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k-MLE, k-Bregman, k-VARs: Theory, Convergence, Computation

Zuogong Yue, Victor Solo

Abstract

We develop hard clustering based on likelihood rather than distance and prove convergence. We also provide simulations and real data examples.

k-MLE, k-Bregman, k-VARs: Theory, Convergence, Computation

Abstract

We develop hard clustering based on likelihood rather than distance and prove convergence. We also provide simulations and real data examples.
Paper Structure (33 sections, 13 theorems, 41 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 33 sections, 13 theorems, 41 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. k-MLE: A General Tool for Clustering
  3. Background
  4. k-MLE Formulation
  5. k-Bregman
  6. Applications
  7. k-MLE Convergence
  8. Convergence to Partial Maxima
  9. Convergence to Local Maxima
  10. Convergence under MLE Uniqueness
  11. Convergence of k-Bregman
  12. Clustering Autocorrelated Time Series with k-VARs
  13. k-VARs Derivation
  14. k-VARs Computation
  15. Initialisation
  16. ...and 18 more sections

Key Result

Lemma 1

$F(\tau)$ is convex.

Figures (4)

  • Figure 1: Comparison of clustering performance of k-VARs with state-of-the-art methods.'k-VARs (rnd)' initializes each cluster with a randomly chosen time series. 'k-VARs (oracle) initialises each cluster with a randomly chosen time series from the true cluster.
  • Figure 2: Log-scale heatmap of BIC scores as a function of number of clusters $K$ and model order $p$. The minimum is 13.15 (surrounded by red lines), attained at $K = 10$ and $p = 4$ (the ground truth is filled in red).
  • Figure 3: Effect of t-distributed driving noise, on clustering performance of k-VARs compared to state-of-the-art methods.
  • Figure 4: Effect of SNR on clustering performance of k-VARs compared to state-of-the-art methods.

Theorems & Definitions (30)

  • Definition 1: k-MLE problem
  • Definition 2: k-MLE algorithm
  • Definition 3: analog constraint set
  • Definition 4: concentrated log-likelihood
  • Lemma 1: Selim1984
  • proof
  • Lemma 2: Theorem 2Selim1984
  • proof
  • Definition 5: k-MLE-c problem
  • Proposition 3: Selim1984
  • ...and 20 more