Exact Inference for Integrated Population Modelling

Abstract

Integrated population modelling is widely used in statistical ecology. It allows data from population time series and independent surveys to be analysed simultaneously. In classical analysis the time‐series likelihood component can be conveniently approximated using Kalman filter methodology. However, the natural way to model systems which have a discrete state space is to use hidden Markov models (HMMs). The proposed method avoids the Kalman filter approximations and Monte Carlo simulations. Subject to possible numerical sensitivity analysis, it is exact, flexible, and allows the use of standard techniques of classical inference. We apply the approach to data on Little owls, where the model is shown to require a one‐dimensional state space, and Northern lapwings, with a two‐dimensional state space. In the former example the method identifies a parameter redundancy which changes the perception of the data needed to estimate immigration in integrated population modelling. The latter example may be analysed using either first‐ or second‐order HMMs, describing numbers of one‐year olds and adults or adults only, respectively. The use of first‐order chains is found to be more efficient, mainly due to the smaller number of one‐year olds than adults in this application. For the lapwing modelling it is necessary to group the states in order to reduce the large dimension of the state space. Results check with Bayesian and Kalman filter analyses, and avenues for future research are identified.

Uncovering ecological state dynamics with hidden Markov models

“…binning abundance states together into larger states; Zucchini et al, 2016, pp. 162-163;Besbeas and Morgan, 2019). Appropriateness will depend on the sensitivity of the inference to the precise value of the state process and is best investigated by varying the coarseness of the approximation.…”

“…For example, behavioural observation could be modelled using a categorical distribution (MacDonald and Raubenheimer, 1995), count using a non‐negative discrete distribution (e.g. Poisson; Besbeas and Morgan, 2019, and measurement using a non‐negative continuous distribution (e.g. zero‐inflated exponential; Woolhiser and Roldan, 1982).…”

“…sex or age classes; see Population level ‐ Developmental state). The state‐dependent distributions can take many different forms depending on the specific observation process, but common choices for count data are binomial or Poisson models (Schmidt et al ., 2015; Besbeas and Morgan, 2019). Sometimes count data alone can be insufficient for describing complex population processes, and this has led to integrated population modelling (Schaub and Abadi, 2011) that uses auxiliary data such as capture–recapture, telemetry or productivity data (Schmidt et al ., 2015; Besbeas and Morgan, 2019).…”

“…Multiple states and processes can be represented: age classes/survival, size classes/growth, sex/birth, genotypes and metapopulations are all states or networks of states with specified connections (Newman et al ., 2014). Such HMMs can be informed by a wide variety of population‐level observations, for example counts of plants (Borgy et al ., 2015) or animals (Schmidt et al ., 2015), as well as auxiliary individual‐level observations (Besbeas and Morgan, 2019). From this general viewpoint, HMMs can be seen as the structure behind open population N‐mixture models (Schmidt et al ., 2015; Cowen et al ., 2017), distance sampling models (Sollmann et al ., 2015) and approximate state‐space population dynamics models (Besbeas and Morgan, 2019).…”

A general framework for modeling population abundance data

“…An adaptive bin width was needed when bootstrapping to test for density dependence as large numbers can arise when simulating from the Gaussian random walk with drift model of Equation . See also Besbeas and Morgan () for a two‐dimensional state‐space application. It is also necessary to determine the range for the state variable.…”