scholarly journals Bayesian inference for continuous-time hidden Markov models with an unknown number of states

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Yu Luo ◽  
David A. Stephens
Keyword(s):  
Bayesian Inference ◽  
Continuous Time ◽  
Markov Models ◽  
Real Data ◽  
Jump Process ◽  
Fixed Number ◽  
Markov Jump ◽  
Model Based Clustering ◽  
Unknown Number ◽  
Canadian Healthcare

AbstractWe consider the modeling of data generated by a latent continuous-time Markov jump process with a state space of finite but unknown dimensions. Typically in such models, the number of states has to be pre-specified, and Bayesian inference for a fixed number of states has not been studied until recently. In addition, although approaches to address the problem for discrete-time models have been developed, no method has been successfully implemented for the continuous-time case. We focus on reversible jump Markov chain Monte Carlo which allows the trans-dimensional move among different numbers of states in order to perform Bayesian inference for the unknown number of states. Specifically, we propose an efficient split-combine move which can facilitate the exploration of the parameter space, and demonstrate that it can be implemented effectively at scale. Subsequently, we extend this algorithm to the context of model-based clustering, allowing numbers of states and clusters both determined during the analysis. The model formulation, inference methodology, and associated algorithm are illustrated by simulation studies. Finally, we apply this method to real data from a Canadian healthcare system in Quebec.

The structure of extremal processes

1973 ◽  
Vol 5 (02) ◽  
pp. 287-307 ◽  
Author(s):  
Sidney I. Resnick ◽  
Michael Rubinovitch
Keyword(s):  
Continuous Time ◽  
Extreme Value Distribution ◽  
Jump Process ◽  
Record Values ◽  
Jump Processes ◽  
Markov Jump ◽  
Additive Structure ◽  
Positive Jump ◽  
Limiting Behaviour ◽  
Extremal Processes

An extremal-Fprocess {Y(t);t≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed likeFin the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration ofF.Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps ofY(t) in (t1,t2] follows a Poisson distribution with parameter logt2/t1. The time transformationt→etgives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. WhenFis an extreme value distribution theY-process has special properties. The most important is that ifF(x) = exp {—e–x} thenY(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour ofY(t) and its jump times.

The structure of extremal processes

1973 ◽  
Vol 5 (2) ◽  
pp. 287-307 ◽  
Author(s):  
Sidney I. Resnick ◽  
Michael Rubinovitch
Keyword(s):  
Continuous Time ◽  
Extreme Value Distribution ◽  
Jump Process ◽  
Record Values ◽  
Jump Processes ◽  
Markov Jump ◽  
Additive Structure ◽  
Positive Jump ◽  
Limiting Behaviour ◽  
Extremal Processes

An extremal-F process {Y (t); t ≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration of F. Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps of Y (t) in (t1, t2] follows a Poisson distribution with parameter log t2/t1. The time transformation t→ et gives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. When F is an extreme value distribution the Y-process has special properties. The most important is that if F(x) = exp {—e–x} then Y(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y(t) and its jump times.

scholarly journals Bayesian inference for Markov jump processes with informative observations

2015 ◽  
Vol 14 (2) ◽  
Author(s):  
Andrew Golightly ◽  
Darren J. Wilkinson
Keyword(s):  
Bayesian Inference ◽  
Sequential Monte Carlo ◽  
Transition Probabilities ◽  
Jump Process ◽  
Jump Processes ◽  
Markov Jump ◽  
Simulation Based ◽  
Intermediate Time ◽  
Computationally Intensive ◽  
Particle Mcmc

AbstractIn this paper we consider the problem of parameter inference for Markov jump process (MJP) representations of stochastic kinetic models. Since transition probabilities are intractable for most processes of interest yet forward simulation is straightforward, Bayesian inference typically proceeds through computationally intensive methods such as (particle) MCMC. Such methods ostensibly require the ability to simulate trajectories from the conditioned jump process. When observations are highly informative, use of the forward simulator is likely to be inefficient and may even preclude an exact (simulation based) analysis. We therefore propose three methods for improving the efficiency of simulating conditioned jump processes. A conditioned hazard is derived based on an approximation to the jump process, and used to generate end-point conditioned trajectories for use inside an importance sampling algorithm. We also adapt a recently proposed sequential Monte Carlo scheme to our problem. Essentially, trajectories are reweighted at a set of intermediate time points, with more weight assigned to trajectories that are consistent with the next observation. We consider two implementations of this approach, based on two continuous approximations of the MJP. We compare these constructs for a simple tractable jump process before using them to perform inference for a Lotka-Volterra system. The best performing construct is used to infer the parameters governing a simple model of motility regulation in

A stochastic Expectation–Maximisation (EM) algorithm for construction of mortality tables

2017 ◽  
Vol 12 (1) ◽  
pp. 1-22
Author(s):  
Luz Judith R. Esparza ◽  
Fernando Baltazar-Larios
Keyword(s):  
Real Data ◽  
Jump Process ◽  
Ageing Process ◽  
Physiological Age ◽  
Phase Type Distribution ◽  
Markov Jump ◽  
Death Time ◽  
Intensity Matrix ◽  
Expectation Maximisation ◽  
Phase Type

AbstractIn this paper, we present an extension of the model proposed by Lin & Liu that uses the concept of physiological age to model the ageing process by using phase-type distributions to calculate the probability of death. We propose a finite-state Markov jump process to model the hypothetical ageing process in which it is possible the transition rates between non-consecutive physiological ages. Since the Markov process has only a single absorbing state, the death time follows a phase-type distribution. Thus, to build a mortality table the challenge is to estimate this matrix based on the records of the ageing process. Considering the nature of the data, we consider two cases: having continuous time information of the ageing process, and the more interesting and realistic case, having reports of the process just in determined times. If the ageing process is only observed at discrete time points we have a missing data problem, thus, we use a stochastic Expectation–Maximisation (SEM) algorithm to find the maximum likelihood estimator of the intensity matrix. And in order to do that, we build Markov bridges which are sampled using the Bisection method. The theory is illustrated by a simulation study and used to fit real data.

scholarly journals Hidden Semi-Markov Models for Predictive Maintenance

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Francesco Cartella ◽  
Jan Lemeire ◽  
Luca Dimiccoli ◽  
Hichem Sahli
Keyword(s):  
Markov Models ◽  
Real Data ◽  
Information Criterion ◽  
Absolute Error ◽  
Predictive Maintenance ◽  
Average Absolute Error ◽  
Current State ◽  
Automatic Model Selection ◽  
State Duration ◽  
Useful Lifetime

Realistic predictive maintenance approaches are essential for condition monitoring and predictive maintenance of industrial machines. In this work, we propose Hidden Semi-Markov Models (HSMMs) with (i) no constraints on the state duration density function and (ii) being applied to continuous or discrete observation. To deal with such a type of HSMM, we also propose modifications to the learning, inference, and prediction algorithms. Finally, automatic model selection has been made possible using the Akaike Information Criterion. This paper describes the theoretical formalization of the model as well as several experiments performed on simulated and real data with the aim of methodology validation. In all performed experiments, the model is able to correctly estimate the current state and to effectively predict the time to a predefined event with a low overall average absolute error. As a consequence, its applicability to real world settings can be beneficial, especially where in real time the Remaining Useful Lifetime (RUL) of the machine is calculated.

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