[PPT] - Exact Bayesian Inference for Animal Movement in Continuous Time PowerPoint Presentation

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Exact Bayesian Inference for Animal Movement in Continuous Time

Paul Blackwell

University of Sheffield

December 2015

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Outline

◮ Animal movement in continuous time ◮ Switching diffusion models ◮ Introduction to applied examples ◮ Exact simulation from continuous time models ◮ Exact Bayesian inference using MCMC ◮ Revisit examples ◮ Extension to multiple animals

Mostly joint work with Mu Niu, now at University of Glasgow; was at Sheffield, funded via National Centre for Statistical Ecology.

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Animal movement

Interested in understanding movement behaviour of individuals. Increasing amount of data from radio-tracking, GPS tagging, etc. Typically data consist of ‘fixes’ at regular times—interval varies within and between studies, even on the same species. View movement as a continuous-time process observed discretely.

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Diffusion models

Movement models could be defined through stochastic differential equations, but here build from simple tractable cases:

◮ Brownian motion (perhaps with constant drift α);

x(s + t)|x(s) ∼ MVN(x(s) + tα, tΣ)

◮ Ornstein-Uhlenbeck (OU) process—essentially Brownian

motion with attraction towards a centre µ (stationary). x(s + t)|x(s) ∼ MVN(µ + eBt(x(s) − µ), V − eBtVeB′t)

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Multi-state/switching diffusion models

For realism we need behavioural switching: animals switch between behavioural states with different movement characteristics. For example, wild boar seem to spend most of the night foraging, then head back to a nest, then spend the day sleeping. Want to allow rate of switching to depend on spatial covariates, and perhaps also time of day (seasonality) or time since switching (semi-Markov case).

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Let x(t), s(t) represent position and behavioural state at time t. Let λij(t, x) represent switching rate from behaviour i to behaviour j at time t, location x, so Pr(s(t+δt)=j|s(t)=i) = λij(t, x)δt + o(δt), i = j. If λij(t, x) ≡ λij, the behaviour just follows a continuous time Markov chain. Realistically, we want to let λij(t, x) depend on t, x.

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Example: fishers

Data on a fisher, Martes pennanti.

Picture by Scott LaPoint.

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Data on a fisher, Martes pennanti, in urban USA, from Scott LaPoint, Max Planck Institute for Ornithology. Locations recorded every 10 minutes using a GPS collar. US National Land Cover Database gives discrete set of known habitat types, 30m grid: ‘developed open space’ (central band, orange); ‘deciduous forest’ (mostly bottom left, green); ‘woody wetlands’ (mostly top right, blue).

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20m

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Simple movement model: finite set of known different habitats; individual switches to an appropriate mode of movement in each. n habitats, n behaviours; time homogeneous. λij(t, x) = λijk when x ∈ Ak, k = 1, . . . , n. λijk = 0 when j = k. n OU processes with common µ, separate Bi = biI, Vi = viI.

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Example: wild boar

Data on a female wild boar, Sus scrofa.

Picture by Michael G¨ abler.

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Data on a female wild boar, Sus scrofa, from Mark Lambert, Animal and Plant Health Agency. Nocturnal population in the UK. GPS collars, with fixes every 15 or 30 minutes (night/day). More frequent observations correspond roughly to period of activity.

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200m

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An individual uses n nest sites; use model with 3 types of state, 3n actual states. Resting, states 1 to n at nests 1 to n; foraging, states n + 1 to 2n; returning to nest, states 2n + 1 to 3n. Movement in each state is represented by a different OU process; n different centres of attraction, 3 different dynamics.

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Transition rates for wild boar

Transition, resting to foraging: similar—not identical—time each day. hα(t) = h(t mod 1; tα, α), i = 1, . . . , n where h(·; tα, α) is the hazard function of a logistic distribution. Transition, foraging to returning: same form, different parameters. Transition, returning to resting: occurs rapidly once close enough to nest.

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Generator Λ(t, x) = (λij(t, x)).                  −hα(t) hα(t)/n · · · hα(t)/n ... . . . . . . −hα(t) hα(t)/n · · · hα(t)/n −hβ(t) hβ(t) ... ... −hβ(t) hβ(t) γI{x∈Bi} −γI{x∈Bi} ... ... γI{x∈Bi} −γI{x∈Bi}                 

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Simulation

Let λj(t, x) =

i=j λij(t, x) represent rate of switching away from

behaviour j given t, x. But if λj(t, x) really depends on x, then we don’t know the time of the switch unless we know x(t), at least up until the switch occurs. Obvious approach is to approximate by a discrete-time representation—short time-steps. (How short? How approximate?)

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Exact solution

Provided rates bounded, simulation can be done exactly i.e. no time discretisation. Let κ = maxj,t,x{λj(t, x)}. Start at s(0) = j, x(0) = x. Let T ∼ Exponential(κ); this is the first potential switch. Simulate X( T) forward from x with movement model j. There is an actual switch at T with probability λj( T, x( T)) κ . Can continue as a dynamic thinning of a Poisson(κ) process.

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(a) 19

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(b)

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(c)

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(d)

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Statistical inference

Usual statistical approach is to treat time as discrete or, at best, approximate the continuous-time model at times of observations. We can use the previous algorithm to simulate path and behaviour conditional on discrete observations, for exact Bayesian inference. Simple Markov chain Monte Carlo algorithm: just simulate as before, and accept with probability that depends on f (x(t)|x(TM), s(TM)) i.e. on how likely data point is, given forward simulation to previous potential switch. Reject if behaviour is wrong!

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Smarter proposals

MCMC proposals don’t have to come from ‘forward simulation’. Instead, can propose behaviour over time from a simpler spatially homogeneous ‘proxy process’. Formally, the full model needs to be absolutely continuous w.r.t. proxy process times Brownian motion. If proxy process is spatially homogeneous, don’t need to generate locations as we go along. Suggests a Markov chain on states, with its transitions—our ‘potential switches’—spread over time using a Poisson process. Formally, a Markov chain subordinated to the Poisson process.

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Generate behaviour from proxy process; generate locations given behaviour and observed locations using the conditioned movement model e.g. (time-inhomogeneous) Brownian/OU bridge. Compare proxy process with true behaviour model, along with probability of observed locations, to get acceptance probability. Pragmatically:

◮ faster to generate; ◮ conditions more efficiently on observed locations; ◮ easier to condition on observed behaviour; ◮ still exact.

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Fisher results

Clear difference between open space (habitat 1) and others. Model with habitat effect fits better than one without; ∆DIC ≈ 154. Less clear whether difference between habitats 2 and 3. Differences robust to details of model.

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−4 −2 2 4 6 −4 −2 2 4

log(vi) log(bi)

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Wild boar results

For this analysis, nest locations taken to be known. Clear difference between all three types of state. Clear differences between rates and timing of types of transitions. Most observations have ‘known’ behaviour; some uncertainty. Interpretation of types of states not so clear.

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6 8 10 12 14 −6 −4 −2

log(vi) log(bi)

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Extension: multiple animals

Leader, real or conceptual: (Lx, Ly)

′ c.f. Langrock et al. (2014).

Followers: (F x

j , F y j )

′, j = 1, . . . , n.

Whole system: Y = (Lx, F x

1 , . . . , F x n , Ly, F y 1 , . . . , F y n )

′.

Followers are attracted linearly to the leader; OU process with a variable centre. If leader also follows OU process, so does Y, in R2(n+1). Parsimonious way to represent dependence.

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