta distribution. The upper row shows three bivariate beta distributions with different parameters. The bottom row shows the marginal distribution of $x_1$ (i.e., the variable on the horizontal axis in the top row). This distribution can be symmetric unimodal (e.g., the left subfigure), skewed unimodal (e.g., the middle subfigure), or bimodal (e.g., the right subfigure).
  • Figure 2: A comparison of the support of the Dirichlet distribution (left) and our multivariate beta distribution (right) with 3 variates. The Dirichlet distribution is only defined on $x_i \in (0, 1)$ such that $x_1 + x_2 + x_3 = 1$ (the standard 2-simplex in $R^3$). On the contrary, our multivariate beta distribution is defined on $(0,1)^3$ (the unit cube in $R^3$), which is a superset of the Dirichlet distribution.
  • Figure 3: Graphical representation of the multivariate beta mixture model
  • Figure 4: Clustering algorithms on synthetic datasets
  • Figure 5: Upper row: data points clustered by the MBMM; bottom row: PDFs based on the fitted parameters.
  • ...and 1 more figures