Hidden Markov Models for Extended Batch Data

Abstract

Summary. Batch marking provides an important and efficient way to estimate the survival probabilities and population sizes of wild animals. It is particularly useful when dealing with animals that are difficult to mark individually. For the first time, we provide the likelihood for extended batch-marking experiments. It is often the case that samples contain individuals that remain unmarked, due to time and other constraints, and this information has not previously been analyzed. We provide ways of modeling such information, including an open N-mixture approach. We demonstrate that models for both marked and unmarked individuals are hidden Markov models; this provides a unified approach, and is the key to developing methods for fast likelihood computation and maximization. Likelihoods for marked and unmarked individuals can easily be combined using integrated population modeling. This allows the simultaneous estimation of population size and immigration, in addition to survival, as well as efficient estimation of standard errors and methods of model selection and evaluation, using standard likelihood techniques. Alternative methods for estimating population size are presented and compared. An illustration is provided by a weather-loach data set, previously analyzed by means of a complex procedure of constructing a pseudo likelihood, the formation of estimating equations, the use of sandwich estimates of variance, and piecemeal estimation of population size. Simulation provides general validation of the hidden Markov model methods developed and demonstrates their excellent performance and efficiency. This is especially notable due to the large numbers of hidden states that may be typically required

Exact Inference for Integrated Population Modelling

“…Discrete time‐series data can in principle be fitted exactly by classical inference using the efficient machinery of HMMs, without the need of the approximations used in Kalman filter analysis; see Cowen et al (), King (), King (), and Zucchini et al () for general introductions and applications of HMMs. Exact analysis also facilitates extensions such as incorporation of density dependence in ; cf Besbeas and Morgan ().…”

Estimating Northern Fur Seal Pup Production: A Case Study of Batch Mark Abundance Estimation

“…Although though there was not great computational savings, it was quite encouraging that the continuous-value approximate hierarchical model produced very similar results to the MCMC sampling of the exact model. Brintz, Fuentes, and Madsen (2018) approximation for open-population batch mark models such as those considered by Cowen et al (2017) or the known-fate assisted N -mixture models proposed by Schmidt et al (2015).…”

“…where the sum is over all possible values of the high-dimensional, unobserved X. Numerically calculating the integrated likelihood over every iteration of a maximization routine would be daunting. Cowen et al (2017) illustrate an efficient numerical method for an open population model using a hidden Markov model formulation. However, the latent structure here is not time indexed in a way that makes HMM formulation straightforward.…”

“…The SE(N ) for these rookeries is then calculated as SE(N ) = var(Â) + var (D) In this paper we propose an updated, model-based, methodology for analyzing the same batch mark data that is currently collected. The model-based approach is similar in spirit to the open N -mixture batch sampling models of Cowen et al (2017) but we consider the populations to be closed over the study, however, different proportions of the population are unavailable for sampling in each resight occasion. Thus, abundance does not follow the open population models of Dail and Madsen (2011), but rather in each resight occasion animals have a probability of not being detectable at all.…”

The score test for the two‐sample occupancy model

“…This is regularly done to estimate standard errors in complex statistical modelling Cowen et al . (). In addition, McCrea & Morgan () evaluated score tests using the expected information, adopting a property of multinomial distributions, and used numerical approximations to first‐order derivatives, rather than evaluate the expected information directly.…”