Uncovering ecological state dynamics with hidden Markov models

Abstract

Ecological systems can often be characterised by changes among a finite set of underlying states pertaining to individuals, populations, communities or entire ecosystems through time. Owing to the inherent difficulty of empirical field studies, ecological state dynamics operating at any level of this hierarchy can often be unobservable or 'hidden'. Ecologists must therefore often contend with incomplete or indirect observations that are somehow related to these underlying processes. By formally disentangling state and observation processes based on simple yet powerful mathematical properties that can be used to describe many ecological phenomena, hidden Markov models (HMMs) can facilitate inferences about complex system state dynamics that might otherwise be intractable. However, HMMs have only recently begun to gain traction within the broader ecological community. We provide a gentle introduction to HMMs, establish some common terminology, review the immense scope of HMMs for applied ecological research and provide a tutorial on implementation and interpretation. By illustrating how practitioners can use a simple conceptual template to customise HMMs for their specific systems of interest, revealing methodological links between existing applications, and highlighting some practical considerations and limitations of these approaches, our goal is to help establish HMMs as a fundamental inferential tool for ecologists.

Keywords: Behavioural ecology; community ecology; ecosystem ecology; hierarchical model; movement ecology; observation error; population ecology; state-space model; time series.

Figure 1

System state processes that can be difficult to observe directly, but can be uncovered from common ecological observation processes using hidden Markov models. The state process (blue) can pertain to any level within the ecological hierarchy (‘Individual’, ‘Population’, ‘Community’ or ‘Ecosystem’) and for convenience is categorised as primarily ‘Existential’, ‘Developmental’ or ‘Spatial’ in nature. The observation process (green) can provide information about state processes at different levels of the hierarchy (green lines) and includes capture–recapture, DNA sampling, animal‐borne telemetry, count surveys, presence–absence surveys and/or abiotic measurements. Observation and state processes from lower levels can be integrated for inferences at higher levels. For example, community‐level biodiversity data could be combined with environmental data to describe ecosystem‐level processes.

Figure 2

Dependence structure of a basic hidden Markov model, with an observed sequence $X_1,\dots ,X_T$ arising from an unobserved sequence of underlying states $S_1,\dots ,S_T$.

Figure 3

Estimated state‐dependent distributions (top row) and Viterbi‐decoded states from a two‐state HMM fitted to counts of feeding lunges performed by a blue whale during a sequence of $T=53$ consecutive dives. Here the most likely state sequence identifies periods of ‘low’ (state 1; blue) and ‘high’ (state 2; black) feeding activity.

Figure 4

Graphical models associated with different extensions of the basic HMM formulation: (a) state sequence with memory order 2; (b) influence of covariate vectors ${\mathbf{z}}_1,\dots ,{\mathbf{z}}_T$ on state dynamics; (c) observations depending on both states and previous observations; (d) bivariate observation sequence, conditionally independent given the states.

Figure 5

Illustration of a possible workflow when using an HMM to infer behavioural modes from the vector of dynamic body acceleration data of a striated caracara (Phalcoboenus australis) over a period of 60 min (see Fahlbusch & Harrington, 2019, for data details). Four behavioural modes were identified and biologically interpreted to be associated with resting (yellow), minimal activity (orange), moderate activity (blue) and flying (green).

Figure 6

Number of publications (left axis) and total number of times these publications were cited (right axis) per year based on a Web of Science search for ‘hidden Markov’ conducted within the categories of ‘Biology’, ‘Ecology’, ‘Marine Freshwater Biology’ and ‘Zoology’ on 7 July 2020.