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A Hidden Markov Model (HMM) is a couple of stochastic processes: A hidden Markov process and an observed emission process. Generally, the HMMs are used to study the hidden behavior of random systems through some observed emission sequences generated by the phenomenon under study. In this frame-work, we propose to solve the likelihood and the decoding problems of HMMs whose state space is composed of a continuous part and a discrete part. We adapt forward, backward and Viterbi algorithms to the case of our proposal. Numerical examples and Monte Carlo simulations are considered to show the efficiency and the adaptation of the algorithms for the proposed model. [ABSTRACT FROM AUTHOR]

The partitioning algorithm is an iterative procedure that computes explicitly the steady-state probability of a finite Markov chain 푋. In this paper, we propose to adapt this algorithm to the case where the state space E := C ∪ D is composed of a continuous part 퐶 and a finite part 퐷 such that C ∩ D = ∅ . In this case, the steady-state probability 휋 of 푋 is a convex combination of two steady-state probabilities π C and π D of two Markov chains on 퐶 and 퐷 respectively. The obtained algorithm allows to compute explicitly π D . If π C cannot be computed explicitly, our algorithm approximates it by numerical resolution of successive integral equations. Some numerical examples are studied to show the usefulness and proper functioning of our proposal. [ABSTRACT FROM AUTHOR]

In this work, we interest to the estimation of the steady-state probabilities of the two processes (the hidden Markov chain and the emission process) composing a standard Hidden Markov Model. Since the estimation of steady-state Markov chain is well known, more attention will be given to the estimation of the steady-state of the emission process. Among different methods, we will focus more on the technique based on the regenerative notion, often used in steady-state Markov chain simulation. Numerical Monte Carlo simulations are carried to show the usefulness of our proposal and to appreciate the quality of different proposed estimators. [ABSTRACT FROM AUTHOR]

In this work, we propose to combine the standard coupling from the past and the multi-gamma coupler algorithms to allow perfect simulation of the steady-state probability of a Markov chain (MC), whose state space E is composed of a continuous part C and a finite part D. We show that for some families of mixed transition kernels of a MC on E, the computation of the steady-state probability returns to the computation of the steady-state probabilities of two MCs: X c on C and X d on D. A basic numerical Monte Carlo example is studied to show the smooth running of the proposed hybrid algorithm. [ABSTRACT FROM AUTHOR]

This paper describes an online solution for visualizing the simulation of a discrete time Markov chain using an animated graph. Based on the D3.js library, the development of this solution offers the user the choice between a presentation based on the concept of a probabilistic graph oriented and a presentation of the transitions in a matrix framework. As an application the solution helps to describe the dynamics of individual policyholders in a Bonus-Malus system of automobile insurance. [ABSTRACT FROM AUTHOR]