State-space Hidden Markov Model Research Articles

Between a rock and a hard place: rugged terrain features and human disturbance affect behaviour and habitat use of Sumatran elephants in Aceh, Sumatra, Indonesia

Asian elephants are threatened throughout their range due to habitat loss, fragmentation, and conflict with humans. Limited data on spatial and temporal use of habitats in largely intact forests by elephants in Sumatra is a major hindrance for conservation and land use planning, and consequently human–elephant conflict mitigation. We analysed GPS data from collared elephants to investigate their use of habitat in relation to available land cover and human disturbance in Aceh, Sumatra, Indonesia. We applied remote sensing techniques to extract environmental variables to use in state-space Hidden Markov Models and integrated Step Selection Function to analyse foraging strategies and land use selectivity, while using dynamic Brownian Bridge Movement Models and Minimum Convex Polygons to determine used and available habitat features within their home ranges. Our results show that Sumatran elephants in Aceh preferred areas with lower elevation (< 200 m) and slopes (0–10°), with minimal terrain ruggedness, which also happen to be areas selected by humans. Elephants closely adhered to rivers and mountain valleys when utilizing homogenous dense natural forests, and expanded from the rivers in heterogenous forests toward open, resource-dense, land use types. Overall, slope, vegetation, and human disturbance had the largest impact on each of the clans’ foraging strategies. Areas closer to human settlements were used more by night and less by day, indicating that human presence and activities influence elephant habitat use and avoidance. Our findings conclude future management decisions should focus on protecting the remaining lowland forests and preventing further urban encroachment in areas with lower elevation and slope, particularly in proximity to rivers which act as corridors between natural habitats, in order to mitigate human-elephant conflict and protect the species from extinction.

Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models

Finite state space hidden Markov models are flexible tools to model phenomena with complex time dependencies: any process distribution can be approximated by a hidden Markov model with enough hidden states. We consider the problem of estimating an unknown process distribution using nonparametric hidden Markov models in the misspecified setting, that is when the data-generating process may not be a hidden Markov model. We show that when the true distribution is exponentially mixing and satisfies a forgetting assumption, the maximum likelihood estimator recovers the best approximation of the true distribution. We prove a finite sample bound on the resulting error and show that it is optimal in the minimax sense--up to logarithmic factors--when the model is well specified.

Integrating Archival Tag Data and a High-Resolution Oceanographic Model to Estimate Basking Shark (Cetorhinus maximus) Movements in the Western Atlantic

Basking shark (Cetorhinus maximus) populations are considered ‘vulnerable’ globally and ‘endangered’ in the northeast Atlantic by the International Union for the Conservation of Nature. Much of our knowledge of this species comes from surface observations in coastal waters, yet recent evidence suggests the majority of their lives may be spent in the deep ocean. Depth preferences of basking sharks have significantly limited movement studies that used pop-up satellite archival transmitting (PSAT) tags as conventional light-based geolocation is impossible for tagged animals that spend significant time below the photic zone. We tagged 57 basking sharks with PSAT tags in the NW Atlantic from 2004-2011. Many individuals spent several months at meso- and bathy-pelagic depths where accurate light-level geolocation was impossible during fall, winter and spring. We applied a newly-developed geolocation approach for the PSAT data by comparing three-dimensional depth-temperature profile data recorded by the tags to modeled in-situ oceanographic data from the high-resolution HYbrid Coordinate Ocean Model (HYCOM). Observation-based likelihoods were leveraged within a state-space hidden Markov model (HMM). The combined tracks revealed that basking sharks moved from waters around Cape Cod, MA to as far as the SE coast of Brazil (20°S), a total distance of over 17,000km. Moreover, 59% of tagged individuals with sufficient deployment durations (> 250 days) demonstrated seasonal fidelity to Cape Cod and the Gulf of Maine, with one individual returning to within 60 km of its tagging location one year later. Tagged sharks spent most of their time at epipelagic depths during summer months around Cape Cod and in the Gulf of Maine. During winter months, sharks spent extended periods at depths of at least 600 m while moving south to the Sargasso Sea, the Caribbean Sea, or the western tropical Atlantic. Our work demonstrates the utility of applying advances in oceanographic modeling to understanding habitat use of highly migratory, often meso- and bathy-pelagic, ocean megafauna. The large-scale movement patterns of tagged sharks highlight the need for international cooperation when designing and implementing conservation strategies to ensure that the species recovers from the historical effects of over-fishing throughout the North Atlantic Ocean.

Full molecular trajectories of RNA polymerase at single base-pair resolution

In recent years, highly stable optical tweezers systems have enabled the characterization of the dynamics of molecular motors at very high resolution. However, the motion of many motors with angstrom-scale dynamics cannot be consistently resolved due to poor signal-to-noise ratio. Using an acousto-optic deflector to generate a "time-shared" dual-optical trap, we decreased low-frequency noise by more than one order of magnitude compared with conventional dual-trap optical tweezers. Using this instrument, we implemented a protocol that synthesizes single base-pair trajectories, which are used to test a Large State Space Hidden Markov Model algorithm to recover their individual steps. We then used this algorithm on real transcription data obtained in the same instrument to fully uncover the molecular trajectories of Escherichia coli RNA polymerase. We applied this procedure to reveal the effect of pyrophosphate on the distribution of dwell times between consecutive polymerase steps.

Consistent Estimation of the Filtering and Marginal Smoothing Distributions in Nonparametric Hidden Markov Models

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the $\mathrm {L}^{1}$ -risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the $\mathrm {L}^{2}$ -risk of the nonparametric spectral estimators. In the case where the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.

The Iterated Auxiliary Particle Filter

ABSTRACTWe present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of “twisted” models: each member is specified by a sequence of positive functions and has an associated -auxiliary particle filter that provides unbiased estimates of L. We identify a sequence that is optimal in the sense that the -auxiliary particle filter’s estimate of L has zero variance. In practical applications, is unknown so the -auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm.

On the modelling of nested risk-neutral stochastic processes with applications in insurance

ABSTRACTWe propose a modelling framework for risk-neutral stochastic processes nested in a real-world stochastic process. The framework is important for insurers that deal with the valuation of embedded options and in particular at future points in time. We make use of the class of State Space Hidden Markov models for modelling the joint behaviour of the parameters of a risk-neutral model and the dynamics of option market instruments. This modelling concept enables us to perform non-linear estimation, forecasting and robust calibration. The proposed method is applied to the Heston model for which we find highly satisfactory results. We use the estimated Heston model to compute the required capital of an insurance company under Solvency II and we find large differences compared to a basic calibration method.

HMMs in Protein Fold Classification.

The limitation of most HMMs is their inherent high dimensionality. Therefore we developed several variations of low complexity models that can be applied even to protein families with a few members. In this chapter we present these variations. All of them include the use of a hidden Markov model (HMM), with a small number of states (called reduced state-space HMM), which is trained with both amino acid sequence and secondary structure of proteins whose 3D structure is known and it is used for protein fold classification. We used data from Protein Data Bank and annotation from SCOP database for training and evaluation of the proposed HMM variations for a number of protein folds that belong to major structural classes. Results indicate that the variations have similar performance, or even better in some cases, on classifying proteins than SAM, which is a widely used HMM-based method for protein classification. The major advantage of the proposed variations is that we employed a small number of states and the algorithms used for training and scoring are of low complexity and thus relatively fast. The main variations examined include a version of the reduced state-space HMM with seven states (7-HMM), a version of the reduced state-space HMM with three states (3-HMM) and an optimized version of the reduced state-space HMM with three states, where an optimization process is applied to its scores (optimized 3-HMM).

Assessment of optimized Markov models in protein fold classification.

Protein fold classification is a challenging task strongly associated with the determination of proteins' structure. In this work, we tested an optimization strategy on a Markov chain and a recently introduced Hidden Markov Model (HMM) with reduced state-space topology. The proteins with unknown structure were scored against both these models. Then the derived scores were optimized following a local optimization method. The Protein Data Bank (PDB) and the annotation of the Structural Classification of Proteins (SCOP) database were used for the evaluation of the proposed methodology. The results demonstrated that the fold classification accuracy of the optimized HMM was substantially higher compared to that of the Markov chain or the reduced state-space HMM approaches. The proposed methodology achieved an accuracy of 41.4% on fold classification, while Sequence Alignment and Modeling (SAM), which was used for comparison, reached an accuracy of 38%.

Sequential Monte Carlo smoothing for general state space hidden Markov models

Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles.

Improving the protein fold recognition accuracy of a reduced state-space hidden Markov model

Fold recognition is a challenging field strongly associated with protein function determination, which is crucial for biologists and the pharmaceutical industry. Hidden Markov models (HMMs) have been widely used for this purpose. In this paper we demonstrate how the fold recognition performance of a recently introduced HMM with a reduced state-space topology can be improved. Our method employs an efficient architecture and a low complexity training algorithm based on likelihood maximization. The fold recognition performance of the model is further improved in two steps. In the first step we use a smaller model architecture based on the {E,H,L} alphabet instead of the DSSP secondary structure alphabet. In the second step secondary structure information (predicted or true) is additionally used in scoring the test set sequences. The Protein Data Bank and the annotation of the SCOP database are used for the training and evaluation of the proposed methodology. The results show that the fold recognition accuracy is substantially improved in both steps. Specifically, it is increased by 2.9% in the first step to 22%. In the second step it further increases and reaches up to 30% when predicted secondary structure information is additionally used and it increases even more and reaches up to 34.7% when we use the true secondary structure. The major advantage of the proposed improvements is that the fold recognition performance is substantially increased while the size of the model and the computational complexity of scoring are decreased.