Efficient methods for fitting nonlinear non-Gaussian state space models of wildlife population dynamics

State-space models (SSMs) are a popular and flexible framework for modelling time series due to their ability to separate changes in the underlying state of a system from the noisy observations made on these states. This thesis explores methods for estimating states and model parameters in non-linear and non-Gaussian Bayesian SSMs. We focus on models of wildlife population dynamics, in particular a case study of the UK grey seal population. Calculation of the likelihood is fundamental to Bayesian analysis, but direct calculation is typically intractable for non-linear non-Gaussian SSMs. We use a class of simulation-based methods, Sequential Monte Carlo (SMC), which build on repeated importance sampling of simulated states to deliver an unbiased estimate of the likelihood. We find that variance of the estimated likelihood can be high and explore techniques for variance reduction. For parameter inference, we use particle marginal Metropolis-Hastings (PMMH), which embeds the SMC likelihood within a Markov chain Monte Carlo (MCMC) algorithm. Careful balance is needed between computational effort expended on the SMC step and the number of MCMC samples. A much faster alternative is the Kalman filter, designed for linear and Gaussian SSMs. We applied the Kalman filter to an approximation of the seal model. The posterior distribution obtained was often close to the true posterior, while reducing computation time by a factor of 1790. We show the seal model suffers from identifiability issues which cannot be resolved by increasing the accuracy of the observations or allowing more flexibility in the underlying biological process with random effects. However, estimation of underlying states (i.e., population sizes) is unaffected by these issues. A reduction in PMMH computation time can be achieved by exploiting the structure of the state model: separately estimating likelihood components in each of the 4 seal regions led to a 5-fold increase in speed."This work was supported by the Engineering and Physical Sciences Research Council, via a Doctoral Training Partnership grant to the University of St Andrews; and by the University of St Andrews (School of Mathematics and Statistics)."--Funding Acknowledgement

A guide to state-space modeling of ecological time series

State–space models (SSMs) are an important modeling framework for analyzing ecological time series. These hierarchical models are commonly used to model population dynamics, animal movement, and capture–recapture data, and are now increasingly being used to model other ecological processes. SSMs are popular because they are flexible and they model the natural variation in ecological processes separately from observation error. Their flexibility allows ecologists to model continuous, count, binary, and categorical data with linear or nonlinear processes that evolve in discrete or continuous time. Modeling the two sources of stochasticity separately allows researchers to differentiate between biological variation and imprecision in the sampling methodology, and generally provides better estimates of the ecological quantities of interest than if only one source of stochasticity is directly modeled. Since the introduction of SSMs, a broad range of fitting procedures have been proposed. However, the variety and complexity of these procedures can limit the ability of ecologists to formulate and fit their own SSMs. We provide the knowledge for ecologists to create SSMs that are robust to common, and often hidden, estimation problems, and the model selection and validation tools that can help them assess how well their models fit their data. We present a review of SSMs that will provide a strong foundation to ecologists interested in learning about SSMs, introduce new tools to veteran SSM users, and highlight promising research directions for statisticians interested in ecological applications. The review is accompanied by an in-depth tutorial that demonstrates how SSMs can be fitted and validated in R. Together, the review and tutorial present an introduction to SSMs that will help ecologists to formulate, fit, and validate their models

Report of the NAMMCO-ICES Workshop on Seal Modelling (WKSEALS 2020)

To support sustainable management of apex predator populations, it is important to estimate population size and understand the drivers of population trends to anticipate the consequences of human decisions. Robust population models are needed, which must be based on realistic biological principles and validated with the best available data. A team of international experts reviewed age-structured models of North Atlantic pinniped populations, including Grey seal (Halichoerus grypus), Harp seal (Pagophilus groenlandicus), and Hooded seal (Cystophora cristata). Statistical methods used to fit such models to data were compared and contrasted. Differences in biological assumptions and model equations were driven by the data available from separate studies, including observation methodology and pre-processing. Counts of pups during the breeding season were used in all models, with additional counts of adults and juveniles available in some. The regularity and frequency of data collection, including survey counts and vital rate estimates, varied. Important differences between the models concerned the nature and causes of variation in vital rates (age-dependent survival and fecundity). Parameterisation of age at maturity was detailed and time-dependent in some models and simplified in others. Methods for estimation of model parameters were reviewed and compared. They included Bayesian and maximum likelihood (ML) approaches, implemented via bespoke coding in C, C++, TMB or JAGS. Comparative model runs suggested that as expected, ML-based implementations were rapid and computationally efficient, while Bayesian approaches, which used MCMC or sequential importance sampling, required longer for inference. For grey seal populations in the Netherlands, where preliminary ML-based TMB results were compared with the outputs of a Bayesian JAGS implementation, some differences in parameter estimates were apparent. For these seal populations, further investigations are recommended to explore differences that might result from the modelling framework and model-fitting methodology, and their importance for inference and management advice. The group recommended building on the success of this workshop via continued collaboration with ICES and NAMMCO assessment groups, as well as other experts in the marine mammal modelling community. Specifically, for Northeast Atlantic harp and hooded seal populations, the workshop represents the initial step towards a full ICES benchmark process aimed at revising and evaluating new assessment models.Publisher PDFPeer reviewe

Modelling the population size and dynamics of the British grey seal

1. Grey seals (Halichoerus grypus) were the first mammals to be protected by an Act of Parliament in the UK and are currently protected under UK, Scottish, and EU conservation legislation. Reporting requirements under each of these statutes requires accurate and timely population estimates. Monitoring is principally conducted by aerial surveys of the breeding colonies; these are used to produce estimates of annual pup production. Translating these data to estimates of adult population size requires information about demographic parameters such as fecundity and sex ratio.2. An age‐structured population dynamics model is presented, which includes density dependence in pup survival, with separate carrying capacities in each of the four breeding regions considered (North Sea, Inner Hebrides, Outer Hebrides, and Orkney). This model is embedded within a Bayesian state–space modelling framework, allowing the population model to be linked to available data and the use of informative prior distributions on demographic parameters. A computer‐intensive fitting algorithm is presented based on particle filtering methods.3. The model is fitted to region‐level pup production estimates from 1984 to 2010 and an independent estimate of adult population size, derived from aerial surveys of hauled‐out seals in 2008. The fitted model is used to estimate total population size from 1984 to 2010.4. The population in the North Sea region has increased at a near‐constant rate; growth in the other three regions began to slow in the mid‐1990s and these populations appear to have reached carrying capacity. The total population size of seals aged 1 year or older in 2010 was estimated to be 116 100 (95% CI 98 400–138 600), an increase of <1% on the previous year.5. The modelling and fitting methods are widely applicable to other wildlife populations where diverse sources of information are available and inference is required for the underlying population dynamics

A guide to state–space modeling of ecological time series

State–space models (SSMs) are an important modeling framework for analyzing ecological time series. These hierarchical models are commonly used to model population dynamics, animal movement, and capture-recapture data, and are now increasingly being used to model other ecological processes. SSMs are popular because they are flexible and they model the natural variation in ecological processes separately from observation error. Their flexibility allows ecologists to model continuous, count, binary, and categorical data with linear or nonlinear processes that evolve in discrete or continuous time. Modeling the two sources of stochasticity separately allows researchers to differentiate between biological variation and imprecision in the sampling methodology, and generally provides better estimates of the ecological quantities of interest than if only one source of stochasticity is directly modeled. Since the introduction of SSMs, a broad range of fitting procedures have been proposed. However, the variety and complexity of these procedures can limit the ability of ecologists to formulate and fit their own SSMs. We provide the knowledge for ecologists to create SSMs that are robust to common, and often hidden, estimation problems, and the model selection and validation tools that can help them assess how well their models fit their data. We present a review of SSMs that will provide a strong foundation to ecologists interested in learning about SSMs, introduce new tools to veteran SSM users, and highlight promising research directions for statisticians interested in ecological applications. The review is accompanied by an in-depth tutorial that demonstrates how SSMs can be fitted and validated in R. Together, the review and tutorial present an introduction to SSMs that will help ecologists to formulate, fit, and validate their models

Report of the NAMMCO-ICES Workshop on Seal Modelling (WKSEALS 2020)

To support sustainable management of apex predator populations, it is important to estimate population size and understand the drivers of population trends to anticipate the consequences of human decisions. Robust population models are needed, which must be based on realistic biological principles and validated with the best available data. A team of international experts reviewed age-structured models of North Atlantic pinniped populations, including Grey seal (Halichoerus grypus), Harp seal (Pagophilus groenlandicus), and Hooded seal (Cystophora cristata). Statistical methods used to fit such models to data were compared and contrasted. Differences in biological assumptions and model equations were driven by the data available from separate studies, including observation methodology and pre-processing. Counts of pups during the breeding season were used in all models, with additional counts of adults and juveniles available in some. The regularity and frequency of data collection, including survey counts and vital rate estimates, varied. Important differences between the models concerned the nature and causes of variation in vital rates (age-dependent survival and fecundity). Parameterisation of age at maturity was detailed and time-dependent in some models and simplified in others. Methods for estimation of model parameters were reviewed and compared. They included Bayesian and maximum likelihood (ML) approaches, implemented via bespoke coding in C, C++, TMB or JAGS. Comparative model runs suggested that as expected, ML-based implementations were rapid and computationally efficient, while Bayesian approaches, which used MCMC or sequential importance sampling, required longer for inference. For grey seal populations in the Netherlands, where preliminary ML-based TMB results were compared with the outputs of a Bayesian JAGS implementation, some differences in parameter estimates were apparent. For these seal populations, further investigations are recommended to explore differences that might result from the modelling framework and model-fitting methodology, and their importance for inference and management advice. The group recommended building on the success of this workshop via continued collaboration with ICES and NAMMCO assessment groups, as well as other experts in the marine mammal modelling community. Specifically, for Northeast Atlantic harp and hooded seal populations, the workshop represents the initial step towards a full ICES benchmark process aimed at revising and evaluating new assessment models