On modifications to the Poisson-triggered hidden Markov paradigm through partitioned empirical recurrence rates ratios and its applications to natural hazards monitoring

Hidden Markov models (HMMs), especially those with a Poisson density governing the latent state-dependent emission probabilities, have enjoyed substantial and undeniable success in modeling natural hazards. Classifications among these hazards, induced through quantifiable properties such as varying intensities or geographic proximities, often exist, enabling the creation of an empirical recurrence rates ratio (ERRR), a smoothing statistic that is gradually gaining currency in modeling literature due to its demonstrated ability in unearthing interactions. Embracing these tools, this study puts forth a refreshing monitoring alternative where the unobserved state transition probability matrix in the likelihood of the Poisson based HMM is replaced by the observed transition probabilities of a discretized ERRR. Analyzing examples from Hawaiian volcanic and West Atlantic hurricane interactions, this work illustrates how the discretized ERRR may be interpreted as an observed version of the unobserved hidden Markov chain that generates one of the two interacting processes. Surveying different facets of traditional inference such as global state decoding, hidden state predictions, one-out conditional distributions, and implementing related computational algorithms, we find that the latest proposal estimates the chances of observing a high-risk period, one threatening several hazards, more accurately than its established counterpart. Strongly intuitive and devoid of forbidding technicalities, the new prescription launches a vision of surer forecasts and stands versatile enough to be applicable to other types of hazard monitoring (such as landslides, earthquakes, floods), especially those with meager occurrence probabilities.

Epilepsy as a dynamic disease: A Bayesian model for differentiating seizure risk from natural variability

Objective

A fundamental challenge in treating epilepsy is that changes in observed seizure frequencies do not necessarily reflect changes in underlying seizure risk. Rather, changes in seizure frequency may occur due to probabilistic variation around an underlying seizure risk state caused by normal fluctuations from natural history, leading to seizure unpredictability and potentially suboptimal medication adjustments in epilepsy management. However, no rigorous statistical approach exists to systematically distinguish expected changes in seizure frequency due to natural variability from changes in underlying seizure risk.

Methods

Using data from SeizureTracker.com, a patient‐reported seizure diary tool containing over 1.2 million recorded seizures across 8 years, a novel epilepsy seizure risk assessment tool (EpiSAT) employing a Bayesian mixed‐effects hidden Markov model for zero‐inflated count data was developed to estimate changes in underlying seizure risk using patient‐reported seizure diary and clinical measurement data. Accuracy for correctly assessing underlying seizure risk was evaluated through a simulation comparison. Implications for the natural history of tuberous sclerosis complex (TSC) were assessed using data from SeizureTracker.com.

Results

EpiSAT led to significant improvement in seizure risk assessment compared to traditional approaches relying solely on observed seizure frequencies. Applied to TSC, four underlying seizure risk states were identified. The expected duration of each state was <12 months, providing a data‐driven estimate of the amount of time a person with TSC would be expected to remain at the same seizure risk level according to the natural course of epilepsy.

Significance

We propose a novel Bayesian statistical approach for evaluating seizure risk on an individual patient level using patient‐reported seizure diaries, which allows for the incorporation of external clinical variables to assess impact on seizure risk. This tool may improve the ability to distinguish true changes in seizure risk from natural variations in seizure frequency in clinical practice. Incorporation of systematic statistical approaches into antiepileptic drug (AED) management may help improve understanding of seizure unpredictability as well as timing of treatment interventions for people with epilepsy.

A simulation-based method for model evaluation

We wish to evaluate and compare models that are non-nested and fit to data using different fitting criteria. We first estimate parameters in all models by optimizing goodness-of-fit to a dataset. Then, to assess a candidate model, we simulate a population of datasets from it and evaluate the goodness-of-fit of all the models, without re-estimating parameter values. Finally, we see whether the vector of goodness-of-fit criteria for the original data is compatible with the multivariate distribution of these criteria for the simulated datasets. By simulating from each model in turn, we determine whether any, or several, models are consistent with the data. We apply the method to compare three models, fit at different temporal resolutions to binary time series of animal behaviour data, concluding that a semi-Markov model gives a better fit than latent Gaussian and hidden Markov models.

Discrete- and continuous-time probabilistic models and algorithms for inferring neuronal UP and DOWN states

UP and DOWN states, the periodic fluctuations between increased and decreased spiking activity of a neuronal population, are a fundamental feature of cortical circuits. Understanding UP-DOWN state dynamics is important for understanding how these circuits represent and transmit information in the brain. To date, limited work has been done on characterizing the stochastic properties of UP-DOWN state dynamics. We present a set of Markov and semi-Markov discrete- and continuous-time probability models for estimating UP and DOWN states from multiunit neural spiking activity. We model multiunit neural spiking activity as a stochastic point process, modulated by the hidden (UP and DOWN) states and the ensemble spiking history. We estimate jointly the hidden states and the model parameters by maximum likelihood using an expectation-maximization (EM) algorithm and a Monte Carlo EM algorithm that uses reversible-jump Markov chain Monte Carlo sampling in the E-step. We apply our models and algorithms in the analysis of both simulated multiunit spiking activity and actual multi- unit spiking activity recorded from primary somatosensory cortex in a behaving rat during slow-wave sleep. Our approach provides a statistical characterization of UP-DOWN state dynamics that can serve as a basis for verifying and refining mechanistic descriptions of this process.

Computer modelling of epilepsy

Epilepsy is a complex set of disorders that can involve many areas of the cortex, as well as underlying deep-brain systems. The myriad manifestations of seizures, which can be as varied as déjà vu and olfactory hallucination, can therefore give researchers insights into regional functions and relations. Epilepsy is also complex genetically and pathophysiologically: it involves microscopic (on the scale of ion channels and synaptic proteins), macroscopic (on the scale of brain trauma and rewiring) and intermediate changes in a complex interplay of causality. It has long been recognized that computer modelling will be required to disentangle causality, to better understand seizure spread and to understand and eventually predict treatment efficacy. Over the past few years, substantial progress has been made in modelling epilepsy at levels ranging from the molecular to the socioeconomic. We review these efforts and connect them to the medical goals of understanding and treating the disorder.

A stochastic framework for evaluating seizure prediction algorithms using hidden Markov models

Responsive, implantable stimulation devices to treat epilepsy are now in clinical trials. New evidence suggests that these devices may be more effective when they deliver therapy before seizure onset. Despite years of effort, prospective seizure prediction, which could improve device performance, remains elusive. In large part, this is explained by lack of agreement on a statistical framework for modeling seizure generation and a method for validating algorithm performance. We present a novel stochastic framework based on a three-state hidden Markov model (HMM) (representing interictal, preictal, and seizure states) with the feature that periods of increased seizure probability can transition back to the interictal state. This notion reflects clinical experience and may enhance interpretation of published seizure prediction studies. Our model accommodates clipped EEG segments and formalizes intuitive notions regarding statistical validation. We derive equations for type I and type II errors as a function of the number of seizures, duration of interictal data, and prediction horizon length and we demonstrate the model's utility with a novel seizure detection algorithm that appeared to predicted seizure onset. We propose this framework as a vital tool for designing and validating prediction algorithms and for facilitating collaborative research in this area.

Monitoring epidemiologic surveillance data using hidden Markov models

The analysis of routinely collected surveillance data is an important challenge in public health practice. We present a method based on a hidden Markov model for monitoring such time series. The model characterizes the sequence of measurements by assuming that its probability density function depends on the state of an underlying Markov chain. The parameter vector includes distribution parameters and transition probabilities between the states. Maximum likelihood estimates are obtained with a modified EM algorithm. Extensions are provided to take into account trend and seasonality in the data. The method is demonstrated on two examples: the first seeks to characterize influenza-like illness incidence rates with a mixture of Gaussian distributions, and the other, poliomyelitis counts with mixture of Poisson distributions. The results justify a wider use of this method for analysing surveillance data.

On the EM algorithm for overdispersed count data

In this paper, we consider the use of the EM algorithm for the fitting of distributions by maximum likelihood to overdispersed count data. In the course of this, we also provide a review of various approaches that have been proposed for the analysis of such data. As the Poisson and binomial regression models, which are often adopted in the first instance for these analyses, are particular examples of a generalized linear model (GLM), the focus of the account is on the modifications and extensions to GLMs for the handling of overdispersed count data.