[Paper Review] Order estimation for non-parametric Hidden Markov Models

Luc Lehéricy|arXiv (Cornell University)|Apr 16, 2016

Blind Source Separation Techniques14 references3 citations

TL;DR

This paper proposes two consistent methods for estimating the order of non-parametric hidden Markov models: one based on matrix rank estimation from the joint distribution of two consecutive observations, and another using a penalized least squares criterion. Both methods achieve strong consistency and an oracle inequality for parameter estimation, with numerical comparisons showing their effectiveness across various scenarios.

ABSTRACT

We propose and study two pratically tractable methods to estimate the order of non-parametric hidden Markov models in addition to their parameters. The first one relies on estimating the rank of a matrix derived from the law of two consecutive observations while the second one selects the order by minimizing a penalized least squares criterion. We show strong consistency of both methods and prove an oracle inequality on the least squares estimators of the model parameters. We numerically compare their ability to select the right order in several situations and discuss their algorithmic complexity.

Motivation & Objective

• To address the challenge of order estimation in non-parametric hidden Markov models, where the number of hidden states is unknown and must be inferred from data.
• To develop computationally tractable methods that simultaneously estimate the model order and parameters without requiring parametric assumptions on the emission distributions.
• To establish theoretical guarantees, including strong consistency and an oracle inequality, for the proposed estimators.
• To compare the empirical performance and algorithmic complexity of the two methods across diverse data-generating scenarios.

Proposed method

• The first method estimates the order by computing the rank of a matrix constructed from the joint distribution of two consecutive observations, leveraging the fact that the rank corresponds to the number of hidden states in the model.
• The second method selects the model order by minimizing a penalized least squares criterion, which balances goodness of fit and model complexity to avoid overfitting.
• Both methods are designed to be practically implementable using empirical estimates of the joint observation distribution from observed data.
• Theoretical analysis establishes strong consistency of the order estimators under mild regularity conditions.
• An oracle inequality is derived for the least squares estimators of the model parameters, showing their performance is close to that of the best possible model with known order.
• The methods are numerically evaluated on synthetic data to compare their accuracy in order selection and computational efficiency.

Experimental results

Research questions

• RQ1 Can the rank of a matrix derived from the joint distribution of two consecutive observations reliably estimate the hidden Markov model order?
• RQ2 Does minimizing a penalized least squares criterion yield a consistent estimator of the model order in non-parametric HMMs?
• RQ3 How do the two proposed methods compare in terms of accuracy and computational complexity across different model configurations?
• RQ4 What theoretical guarantees, such as consistency and oracle inequalities, can be established for the parameter and order estimators?
• RQ5 Under what conditions do the proposed methods achieve optimal performance relative to the true model order?

Key findings

• The matrix rank-based method provides a consistent estimator of the hidden Markov model order under regularity conditions on the observation distribution.
• The penalized least squares method also yields a strongly consistent order estimator, with theoretical support from an oracle inequality on the parameter estimation error.
• Numerical experiments demonstrate that both methods effectively recover the true model order across various simulation settings, with performance varying by model complexity and sample size.
• The penalized least squares approach generally shows better stability and lower variance in order selection compared to the rank-based method in finite samples.
• The algorithmic complexity of both methods is tractable, with the rank-based method being faster but potentially less robust to estimation errors in the joint distribution.
• The theoretical results confirm that the parameter estimators achieve near-optimal performance, as quantified by the oracle inequality, even when the model order is unknown and estimated.