Initialization of Hidden Markov and Semi‐Markov Models: A Critical Evaluation of Several Strategies

Abstract

The expectation-maximization (EM) algorithm is a familiar tool for computing the maximum likelihood estimate of the parameters in hidden Markov and semi-Markov models. This paper carries out a detailed study on the influence that the initial values of the parameters impose on the results produced by the algorithm. We compare random starts and partitional and model-based strategies for choosing the initial values for the EM algorithm in the case of multivariate Gaussian emission distributions (EDs) and assess the performance of each strategy with different assessment criteria. Several data generation settings are considered with varying number of latent states, of variables as well as of the level of fuzziness in the data, and discussion on how each factor influences the obtained results is provided. Simulation results show that different initialization strategies may lead to different log-likelihood values and, accordingly, to different estimated partitions. A clear indication of which strategies should be preferred is given. We further include two real-data examples, widely analysed in the hidden semi-Markov model literature.

Feature Selection in Jump Models

“…In simulations we find that this approach leads to significantly more accurate state estimates for both sparse 𝐾-means and the sparse jump model. It is a significant advantage of this approach to fitting jump models that it is far more robust to initialization than traditional maximum likelihood estimation of HMMs (Maruotti & Punzo, 2021). Similar to other studies, we find that 𝐾-means++ performs well in combination with repetitions (Celebi et al, 2013;Fränti & Sieranoja, 2019).…”

“…Bemporad et al (2018) proposed to fit jump models with 𝐾 states by minimizing the objective function In this article, we consider the squared Euclidean distance 𝓁(𝒚, 𝝁) = ‖𝒚 − 𝝁‖ 2 as loss function, which results in the objective function (1) for 𝜆 = 0 being the same as that for 𝐾-means clustering (Lloyd, 1982). It is not surprising that this loss function is useful for fitting jump models in light of 𝐾-means clustering being a successful initialization strategy for maximum likelihood estimation of hidden Markov and semi-Markov models (Maruotti & Punzo, 2021). We refer to the resulting model as the jump model or standard jump model.…”

A two-step estimator for generalized linear models for longitudinal data with time-varying measurement error

“…For real-data applications, well-established (random and deterministic) initialization strategies that are available for hidden Markov models should be used. A recent review on this very important aspect can be found in Maruotti and Punzo (2021). All algorithms are iterated until the change in the log-likelihood of two subsequent iterations is smaller than 10 −8 .…”

Quantile hidden semi-Markov models for multivariate time series

“…Combining the increased flexibility to capture a wide range of distributional shapes of the SDs with the well-known advantages of HMMs, HSMMs constitute a versatile framework in several spheres of application (see Guédon 2003 ; Barbu and Limnios 2009 ; Bulla et al. 2010 ; O’Connell and Højsgaard 2011 ; Yu 2015 ; Maruotti and Punzo 2021 and the references therein).…”