uncovering animal movement decisions from positional data jonathan potts, postdoctoral fellow,...
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Uncovering animal movement decisions from positional data
Jonathan Potts, Postdoctoral Fellow, University of Alberta, September 2013
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From decision to data
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From decision to data
Movement
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From decision to data
Direct interactions
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From decision to data
Mediated interactions
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From decision to data
Environmental interactions
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From decision to data
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Movement: correlated random walkExample step length distribution:
Example turning angle distribution:
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The step selection function
• is the step length distribution,• is the turning angle distribution• is a weighting function• E is information about the environment
Fortin D, Beyer HL, Boyce MS, Smith DW, Duchesne T, Mao JS (2005) Wolves influence elk movements: Behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86:1320-1330.
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Example : Amazonian bird flocks 𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
Potts JR, Mokross K, Stouffer PC, Lewis MA (in revision) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology
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Hypotheses
1. Birds are more likely to move to higher canopies:
𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
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Hypotheses
1. Birds are more likely to move to higher canopies:
2. In addition, birds are more likely to move to lower ground:
(
𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
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Maximum likelihood technique
1. Find the that maximises:
where and are, respectively, the sequence of positions and trajectories from the data, and
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Maximum likelihood technique
2. Find the that maximises:
where is the value of that maximises the likelihood function on the previous page, and
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Resulting model
Step length distribution
Turning angle distribution
Canopy height at end of step
Topographical height at end of step
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Coupled step selection functionsOne step selection function for each agent and include an interaction term :
where represents both the population positions and any traces of their past positions left either in the environment or in the memory of agent .
Potts JR, Mokross K, Lewis MA (in revision) A unifying framework for quantifying the nature of animal interactions Ecol Lett
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Unifying collective behaviour and resource selection
Potts JR, Mokross K, Lewis MA (in revision) A unifying framework for quantifying the nature of animal interactions, Ecol Lett
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Collective/territorial models: from process to pattern
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality, Plos Comput Biol, 7(3):e1002008
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Collective/territorial models: from process to pattern
Deneubourg JL, Goss S, Franks N, Pasteels JM (1989) The blind leading the blind: Modeling chemically mediated army ant raid patterns. J Insect Behav, 2, 719-725Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality. Plos Comput Biol, 7(3):e1002008Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel Type of Phase Transition in a System of Self-Driven Particles. Phys Rev Lett, 75, 1226-1229
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Coupled step selection functions
Resource/step-selection models: Detecting the mechanisms
Model 1 Model 2 Model 3 Model 4
Positional data
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Detecting the territorial mechanism: the example of Amazonian birds
Territorial marking (vocalisations): if any flock is at position at time totherwise.
Hypothesis 1 (tendency not to go into another’s territory):
Hypothesis 2 (tendency to retreat after visiting another’s territory):
where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.
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Detecting the territorial mechanism: the example of Amazonian birds
Territorial marking (vocalisations): if any flock is at position at time totherwise.
Hypothesis 1 (tendency not to go into another’s territory):
Hypothesis 2 (tendency to retreat after visiting another’s territory):
where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.
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Amazon birds: space use patterns
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Interaction vs. no interaction
between competing models
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Classical mechanistic modelling
• Use maths/simulations to show:Process A => Pattern B
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Classical mechanistic modelling
• Use maths/simulations to show:Process A => Pattern B
• Observe pattern B
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Classical mechanistic modelling
• Use maths/simulations to show:Process A => Pattern B
• Observe pattern B• Conclude process A is causing B
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Classical mechanistic modelling
• Use maths/simulations to show:Process A => Pattern B
• Observe pattern B• Conclude process A is causing B• Logical fallacy: A=>B does not mean B=>A
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Classical mechanistic modelling
• Use maths/simulations to show:Process A => Pattern B
• Observe pattern B• Conclude process A is causing B• Logical fallacy: A=>B does not mean B=>A• Guilty! Potts JR, Harris S, Giuggioli L (2013)
American Naturalist
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New approach
• Use maths/simulations to show:Process A => Pattern B
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New approach
• Use maths/simulations to show:Process A => Pattern B
• Observe process A
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New approach
• Use maths/simulations to show:Process A => Pattern B
• Observe process A• See if pattern B follows
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New approach
• Use maths/simulations to show:Process A => Pattern B
• Observe process A• See if pattern B follows• If not, process A is insufficient for describing
data: i.e. need better model
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New approach
• Use maths/simulations to show:Process A => Pattern B
• Observe process A• See if pattern B follows• If not, process A is insufficient for describing
data: i.e. need better model• Contrapositive: A=>B means not-B=>not-A• Correct logic
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Amazon birds: space use patterns
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How close is a movement model
to reality?
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How close is a movement model
to data?
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Try to mimic regression approaches
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Try to mimic regression approaches
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Look at the residuals
Zuur et al. (2009) Mixed effects models and extensions in ecology with R. Springer Verlag
“Residual”: the (vertical) distance between the prediction and data
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More complicated than regression
• predicted positions given by the contours• is the actual place the animal moves to
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More complicated than regression
• predicted positions given by the contours• is the actual place the animal moves to
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Earth mover`s distance: a generalised residual
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Earth mover`s distance: a generalised residual
• is the actual place the animal moves to
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Earth mover`s distance: a generalised residual
• is the actual place the animal moves to
∫Ω
❑
𝑓 (𝑥|𝑦 ,𝜃 ,𝐸 )∨𝑥−𝑥0∨𝑑𝑥
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How to use the Earth Mover`s distance
Simulated movement in artificial landscape with two layers:
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Earth mover`s distance and direction
Earth mover’s distance:
is the actual place the animal moves to
Direction where:
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Wagon wheels
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Wagon wheels of Earth Mover`s distance: include direction
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Dharma wheel
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Dharma wheels of Earth Mover`s Distance
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Using simulated data with a = 1.5, b = 0x-axis: value of layer 1y-axis: earth mover`s distance (EMD)Left: EMD from model with a = b = 0Right: EMD from model with a = 1.5, b = 0
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points• Simulate your model for N steps and repeat M times, where M is
nice and big
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points• Simulate your model for N steps and repeat M times, where M
is nice and big• For each simulation, generate the Earth Movers distances to
give M dharma wheels
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points• Simulate your model for N steps and repeat M times, where
M is nice and big• For each simulation, generate the Earth Movers distances
to give M dharma wheels• Each spoke of the dharma wheel then has a mean and
standard deviation (SD)
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points• Simulate your model for N steps and repeat M times,
where M is nice and big• For each simulation, generate the Earth Movers distances
to give M dharma wheels• Each spoke of the dharma wheel then has a mean and
standard deviation (SD)• Generate a dharma wheel for the data
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A scheme for testing how close your model is to “reality” (i.e. data)
• Suppose you have N data points• Simulate your model for N steps and repeat M times,
where M is nice and big• For each simulation, generate the Earth Movers
distances to give M dharma wheels• Each spoke of the dharma wheel then has a mean and
standard deviation (SD)• Generate a dharma wheel for the data• If any spoke of the data dharma wheel is not of length
mean plus/minus 1.96*SD from the simulated dharma wheel then reject null hypothesis that model describes the data well
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Normalised earth mover`s distance
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Acknowledgements
Mark Lewis (University of Alberta)
Karl Mokross (Louisiana State)
Marie Auger-Méthé (UofA)
Phillip Stouffer (Louisiana State)
Members of the Lewis Lab
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Movement and interaction data
Mathematical analysis Simulations/IBMs
Coupled step selection functions
Conclusion
“To develop a statisticalmechanics for ecological systems” Simon Levin, 2011
Spatial patterns
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Thanks for listening!