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Geolocation of Icelandic Cod using a modified Particle Filter Method
David BrickmanVilhjamur Thorsteinsson
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What does one do when
…
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• Note that the T simulation is good, but the recapture estimate is way off
• Note that track goes into deep water – not considered likely for Icelandic Cod
• Varying parameters improves results but not by that much.
The best that a “standard”
particle filter can do is
DST recap position
model recap position
DST tag position
model simulation
600m200m
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Why does this occur??
• T field around Iceland is ~ parabolic so that particles drifting from tag location, and trying to follow T data, have 2 possible directions to choose.
Temperature field ~ parabolic
Climatological September T at 100m
• Aside: T field for this study comes from a state-of-the-art circulation model for the Iceland region developed by Kai Logemann
(Logemann and Harms Ocean Sci., 2, 291–304, 2006)
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OPTIONS 1. Accept that this is the best that the PF
method can do and
• Do nothing
• Hide these results (~5 out of 27)
2. See whether modifications to the PF method can produce better simulations
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The Data: 27 useable DSTs
Example of tag being inserted into cod fish
(from Star-Oddi website)
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Example of DST data
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Movement model:
Udtdd
)(Vxxxx n1n
Where
• xn = (lon,lat) position at time n = the “state”
• V = (max) swim velocity = model parameter
• U = random # from uniform distribution
• dx = change in (lon,lat) position
• dt = timestep
“Standard” Particle Filter (PF-1)(Andersen et al. 2007, CJFAS 64:618-627)
dxVdt
xn
xn+1
• particle z-level = DST z-level
Particles start at the initial tagging position, and evolve according to a
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Observation model:
nn gy x
Where
• yn = observation at time n (i.e. temperature) from the DST
• NB: last time includes the “recapture” observation
• = error
• g(x) is the model temperature field at x derived from a numerical circulation model
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Error Model -- Particle Filter Weight:• Standard assumptions for a SIR filter yield:
ni
nni yPw x|
model| niobs
nni
n TTyP x
• The probability of the observation given the state is
• and following Andersen et al. (and others):
model;parameter
),0;(n
iobsnn
iT
Tn
ini
TTT
TNw
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Particle Filter: “PF-1”• At t=0 NP particles are seeded at the (known) DST tagging position
• Each particle evolves according to the movement model
• (A) At each timestep evaluate P particle filter weights w
• (B) Sample with replacement NP particles from w, preferentially choosing those with higher probability (i.e lower error). Use the standard SIR cumulative distribution method.
• (C) Propagate these particles to the next step
• Repeat A-C
• Continue to end of series, at which time the recapture position is an (important) observation to be incorporated into P.
• NB: no backward smoothing procedure coded.
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Example of a Good Result
Standard PF
PF-1
However, note that offshelf drift is not
considered biologically realistic
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Modifications to standard PFTwo modifications added:
• “Attractor” function: To increase the influence of the final (recapture -- R) position, a time-dependent term was added to the error model:
a
ni
ni RWa
time0)-(timetanh1~2x
distance from recapture position
factor that increases as final time is approached
• Allows a future observation to influence present state
• Adds 2 parameters: time0 and a
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Interpretation of Attractor term
Consider 2 particles returning the same T (i.e. T-error) – late in the simulation
• The estimate reported by particle 2 is considered more likely because it is closer to the recapture position.
1 2
recap position
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• Demersal error term:
• Intended to correct the tendancy for particles to follow increasing temperatures by drifting southward
• Observed in many simulations but considered biophysically unlikely.
• where zni is the depth of the i-th particle at time n (=
DST depth) and zbtm(xni) is the model bottom depth at
location xni
• d is a vertical scale parameter
d
niWd
)(z-z-exp1~
nibtm
ni x
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Interpretation of Demersal term
• Assumes that the school of fish are clustered within d of the bottom and penalizes those fish that do not fit into this “demersal” vertical distribution.
Consider 2 particles at the same depth, reporting the same T
• the estimate reported by particle 2 considered more likely as that fish is exhibiting a more demersal behavior.
• Action ~ negative diffusion
12
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• New terms incorporated in an error distribution (at every timestep, for each particle):
• E = {T-error + attractor-term + demersal-term}
i.e. additive error distribution of un-normalized error terms, sampled using a SIR-type procedure;
New Error Model
Preferentially choose particles with lowest error
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How to think about this -- Heuristically
• For this type of problem (i.e. DST) the backward smoothing procedure is essential as it is the way that the recapture observation influences the solution:
• Up to recap obs, PF yields optimal “local” solution
• Use of backward smoothing produces optimal “global” solution.
• E-distribution “attempts” to solve the global problem in one pass through the data.
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• Regarding E -- Consider minimizing a likelihood function over all observations:
E(arg))log(
exp(arg)
)log()|(log1
n
nn
N
n
n
w
pdfw
wyL
• BTW: solved L(y|) for optimal parameters using a Direct Search algorithm.
• DS algorithm: see Kolda et al. 2003, SIAM V.46, no.3, pp.385-482
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Note that the Demersal term could be incorporated into the Movement Model by including bathymetry:
shallow
deep
shallow
deep
dxVdt
xn
xn+1 Present model
Model using bathymetry
dxVdt
xn
xn+1
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Results
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Addition of Attractor function
only
(PF-2)
Cf: no attractor
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Addition of Attractor function
plus
Demersal term
(PF-4)
Cf: attractor only
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Comparison of PF-1 versus PF-4 (NB: Different DST)
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Summary / Conclusion• Standard PF seen to perform poorly on a number of DSTs
• PF method modified by adding:
• Attractor term that “sucked” particles toward the recapture position
• allows future data to influence present result
• Demersal term that favoured particles that adhered to a “gadoid-type” behavior
• keeps particles onshelf
• Attractor + demersal terms can be considered to be rules or behaviors imposed on the particles.
• Result likely depends on Temperature field:
• demersal term may not be necessary
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Forcing better biological behavior (addition of demersal term) resulted in poorer simulation of temperature
timeseries.
i.e. quantitatively WORSE results
“Best” result is subjective
OR
• Modified PFs performed better than standard PF, especially on difficult DSTs.
• However,
When signal processing theory meets fisheries biology adjustments may have to be made