Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

A new look at Markov processes of G/M/1-type

We present a new method for deriving the stationary distribution of an ergodic Markov process of G/M/1-type in continuous-time, by deriving and making use of a new representation for each element of the rate matrices contained in these distributions. This method can also be modified to derive the Laplace transform of each transition function associated with Markov processes of G/M/1-type.     

CONTINUOUS-TIME FILTERS FOR STATE ESTIMATION FROM POINT PROCESS MODELS OF NEURAL DATA

Neural spike trains, the primary communication signals in the brain, can be accurately modeled as point processes. For many years, significant theoretical work has been done on the construction of exact and approximate filters for state estimation from point process observations in continuous-time. We have previously developed approximate filters for state estimation from point process observations in discrete-time and applied them in the study of neural systems. Here, we present a coherent framework for deriving continuous-time filters from their discrete-counterparts. We present an accessible derivation of the well-known unnormalized conditional density equation for state evolution, construct a new continuous-time filter based on a Gaussian approximation, and propose a method for assessing the validity of the approximation following an approach by Brockett and Clark. We apply these methods to the problem of reconstructing arm reaching movements from simulated neural spiking activity from the primary motor cortex. This work makes explicit the connections between adaptive point process filters for analyzing neural spiking activity in continuous-time, and standard continuous-time filters for state estimation from continuous and point process observations.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

Bayesian inference for dynamic social network data

We consider a continuous-time model for the evolution of social networks. A social network is here conceived as a (di-) graph on a set of vertices, representing actors, and the changes of interest are creation and disappearance over time of (arcs) edges in the graph. Hence we model a collection of random edge indicators that are not, in general, independent. We explicitly model the interdependencies between edge indicators that arise from interaction between social entities. A Markov chain is defined in terms of an embedded chain with holding times and transition probabilities. Data are observed at fixed points in time and hence we are not able to observe the embedded chain directly. Introducing a prior distribution for the parameters we may implement an MCMC algorithm for exploring the posterior distribution of the parameters by simulating the evolution of the embedded process between observations.     

Continuous single attribute control chart for Markov-dependent processes

Most of the times, the observations related to the quality characteristic of a process do not need to be independent. In such cases, control charts based on the assumption of independence of the observations are not appropriate. When the characteristic under study is qualitative, Markov model serves as a simple model to account for the dependency of the observations. For this purpose, we develop an attribute control chart under 100% inspection for a Markov dependent process by controlling the error probabilities. This chart consists of two sub-charts. For a given sample, depending upon the state of the last observation of previous sample (if any), one of these two will be used. Optimal values of the design parameters of the control chart are obtained. Chart’s performance is studied by using its capability (probability) of detecting a shift in process parameters.     

Costing Mixed Coxian Phase-type Systems with Poisson Arrivals

Previously, we developed a modeling framework which classifies individuals with respect to their length of stay (LOS) in the transient states of a continuous-time Markov model with a single absorbing state; phase-type models are used for each class of the Markov model. We here add costs and obtain results for moments of total costs in (0, t], for an individual, a cohort arriving at time zero and when arrivals are Poisson. Based on stroke patient data from the Belfast City Hospital we use the overall modelling framework to obtain results for total cost in a given time interval.     

Sequential Bayesian inference in hidden Markov stochastic kinetic models with application to detection and response to seasonal epidemics

We study sequential Bayesian inference in stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of continuous-time jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is modeling of seasonal infectious disease outbreaks represented through a compartmental epidemic model. We demonstrate inference in such models with several numerical illustrations and also discuss predictive analysis of epidemic countermeasures using sequential Bayes estimates.     

Simulated Likelihood Approximations for Stochastic Volatility Models

Abstract. This paper deals with parametric inference for continuous-time stochastic volatility models observed at discrete points in time. We consider approximate maximum likelihood estimation: for the k th-order approximation, we pretend that the observations form a k th-order Markov chain, find the corresponding approximate log-likelihood function, and maximize it with respect to θ . The approximate log-likelihood function is not known analytically, but can easily be calculated by simulation. For each k , the method yields consistent and asymptotically normal estimators. Simulations from a model based on the Cox–Ingersoll–Ross model are used for illustration.     

Regime-switching pure jump processes and applications in the valuation of mortality-linked products

In this paper, we study the pricing of longevity bonds and an insurance contract on multiple lives in a regime-switching market driven by an underlying continuous-time Markov chain. For modeling dependent mortality, we make use of a Markov chain and some shot noise processes with regime switching. By using a martingale method, we give semi-analytical expressions for the price of longevity bonds and the premium of an insurance contract on the kth person to die.     

Coordinate sampler: a non-reversible Gibbs-like MCMC sampler

We derive a novel non-reversible, continuous-time Markov chain Monte Carlo sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process, which is a variant of the Zigzag sampler of Bierkens et al. (Ann Stat 47(3):1288–1320, 2019). In addition to providing a theoretical validation for this new simulation algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size.     

A Monte Carlo method for filtering a marked doubly stochastic Poisson process

The author provides an approximated solution for the filtering of a state-space model, where the hidden state process is a continuous-time pure jump Markov process and the observations come from marked point processes. Each state k corresponds to a different marked point process, defined by its conditional intensity function λ _k (t). When a state is visited by the hidden process, the corresponding marked point process is observed. The filtering equations are obtained by applying the innovation method and the integral representation theorem of a point process martingale. Since the filtering equations belong to the family of Kushner–Stratonovich equations, an iterative solution is calculated. The theoretical solution is approximated and a Monte Carlo integration technique employed to implement it. The sequential method has been tested on a simulated data set based on marked point processes widely used in the statistical analysis of seismic sequences: the Poisson model, the stress release model and the Etas model.     

A functional central limit theorem for Markov additive processes with an application to the closed Lu–Kumar network

A functional central limit theorem for a class of time-homogeneous continuous-time Markov processes (X,Y) is proved. The process X is a positive recurrent Markov process on a countable-state space and the process Y has conditionally independent increments given X. The pair (X,Y) is called a Markov additive process. This paper unifies and generalizes several functional central limit theorems for Markov additive processes. An explicit expression for the variance parameter of the limit process is calculated using the local characteristics of the X process. The functional central limit theorem is then used to prove a heavy traffic limit theorem for the closed Lu–Kumar network.     

A comparison of non-homogeneous Markov regression models with application to Alzheimer's disease progression

Markov regression models are useful tools for estimating the impact of risk factors on rates of transition between multiple disease states. Alzheimer's disease (AD) is an example of a multi-state disease process in which great interest lies in identifying risk factors for transition. In this context, non-homogeneous models are required because transition rates change as subjects age. In this report we propose a non-homogeneous Markov regression model that allows for reversible and recurrent disease states, transitions among multiple states between observations, and unequally spaced observation times. We conducted simulation studies to demonstrate performance of estimators for covariate effects from this model and compare performance with alternative models when the underlying non-homogeneous process was correctly specified and under model misspecification. In simulation studies, we found that covariate effects were biased if non-homogeneity of the disease process was not accounted for. However, estimates from non-homogeneous models were robust to misspecification of the form of the non-homogeneity. We used our model to estimate risk factors for transition to mild cognitive impairment (MCI) and AD in a longitudinal study of subjects included in the National Alzheimer's Coordinating Center's Uniform Data Set. Using our model, we found that subjects with MCI affecting multiple cognitive domains were significantly less likely to revert to normal cognition.     

Estimation in multiple linear regression Berkson model for processes with uncorrelated increments

This paper presents a method of estimating the regression and variance parameters in the multiple linear regression Berkson model for a continuous-time stochastic process with uncorrelated increments. Under minimal conditions, we establish (i) the Gauss–Markov theorem and the quadratic mean—as well as the strong consistency of the proposed estimate of the regression parameter and (ii) the weak consistency of the proposed estimate of the variance parameter.     

State-space models for multivariate longitudinal data of mixed types

We propose a class of state-space models for multivariate longitudinal data where the components of the response vector may have different distributions. The approach is based on the class of Tweedie exponential dispersion models, which accommodates a wide variety of discrete, continuous and mixed data. The latent process is assumed to be a Markov process, and the observations are conditionally independent given the latent process, over time as well as over the components of the response vector. This provides a fully parametric alternative to the quasilikelihood approach of Liang and Zeger. We estimate the regression parameters for time-varying covariates entering either via the observation model or via the latent process, based on an estimating equation derived from the Kalman smoother. We also consider analysis of residuals from both the observation model and the latent process.     

Bayesian analysis of the scatterometer wind retrieval inverse problem: some new approaches

Summary.  The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.     

Penalized Projection Estimator for Volatility Density

Abstract.  In this paper, we consider a stochastic volatility model ( Y _t , V _t ), where the volatility (V _t ) is a positive stationary Markov process. We assume that ( ln V _t ) admits a stationary density f that we want to estimate. Only the price process Y _t is observed at n discrete times with regular sampling interval Δ . We propose a non-parametric estimator for f obtained by a penalized projection method. Under mixing assumptions on ( V _t ), we derive bounds for the quadratic risk of the estimator. Assuming that Δ=Δ _n tends to 0 while the number of observations and the length of the observation time tend to infinity, we discuss the rate of convergence of the risk. Examples of models included in this framework are given.     

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

Abstract

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.     

An EM-Based Viterbi Approximation Algorithm for Mixed-State Latent Factor Models

In this article, a state-space model based on an underlying hidden Markov chain model (HMM) with factor analysis observation process is introduced. The HMM generates a piece-wise constant state evolution process and the observations are produced from the state vectors by a conditionally heteroscedastic factor analysis observation process. More specifically, we concentrate on situations where the factor variances are modeled by univariate Generalized Quadratic Autoregressive Conditionally Heteroscedastic processes (GQARCH). An expectation maximization (EM) algorithm combined with a mixed-state version of the Viterbi algorithm is derived for maximum likelihood estimation. The various regimes, common factors, and their volatilities are supposed unobservable and the inference must be carried out from the observable process. Extensive Monte Carlo simulations show promising results of the algorithms, especially for segmentation and tracking tasks.