Marine Mammal Movements: Data Analysis and Theory David R ... · (i.e. less than 200 fathoms deep -...
Transcript of Marine Mammal Movements: Data Analysis and Theory David R ... · (i.e. less than 200 fathoms deep -...
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Marine Mammal Movements: Data Analysis and Theory
David R. Brillinger Statistics Department
University of California, Berkeley
www.stat.berkeley.edu/~brill/marinepapers.html
Peter Wall Institute University of British Columbia
Vancouver, BC
May 4-8, 2015
East Sooke Park
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Brent Stewart, Hubbs
Fairbanks 1992
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Structure of presentation: 4 examples
Introduction
. “The science of statistics is essentially a branch of Applied Mathematics, and may be regarded as mathematics applied to observational data.” R. A. Fisher (1925)
Scientific work is often motivated by concepts and
methods from physics and mathematics
Pertient names: Newton, Einstein, Smoluchovsky,
Langevin, Wiener, Chandresekhar, Nelson, Kendall, …
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Beginning models abstract and mathematical but they
motivate discrete time, programmable, checkable models.
Defence: providing answers to broad variety of questions.
e.g. interventions existing?, predicted locations, change?
Goal is to estimate a potential function whose gradient
appears in a (linear) model.
Then can use many regression results
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The mammals
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Interpretation of a potential function
Motion described by SDE or gradient of a potential function
rolling ball bearing
types of potential functions: attraction, repulsion, time
dependent, attractor-repellor, periodic, inverse power,
polynomial, isotropic,
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||r||α
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Path/track data from: elephant seals, monk seals, elk,
whale shark tag
Reasons for study: endangered, coexistence possible?,
discovery, management, prediction, change, outside
influence?
Exploratory analytic method
Building on historical background and available statistical
methods (EDA, least squares)
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elk
Data
Free floating tag
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Assuming friction, γ, is high
velocity dR/dt ~ -grad Б (r,t) + Ψ(t)
Stochastic calculus
Langevin-Chandresekhar equation
cp. F = ma
acceleration
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Some results
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Example 1. Northern elephant seal (Mirounga angustirostris)
Were virtually extinct
Size male 2000 kg female 600 kg
Exceptional navigators
Most of year at sea
Double annual migrations
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Able to assess position by astronmical or magnetic field and
correct course???
forage continuously
EDA: discovery (visualization), need for robust/resistant methods
Great circle?
Particle movement models
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Model 1. patricle model Random walk on sphere with drift, (lat,
long) = (θ,φ)
D. G. Kendall model for birds, SDE
Change of variables so heading to North Pole
Equation of motion
Potential function
H(θ, φ) = -½ σ2 log sin θ.
point of attraction North Pole
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Inference
-2 log like
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Surprises . Brownian with trend on sphere, new model,
path can be great circle
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Discussion.
great-circle path hypothesis not contradicted
keep going straight ahead
one northern elephant seal female
Suggests seals can have a destination when departing from an
origin
natural selection has favoured development of neural and
sensory mechanisms
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Example 2. Monk seal (Monachus schauinslandi
Endangered. Now numbers around 1100
Key factor in recent decline poor survival of juveniles
hypothesized related to poor foraging success
Basic motivation to learn where animals go to forage vertically and
geographically.
Information needed for management and conservation purposes.
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Example 2. Monk seal (Monachus schauinslandi
Endangered. Now numbers around 1100
Key factor in decline poor survival of juveniles
Hypothesized related to poor foraging success
Which geographic and vertical marine habitats seals use?
What habitats are essential, with some buffer, to the survival and
vitality of this species?
Are there age and sex differences in habitats used when foraging?
Do seals have individual preferences in foraging locations and does
an individual vary its behavior over different time scales?
how long is a foraging trip?
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EDA bagplot (bivariate boxplot)
Surprise: Penguin Bank
To begin: EDA scatter plot of GIS positions
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Model 2
drt = - grad U(rt )dt + Σ dBt
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Foraging trips
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Second seal male
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Discussion.
Particular marine habitats attractors to foraging monk seals.
Here foraging habitats confined to relatively shallow offshore
bathymetric features
(i.e. less than 200 fathoms deep - Penguin Bank)
Time seal spent foraging appeared constrained by powerful
attractor associated with periodic resting ashore (i.e., terrestrial
haulout habitat).
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Example 3. Elk (cervus elaphus)
US Federal land managers examined effects on Rocky Mountain elk
of forest management, domestic livestock grazing
Starkey Project initiated in northeastern Oregon
9000 ha fenced area
Experiments using locations of elk, deer and cattle
continuously monitored
Problem of interest: description of movement of free-ranging
animals.
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ModelI. gradient system (Skorokhod).
Note ri – rj
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The potential function form employed is
β1x + β2y + β3x2 + β4xy + β5y
2 + β6x3 + β7x
2y + β8xy2 + β9y3
with (x,y) denoting location. The gradient is
1 0 2x y 0 3x 2 2xy y 2 0
( ) 0 1 0 x 2y 0 x 2 2xy 3y 2
matrix multiplied by the transpose of the row vector
(β1 β2 β3 β4 β5 β6 β7 β8 β 9)
XβT : linear combination of β’s
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Attraction plus bias to SE
Note ri – rj
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ModeIII. gradient syetem
Note
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Distance apart
Expanding layers
Note || ri – rj||
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Discusion
The last two models concern a particle being
influenced by another particle of the same type or by a
lagged particle of a different type.
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Further models.
Two models concerning particle being influenced by another
of same type or by a lagged particle of different type (hunter).
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Example 4 driftingtag.pdf
Based on surface drifting movement of a small satellite-linked radio
transmitter tag.
Goal: to compare its movements (direction and velocity with
direction and velocity of sea surface currents estimated
independently from gradient of sea surface height..
Daily estimates of th tag’s locations determined from transmissions
received at irregular times by polar-orbittinf satellites
Second goal developing presictive model using past tag locations,
the currents and winds
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form is V(r) = γ1x + γ2y + γ11x2 + γ12xy + γ22 y
2 + C/dM
where dM = dM(x, y) is the distance from location (x, y) to the
nearest point of a region, M,
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Uses
intervention analysis
prediction
change
association
residual assessment
regression results
. explanatories
measurement error
…
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Difficulties.
Boundaries and other objects
Island/line – use nearest point
slopes
Outliers
Different (iregular) times for different animals
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Future work variance stabilizing transform
1. Acceleration model (Data anyone?)
2. Effects of sound – add terms to movement model
e.g. moving (pressure) wave g(αx + βy – γt)
Does g have an effect? Is there change?
Example filtered sonar signal
3.Study many animals at the same time (herd)
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Summary.
EDA discoveries: great circle, Penguin Bank,
clustering, testing NOAA values, Brownian with trend
on sphere
elephant seal
SDE model
monk seal
potential function model
two elk
three potential models
free floating tag
explanatories derived from a potential
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Acknowledgments
I thank my collaborators A. Ager, C. Littnan, H. Preisler,
and B. Stewart
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Newton’s Second Law [PUT TO END?]
Langevin (1908) “… trajectory of a particle …”
v = dR/dt, mdv/dt = - γ v + Ψ(t)
particle: marine mammal, m: mass
γ: friction coefficient: dv/dt: acceleration
m dv/dt: momentum: Ψ: random forces
Chandresekhar (1943) adds potential
K(R(t),t) = – (δ /δx, δ /δy)Б(R(i),t)
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Stochastic calculus
t: continuous time ti : increasing discrete times
r(t): location at time t
Random walk
r(ti+1) - r(ti) = IN2 (0, (ti+1 – ti )Σ )
Brownian B(t) t continuous
disjoint increments B(I) ~ IN(0, |I|)
dY(t) = Y(t+dt) – Y(t)
Random walk with linear drift SDE
dY(t) = (α + βt)dt + Σ dB(t)
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Variance-stabilizing transform
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Details of computations
Robust/resistant methods
EDA – visualization
lm, mgcv
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Appendix
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δ speed towards the origin.
Potential function
H(φ,θ) = ½σ2 log sin θ - δθ
point of attraction - North Pole
Residuals
Measurement error