The notion of infinite order Markov process is introduced and the Markov property of the flow of information is established.

In this contribution three examples of techniques that can be used for state order estimation of hidden Markov models are given. The methods are also exem-pliied using real laser range data, and the computational burden of the three methods is discussed. Two techniques, Maximum Description Length and Maximum a Posteriori Estimate, are shown to be very similar under certain circumstances. The th...

We describe estimators χ n (X 0 , X 1 ,. .. , X n), which when applied to an unknown stationary process taking values from a countable alphabet X , converge almost surely to k in case the process is a k-th order Markov chain and to infinity otherwise.

We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of...

AbstructWe consider the problem of estimating the order of a stationary ergodic Markov chain. Our focus is on estimators which satisfy a generalized Neyman-Pearson criterion of optimality. Specifically, the optimal estimator minimizes the probability of underestimation among all estimators with probability of overestimation not exceeding a given value. Our main result identifies the best expone...

We introduce the Conditional Mutual Information (CMI) for the estimation of the Markov chain order. For a Markov chain of K symbols, we define CMI of order m, Ic(m), as the mutual information of two variables in the chain being m time steps apart, conditioning on the intermediate variables of the chain. We find approximate analytic significance limits based on the estimation bias of CMI and dev...

We consider the estimation of the number of hidden states (the order) of a discrete-time finite-alphabet hidden Markov model (HMM). The estimators we investigate are related to code-based order estimators: penalized maximum-likelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan. We prove strong consistency of those estimators without assuming an...

We develop two models for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both are based on the mixture transition distribution, which constructs a probability tensor with additive mixing of probabilities from first-order matrices. demonstrate uses proposed models: parsimonious approximation high-order dynamics by lower-order models, order/lag through over-specifi...

The energy produced by a wind farm in given location and its associated income depends both on the characteristics that location—i.e., speed direction—and dynamics of electricity spot price. Because evidence cross-correlations between speed, direction price series their lagged series, we aim to assess hypothetical located central Italy when all interactions are considered. To model these cross ...