Martin A Tanner | Search PubMed

PubMed2005

Mixtures-of-experts of autoregressive time series: asymptotic normality and model specification.

We consider a class of nonlinear models based on mixtures of local autoregressive time series. At any given time point, we have a certain number of linear models, denoted as experts, where the vector of covariates may include lags of the dependent variable. Additionally, we assume the existence of a latent multinomial variable, whose distribution depends on the same covariates as the experts, that determines which linear process is observed. This structure, denoted as mixture-of-experts (ME), is considerably flexible in modeling the conditional mean function, as shown by Jiang and Tanner. In this paper, we present a formal treatment of conditions to guarantee the asymptotic normality of the maximum likelihood estimator (MLE), under stationarity and nonstationarity, and under correct model specification and model misspecification. The performance of common model selection criteria in selecting the number of experts is explored via Monte Carlo simulations. Finally, we present applications to simulated and real data sets, to illustrate the ability of the proposed structure to model not only the conditional mean, but also the whole conditional density.

Algorithms↗

PubMed2002

Factorial hidden Markov models and the generalized backfitting algorithm.

Previous researchers developed new learning architectures for sequential data by extending conventional hidden Markov models through the use of distributed state representations. Although exact inference and parameter estimation in these architectures is computationally intractable, Ghahramani and Jordan (1997) showed that approximate inference and parameter estimation in one such architecture, factorial hidden Markov models (FHMMs), is feasible in certain circumstances. However, the learning algorithm proposed by these investigators, based on variational techniques, is difficult to understand and implement and is limited to the study of real-valued data sets. This chapter proposes an alternative method for approximate inference and parameter estimation in FHMMs based on the perspective that FHMMs are a generalization of a well-known class of statistical models known as generalized additive models (GAMs; Hastie & Tibshirani, 1990). Using existing statistical techniques for GAMs as a guide, we have developed the generalized backfitting algorithm. This algorithm computes customized error signals for each hidden Markov chain of an FHMM and then trains each chain one at a time using conventional techniques from the hidden Markov models literature. Relative to previous perspectives on FHMMs, we believe that the viewpoint taken here has a number of advantages. First, it places FHMMs on firm statistical foundations by relating them to a class of models that are well studied in the statistics community, yet it generalizes this class of models in an interesting way. Second, it leads to an understanding of how FHMMs can be applied to many different types of time-series data, including Bernoulli and multinomial data, not just data that are real valued. Finally, it leads to an effective learning procedure for FHMMs that is easier to understand and easier to implement than existing learning procedures. Simulation results suggest that FHMMs trained with the generalized backfitting algorithm are a practical and powerful tool for analyzing sequential data.

Algorithms↗