Sojourn Distributions Research Articles

Quantile hidden semi-Markov models for multivariate time series.

This paper develops a quantile hidden semi-Markov regression to jointly estimate multiple quantiles for the analysis of multivariate time series. The approach is based upon the Multivariate Asymmetric Laplace (MAL) distribution, which allows to model the quantiles of all univariate conditional distributions of a multivariate response simultaneously, incorporating the correlation structure among the outcomes. Unobserved serial heterogeneity across observations is modeled by introducing regime-dependent parameters that evolve according to a latent finite-state semi-Markov chain. Exploiting the hierarchical representation of the MAL, inference is carried out using an efficient Expectation-Maximization algorithm based on closed form updates for all model parameters, without parametric assumptions about the states’ sojourn distributions. The validity of the proposed methodology is analyzed both by a simulation study and through the empirical analysis of air pollutant concentrations in a small Italian city.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11222-022-10130-1.

Improved state change estimation in dynamic functional connectivity using hidden semi-Markov models

The study of functional brain networks has grown rapidly over the past decade. While most functional connectivity (FC) analyses estimate one static network structure for the entire length of the functional magnetic resonance imaging (fMRI) time series, recently there has been increased interest in studying time-varying changes in FC. Hidden Markov models (HMMs) have proven to be a useful modeling approach for discovering repeating graphs of interacting brain regions (brain states). However, a limitation lies in HMMs assuming that the sojourn time, the number of consecutive time points in a state, is geometrically distributed. This may encourage inaccurate estimation of the time spent in a state before switching to another state. We propose a hidden semi-Markov model (HSMM) approach for inferring time-varying brain networks from fMRI data, which explicitly models the sojourn distribution. Specifically, we propose using HSMMs to find each subject's most probable series of network states and the graphs associated with each state, while properly estimating and modeling the sojourn distribution for each state. We perform a simulation study, as well as an analysis on both task-based fMRI data from an anxiety-inducing experiment and resting-state fMRI data from the Human Connectome Project. Our results demonstrate the importance of model choice when estimating sojourn times and reveal their potential for understanding healthy and diseased brain mechanisms.

Modeling rates of infection with transient maternal antibodies and waning active immunity: Application to Bordetella pertussis in Sweden

Serological surveys provide reliable information from which to calculate forces (instantaneous rates) of infection, but waning immunity and clinical consequences that depend on residual immunity complicate interpretation of results. We devised a means of calculating these rates that accounts for passively acquired maternal antibodies that decay or active immunity that wanes, permitting re-infection. We applied our method to pertussis (whooping cough) in Sweden, where vaccination was discontinued from 1979 to 1995. A national cross-sectional serosurvey of antibodies to pertussis toxin, which peak soon after infection and then decay, was conducted shortly after vaccination resumed. Together with age-specific contact rates in Finland, contemporary forces of infection enable us to evaluate the recent assertion that the probability of infection upon contact is age-independent. We find elevated probabilities among children, adolescents and young adults, whose contacts may be more intimate than others. Products of contact rates and probabilities of infection permit transmission modeling and estimation of the intrinsic reproduction number. In contrast to another recent estimate, ours approximates the ratio of life expectancy and age at first infection. Our framework is sufficiently general to accommodate more realistic sojourn distributions and additional lifetime infections.

BiomvRhsmm: genomic segmentation with hidden semi-Markov model.

High-throughput technologies like tiling array and next-generation sequencing (NGS) generate continuous homogeneous segments or signal peaks in the genome that represent transcripts and transcript variants (transcript mapping and quantification), regions of deletion and amplification (copy number variation), or regions characterized by particular common features like chromatin state or DNA methylation ratio (epigenetic modifications). However, the volume and output of data produced by these technologies present challenges in analysis. Here, a hidden semi-Markov model (HSMM) is implemented and tailored to handle multiple genomic profile, to better facilitate genome annotation by assisting in the detection of transcripts, regulatory regions, and copy number variation by holistic microarray or NGS. With support for various data distributions, instead of limiting itself to one specific application, the proposed hidden semi-Markov model is designed to allow modeling options to accommodate different types of genomic data and to serve as a general segmentation engine. By incorporating genomic positions into the sojourn distribution of HSMM, with optional prior learning using annotation or previous studies, the modeling output is more biologically sensible. The proposed model has been compared with several other state-of-the-art segmentation models through simulation benchmarking, which shows that our efficient implementation achieves comparable or better sensitivity and specificity in genomic segmentation.

Fitting and interpreting continuous‐time latent Markov models for panel data

Multistate models characterize disease processes within an individual. Clinical studies often observe the disease status of individuals at discrete time points, making exact times of transitions between disease states unknown. Such panel data pose considerable modeling challenges. Assuming the disease process progresses accordingly, a standard continuous-time Markov chain (CTMC) yields tractable likelihoods, but the assumption of exponential sojourn time distributions is typically unrealistic. More flexible semi-Markov models permit generic sojourn distributions yet yield intractable likelihoods for panel data in the presence of reversible transitions. One attractive alternative is to assume that the disease process is characterized by an underlying latent CTMC, with multiple latent states mapping to each disease state. These models retain analytic tractability due to the CTMC framework but allow for flexible, duration-dependent disease state sojourn distributions. We have developed a robust and efficient expectation-maximization algorithm in this context. Our complete data state space consists of the observed data and the underlying latent trajectory, yielding computationally efficient expectation and maximization steps. Our algorithm outperforms alternative methods measured in terms of time to convergence and robustness. We also examine the frequentist performance of latent CTMC point and interval estimates of disease process functionals based on simulated data. The performance of estimates depends on time, functional, and data-generating scenario. Finally, we illustrate the interpretive power of latent CTMC models for describing disease processes on a dataset of lung transplant patients. We hope our work will encourage wider use of these models in the biomedical setting.

Estimating parametric semi-Markov models from panel data using phase-type approximations

Inference for semi-Markov models under panel data presents considerable computational difficulties. In general the likelihood is intractable, but a tractable likelihood with the form of a hidden Markov model can be obtained if the sojourn times in each of the states are assumed to have phase-type distributions. However, using phase-type distributions directly may be undesirable as they require estimation of parameters which may be poorly identified. In this article, an approach to fitting semi-Markov models with standard parametric sojourn distributions is developed. The method involves establishing a family of Coxian phase-type distribution approximations to the parametric distribution and merging approximations for different states to obtain an approximate semi-Markov process with a tractable likelihood. Approximations are developed for Weibull and Gamma distributions and demonstrated on data relating to post-lung-transplantation patients.

Bootstrap confidence bands for sojourn distributions in multistate semi-Markov models with right censoring

Transient semi-Markov processes have traditionally been used to describe the transitions of a patient through the various states of a multistate survival model. A survival distribution in this context is a sojourn through the states until passage to a fatal absorbing state or certain endpoint states. Using complete sojourn data, this paper shows how such survival distributions and associated hazard functions can be estimated nonparametrically and also how nonparametric bootstrap pointwise confidence bands can be constructed for them when patients are subject to independent right censoring from each state during the sojourn. Limitations to the estimability of such survival distributions that result from random censoring with bounded support are clarified. The methods are applicable to any sort of sojourn through any finite state process of arbitrary complexity involving feedback into previously occupied states.

Hidden Semi Markov Models for Multiple Observation Sequences: ThemhsmmPackage forR

This paper describes the <strong>R</strong> package <strong>mhsmm</strong> which implements estimation and prediction methods for hidden Markov and semi-Markov models for multiple observation sequences. Such techniques are of interest when observed data is thought to be dependent on some unobserved (or hidden) state. Hidden Markov models only allow a geometrically distributed sojourn time in a given state, while hidden semi-Markov models extend this by allowing an arbitrary sojourn distribution. We demonstrate the software with simulation examples and an application involving the modelling of the ovarian cycle of dairy cows.

Semi‐Markov Models with Phase‐Type Sojourn Distributions

Continuous-time multistate models are widely used for categorical response data, particularly in the modeling of chronic diseases. However, inference is difficult when the process is only observed at discrete time points, with no information about the times or types of events between observation times, unless a Markov assumption is made. This assumption can be limiting as rates of transition between disease states might instead depend on the time since entry into the current state. Such a formulation results in a semi-Markov model. We show that the computational problems associated with fitting semi-Markov models to panel-observed data can be alleviated by considering a class of semi-Markov models with phase-type sojourn distributions. This allows methods for hidden Markov models to be applied. In addition, extensions to models where observed states are subject to classification error are given. The methodology is demonstrated on a dataset relating to development of bronchiolitis obliterans syndrome in post-lung-transplantation patients.

Maximum Likelihood Estimation in Alternating Renewal Processes Under Window Censoring

Consider a process that jumps back and forth between two states, with random times spent in between. Suppose the durations of subsequent on and off states are i.i.d. and that the process has started far in the past, so it has achieved stasis. We estimate the sojourn distributions through maximum likelihood when data consist of several realizations observed over windows of fixed length. For discrete and continuous time Markov chains, we also examine if there is any loss of efficiency incurred when ignoring the stationarity structure in the estimation.

Smoothed nonparametric estimation in window censored semi-Markov processes

Consider a process that jumps among a finite set of states, with random times spent in between. In semi-Markov processes transitions follow a Markov chain and the sojourn distributions depend only on the connecting states. Suppose that the process started far in the past, achieving stationary. We consider non-parametric estimation by modelling the log-hazard of the sojourn times through linear splines; and we obtain maximum penalized likelihood estimators when data consist of several i.i.d. windows. We prove consistency using Grenander's method of sieves.

On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations

It is well known that the sojourn time of Brownian motion B(t), t>0 on the positive half-line, during the interval [0, t] and under the condition B(t)=0, is uniformly distributed, while it has the form of the “corrected arc-sine law” when the condition B(t)>0 is assumed. We find the analogues of these laws for “processes” X(t), t>0 governed by signed measures whose densities are the fundamental solutions of third and fourth-order heat-type equations. Surprisingly, both laws hold for the fourth-order “process.” The uniform law is still valid for the third-order “process” but a different law emerges when the condition X(t)>0 is considered.

Biphasic effect of pentobarbitone on chick myotube nicotinic receptor channel kinetics.

1. The modulatory action of pentobarbitone on chick myotube nicotinic acetylcholine receptor kinetics was studied by the patch clamp technique, particularly focussing on effects at low concentrations. 2. Open time sojourn distributions of foetal-type receptors recorded at room temperature (22-24 degrees C) in cell-attached mode in the presence of 0.2 microM acetylcholine were well described by two exponentials, with fast and slow time constants of 0.53 +/- 0.12 and 16.7 +/- 2.2 ms (means +/- s.e. mean) respectively. 3. The duration of the slow open time constant was increased by low concentrations of pentobarbitone (up to 1 microM), and thereafter decreased with higher concentrations (10-50 microM) in a concentration-dependent manner. 4. Complementary evidence for the stimulatory effect of pentobarbitone on open time was obtained (i) by using a backfill technique where drug concentration at the patch gradually increases over time, and (ii) through use of perfused outside-out preparations where receptors in the same patch were successively exposed to acetylcholine in the absence and presence of pentobarbitone. 5. The dual action of pentobarbitone on channel kinetics probably, indicates that an allosteric interaction mechanism is involved rather than simple steric channel blockade.

Optimal scheduling of examinations for the early detection of disease

The general theory of scheduling examinations for the diagnosis of disease is formulated with respect to the optimal spacing between examinations. This same theory is also applicable, with a change in language, to the inspection of equipment and the scheduling of patients under surveillance with disease. Optimal scheduling programmes are investigated using a weighted utility function which is linear in the probabilities both of finding a case at examination and of being clinically incident between examinations. If disease incidence is independent of time, then a necessary and sufficient condition for the intervals to be equally spaced is that the sensitivity of the examination be unity. The equations for finding the optimal intervals are derived and depend on the distribution of the pre-clinical sojourn times and the sensitivity of the test. If the sojourn distribution is exponential, the optimal intervals are equal except for the first and last intervals.

Two-Stage Models for the Analysis of Cancer Screening Data

Methods are proposed for the analysis of the natural history of disease from screening data when it cannot be assumed that untreated preclinical disease always progresses to clinical disease. The methodology is based on a two-stage model for preclinical disease in which stage 1 lesions may or may not progress to stage 2, but all stage 2 lesions progress to clinical disease. The focus is on joint estimation of the total preclinical duration and the sensitivity of the screening test. A partial likelihood is proposed for the analysis of prospectively collected screening data, and an analogous conditional likelihood is proposed for retrospective data. Some special cases for the joint sojourn distribution of the two stages are considered, including the independent model and limiting models where the duration of stage 2 is short relative to stage 1. The methods are applied to a case-control study of cervical cancer screening in Northeast Scotland.