Robust Hierarchical Bayes State-Space Models for ...

Robust Hierarchical Bayes State-Space Models for Animal Movment Ian D. Jonsen Department of Biology Dalhousie University Halifax, NS, CANADA ESA Montreal, 2005

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Collaborators & Funding

The Sloan Foundation

Ransom Myers Joanna Mills Flemming Mike James Chris Field

Ian Jonsen (Dalhousie University)

Census of Marine Life Future of Marine Animal Populations

NSERC

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Argos Satellite Telemetry Data

Polar orbiting satellites Doppler shift in frequency used to estimate location Requires at least 2 uplinks 6 location quality classes more uplinks = ↑ quality of location extreme values occur

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Argos Satellite Telemetry Data Getting more out of the data

Infer true locations from noisy data Account for error w/out loss of information Infer behaviour, test hypotheses

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

5 / 23

Argos Satellite Telemetry Data Getting more out of the data

Infer true locations from noisy data Account for error w/out loss of information Infer behaviour, test hypotheses

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

6 / 23

State-Space Models

Process model true location xt+1 = f (true location xt , parameters, process variabiliy)

Observation model observed location yt = h(true location xt , observation error)

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

7 / 23

State-Space Models

Process model true location xt+1 = f (true location xt , parameters, process variabiliy)

Observation model observed location yt = h(true location xt , observation error)

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

7 / 23

Likelihood of 1st location

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Apply process model

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Observe location w uncertainty

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Integrate over predicted & observed densities (Bayes Rule)

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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State estimate at t = 1 is prior for next iteration

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Data Filtering & Parameter Estimation Jonsen et al. in press. Ecology

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Navigation Ability: Estimating the “Circle of Confusion” Flemming et al. in review. Environmetrics

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Leatherback Turtle Migration

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Examining Diel Migration Behaviour in Leatherbacks

xt = xt−1 + αd + η

Process

A one-dimensional random walk with drift model Focus only on southward movement

yt = xt +t

Ian Jonsen (Dalhousie University)

Observation

αd = migration rate at day or night

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

16 / 23

Hierarchical Bayes State-Space Model µα1 ∼ N(0, 104 )

σα1 ∼ U(0, 1)

Hyper-priors

µα1

σα1

Hyper-parameters

2 π(α1 ) ∼ N(µα1 , σα ) 1

α1,1

...

Prior

α1,14

ft (xt,1 |xt−1,1 , α1,1 , α2,1 , σ1 )

ht (yt,i,1 |xt,1 , τt,i , νt,i )

yt,1

Ian Jonsen (Dalhousie University)

Parameters

...

...

Process sampling densities

...

Observation sampling densities

...

Data

...

http://ram.biology.dal.ca/∼jonsen/

...

ft (xt,14 |xt−1,14 , α1,14 , α2,14 , σ14 )

ht (yt,i,14 |xt,14 , τt,i , νt,i )

yt,14

ESA Montreal, 2005

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Hierarchical Bayes State-Space Model µα1 ∼ N(0, 104 )

σα1 ∼ U(0, 1)

µα2 ∼ N(0, 104 )

σα2 ∼ U(0, 1)

µσ ∼ U(0, 1)

σσ ∼ U(0, 1)

µα1

σα1

µα2

σα2

µσ

σσ

2 π(α1 ) ∼ N(µα1 , σα ) 1

α1,1

...

α1,14

2 π(α2 ) ∼ N(µα2 , σα ) 2

α2,1

...

α2,14

2 π(σ) ∼ N(µσ , σσ )I(0, )

σ1

...

σ14

ft (xt,1 |xt−1,1 , α1,1 , α2,1 , σ1 )

...

ft (xt,14 |xt−1,14 , α1,14 , α2,14 , σ14 )

ht (yt,i,1 |xt,1 , τt,i , νt,i )

...

ht (yt,i,14 |xt,14 , τt,i , νt,i )

yt,1

...

yt,14

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

Ian Jonsen (Dalhousie University)

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HB SSM Model Results Diel Variation in Migration Rate

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Summary

Hierarchical state-space models optimal way to infer underlying behaviour from error-prone, complex data Models can be fit to other types of sequential movement data (GPS, Archival tags) Got Data??, http://ram.biology.dal.ca/∼jonsen/

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

20 / 23

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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dt = γT(θ)dt−1 + N(0, Σ2t )

2 yt,i = (1 − ji )xt−1 + ji xt + t(0, τt,i , νt,i )

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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Likelihood Surface Plots: Argos errors are Non-Gaussian Robust methods needed

Ian Jonsen (Dalhousie University)

http://ram.biology.dal.ca/∼jonsen/

ESA Montreal, 2005

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