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Cluster weighted models with multivariate skewed distributions for functional data

Cristina Anton, Roy Shivam Ram Shreshtth

Abstract

We propose a clustering method, funWeightClustSkew, based on mixtures of functional linear regression models and three skewed multivariate distributions: the variance-gamma distribution, the skew-t distribution, and the normal-inverse Gaussian distribution. Our approach follows the framework of the functional high dimensional data clustering (funHDDC) method, and we extend to functional data the cluster weighted models based on skewed distributions used for finite dimensional multivariate data. We consider several parsimonious models, and to estimate the parameters we construct an expectation maximization (EM) algorithm. We illustrate the performance of funWeightClustSkew for simulated data and for the Air Quality dataset.

Cluster weighted models with multivariate skewed distributions for functional data

Abstract

We propose a clustering method, funWeightClustSkew, based on mixtures of functional linear regression models and three skewed multivariate distributions: the variance-gamma distribution, the skew-t distribution, and the normal-inverse Gaussian distribution. Our approach follows the framework of the functional high dimensional data clustering (funHDDC) method, and we extend to functional data the cluster weighted models based on skewed distributions used for finite dimensional multivariate data. We consider several parsimonious models, and to estimate the parameters we construct an expectation maximization (EM) algorithm. We illustrate the performance of funWeightClustSkew for simulated data and for the Air Quality dataset.

Paper Structure

This paper contains 17 sections, 3 theorems, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Multivariate skewed distributions
  3. Multivariate functional cluster weighted model
  4. The linear functional regression latent mixture model
  5. Parameter inference for the FLM[$a_{kj},b_k,{\mathbfit{Q}}_k,d_k$] - VVV model
  6. The E-step
  7. The M-step
  8. Initialization and computational details
  9. Numerical Experiments
  10. Simulated data
  11. The daily air quality dataset
  12. Conclusions and future work
  13. Proposition \ref{['proplik']}
  14. Proof of Proposition \ref{['propEM']}
  15. Proof of Proposition \ref{['propM']}
  16. ...and 2 more sections

Key Result

Proposition 1

Let us denote We have

Figures (6)

  • Figure 1: Smooth data from NIG-VG distributions colored by group for one simulation.
  • Figure 2: Smooth data from NIG-NIG distributions colored by group for one simulation.
  • Figure 3: Smooth data from ST-ST distributions colored by group for one simulation.
  • Figure 4: Smooth data simulated from VG-VG distributions colored by group for one simulation.
  • Figure 5: The 355 daily curves for each of the six predictors in the Air Quality dataset colored by group and the group estimated means.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • proof
  • proof