Existence of infinite Viterbi path for pairwise Markov models

Lember, Jüri; Sova, Joonas

doi:10.1016/j.spa.2019.05.004

Cited by 10 publications

(21 citation statements)

References 25 publications

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“…The Viterbi algorithm is used for finding the sequence of Viterbi path‐implied states, which are most likely to produce the observed event sequence. It is one of the methods used by the HMM to estimate the hidden chain (Lember & Sova, 2020). The optimal model trained by the CS‐HMM method is used as the given model of the Viterbi algorithm and the observable sequence is used as the specified observation sequence to predict and assess the CS‐HMM model.…”

Section: Methodsmentioning

confidence: 99%

Dynamic risk assessment of food safety based on an improved hidden Markov model integrating cuckoo search algorithm: A sterilized milk study

Lin

Han

et al. 2021

J Food Process Engineering

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With the growth in the living standard of people, food safety and quality risk assessment has gradually become the focus. Therefore, this article realizes the dynamic risk assessment of food safety based on an improved hidden Markov model (HMM) integrating the cuckoo search algorithm (CS) (CS‐HMM). The CS is used to conduct global search to obtain the initial value of HMM, and then the Baum–Welch algorithm is used for modifying the initial value to obtain a trained risk assessment model. Finally, in terms of several benchmark functions, the accuracy of the CS algorithm is better than the PSO and the GA algorithms for seeking global optimal solution. Moreover, the presented method is applied to evaluate the safety risk of the sterilized milk production. Combined with the characteristics of time and temperature, the deterioration reasons of sterilized milk are analyzed. Moreover, the proposed method has certain theoretical significance and practical value for improving the ability of food quality risk assessment and formulating control measures in advance. Practical applications The presented method is applied to evaluate the safety risk of the sterilized milk production, which are supplied by the food inspection agency of Guizhou province in China. Combined with the characteristics of time and temperature, the deterioration reasons of sterilized milk are analyzed. Furthermore, the improved model breaks out of the local optimal limit, so as to better realize the risk prediction of food and obtain better evaluation results. Based on the analysis of dynamic risk assessment result with the temperature characteristics, the future risk trend can be predicted. Meanwhile, the causes affecting the deterioration of the sterilized milk and the direction to improve the quality of the sterilized milk can be obtained. In this way, risk control measures can be taken more precisely and risks can be reduced more effectively, which is helpful to the production and storage of food products.

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Section: Methodsmentioning

confidence: 99%

Dynamic risk assessment of food safety based on an improved hidden Markov model integrating cuckoo search algorithm: A sterilized milk study

Lin

Han

et al. 2021

J Food Process Engineering

View full text Add to dashboard Cite

show abstract

“…In other words, for every n big enough, there exists at least one Viterbi path so that v(x 1:n ) 1:t = v 1:t . For the definition of infinite Viterbi path in the case Y is infinite, see [6,28].…”

Section: Infinite Viterbi Pathmentioning

confidence: 99%

“…It has been recently proven [28] that under some conditions the infinite Viterbi path exists for almost every realization x 1:∞ of X, allowing to define an infinite Viterbi decoding of X, called the Viterbi process. This was done trough construction of barriers.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

“…Having infinitely many barriers is not necessary for existence of infinite Viterbi path (see [28,Example 1.2]), but the barrier-construction has several advantages. One of them is that it allows to construct the infinite path piecewise, meaning that to determine the first k elements v 1:k of the infinite path it suffices to observe x 1:n for n big enough.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

“…In Subsection 1.3, the segmentation problem, infinite Viterbi path, barriers and many other concepts are introduced and defined. In Subsection 1.4 we state the two barrier construction theorems from [28] (Theorem 1.1 concerning HMM and Theorem 1.2 concerning general PMM) as well as introduce some general Markov chain terminology. In Section 2 we prove our main regeneration-theorem.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

See 2 more Smart Citations

Regenerativity of Viterbi Process for Pairwise Markov Models

Lember¹,

Sova²

2020

J Theor Probab

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For hidden Markov models one of the most popular estimates of the hidden chain is the Viterbi path -the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions the Viterbi path of the PMM can almost surely be extended to infinity, thereby defining the infinite Viterbi decoding of the observation sequence, called the Viterbi process. This was done by constructing a block of observations, called a barrier, which ensures that the Viterbi path goes trough a given state whenever this block occurs in the observation sequence. In this paper we prove that the joint process consisting of Viterbi process and PMM is regenerative. The proof involves a delicate construction of regeneration times which coincide with the occurrences of barriers. As one possible application of our theory, some results on the asymptotics of the Viterbi training algorithm are derived.

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Pairwise Markov Models and Hybrid Segmentation Approach

Kuljus

Lember²

2023

Methodol Comput Appl Probab

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The article studies segmentation problem (also known as classification problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and underlying state sequence form a two-dimensional Markov chain, it is a natural generalization of a hidden Markov model. To demonstrate the richness of the class of PMMs, we examine closer a few examples of rather different types of PMMs: a model for two related Markov chains, a model that allows to model an inhomogeneous Markov chain as a conditional marginal process of a homogeneous PMM, and a semi-Markov model. The segmentation problem assumes that one of the marginal processes is observed and the other one is not, the problem is to estimate the unobserved state path given the observations. The standard state path estimators often used are the so-called Viterbi path (a sequence with maximum state path probability given the observations) or the pointwise maximum a posteriori (PMAP) path (a sequence that maximizes the conditional state probability for given observations pointwise). Both these estimators have their limitations, therefore we derive formulas for calculating the so-called hybrid path estimators which interpolate between the PMAP and Viterbi path. We apply the introduced algorithms to the studied models in order to demonstrate the properties of different segmentation methods, and to illustrate large variation in behaviour of different segmentation methods in different PMMs. The studied examples show that a segmentation method should always be chosen with care by taking into account the purpose of modelling and the particular model of interest.

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Existence of infinite Viterbi path for pairwise Markov models

Cited by 10 publications

References 25 publications

Dynamic risk assessment of food safety based on an improved hidden Markov model integrating cuckoo search algorithm: A sterilized milk study

Dynamic risk assessment of food safety based on an improved hidden Markov model integrating cuckoo search algorithm: A sterilized milk study

Regenerativity of Viterbi Process for Pairwise Markov Models

Pairwise Markov Models and Hybrid Segmentation Approach

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