Modelling Spatial Trajectories

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Modelling Spatial Trajectories David R.
Brillinger Statistics Department University of
California, Berkeley 2? ? 1 Math Stat, Mac,
20 November 2008
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1.1 Introduction. Lotation r (x,y) Time
t Trajectory r(t), -? lt t lt ? Trajectory data
r(ti), i1,...,n, and the ti Random
trajectory r(t,?), ? random
variable Difference, differential equations,
potential functions assessing websites,
player movement, iceberg motion, bird navigation,
hazard exposure, ocean drifters, wildlife
movement, image scanning, ...
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1.2 History and examples 1.2.1 Planetary motion
Brahe (data) Kepler (laws) Newton (DEs)
Lagrange (potential function) G/r0 r
, G gravity
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1.2.2 Brownian motion Robert Brown
(1827) Bachelier (1900) Einstein
(1905), Smoluchowsky (1905) Langevin (1908)
SDE
X "complimentary force", stochastic
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Perrin (1913)
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1.2.3 Monk seals. Hawaiian (Monachus
schauinslandi) 2.2m, 250kg, life span 30
yr Endangered species, 1400 Risks
environmental change, habitat modification,
reduction in prey, humans, random fluctuations
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1.2.4. 2006 World Cup in Germany Argentina
vs. Serbia-Montenegro game
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1.2.5 Elk. Starkey Reserve. Explanatory x(t)
(ATV) Can elk, deer, cows, humans coexist?
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1.2.6 Whale shark. Vulnerable to
extinction Filter feeders eat krill,
Largest living fish Tropical and
water-temparate regions One was 41.50 ft
long, 47,300lb, with girth 23.0 ft.
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OZ
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1.3 Statistical concepts and models. 1.3.1
Displays (x(ti),y(ti)), i1,2,... joined by
straight lines plus background
estimated speed vs. (ti1ti )/2 bivariate
density estimate bagplot
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1.3.2 Autoregressive models simple random
walk rt1 rt et1, t0,1,2, ...
VAR(1) rt1 art et1
M. Moore VARIMA(p,2,0) for iceberg tracks
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1.3.3 Stochastic differential equations (SDEs).
B(t) bivariate Brownian
B(tdt) B(t) dB(t) dr(t)
µ(r(t),t)dt s(r(t),t)dB(t) µ drift
s diffusion
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How is an SDE defined? integral equation r(t)
?0t ?(r(s),s)ds ?0t ?(r(s),s)dB(s)
Take l.i.m of discrete approximation,
tj Sometimes there is a stationary
distribution, p(r)
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1.3.4. The potential function approach
Gradient system µ
-(?H/?x,?H/?y)T - grad H
Steepest descent equation
µ shows the direction of movement
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Sometimes log p(r) ? -2 log H(r)
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1. Meandering particle. H constant 2.
Attraction. To point a, H(r) ar-a2, O-U
½s2log r-a - dr-a , bird navigation To
region, take a nearest point in region 3.
Repulsion. From point a, H(r) r-a-2 From
region, a nearest point 4. Quadratic. H(r) ?1x
?2y ?11x2 ?12xy ?22y2
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• SDEs - approximation/simulation
• (r(ti1) r(ti))/(ti1 ti) ?(r(ti)) sZi1
• (r(ti1) r(ti))/(ti1 ti) -(?H/?x,?H/?y)T
  sZi1
• Zi i.i.d. N2
• Data. tj,(x(tj),y(tj)), where r(t)
  (x(t),y(t))T

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1.4 Inference methods. Suppose µ has form µ(r)
g(r)ßT Stack (r(ti1) - r(ti))/?ti1-ti to
form Yn Stack µ(ri,ti)?ti1-ti to form Xn Stack
sZi1 to form en then Yn Xnß en Consider
sZi1 form martingale difference sequence wrt
s-field Fi generated by r(t1),,r(ti)
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Theorem A.1. Lai and Wei (1982). Suppose
supn E(enaFn-1) lt 8 a.s. for some a gt
2. Suppose limn?8 varenFn-1) s2 a.s. for
some nonstochastic s. Write X Assume that xn
Fn-1-measurable r.v. and existance of non-random
positive definite symmetric L by L matrix Bn for
which Bn-1(XnTXn)1/2?I and sup1in Bn-1xn?0
in probability.Then
(XnTXn)1/2(b-ß) ?N(0, s2I), in distribution as
n?8. 
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CI for f(r)T ß Theorem A.2. Under the assumptions
of Theorem A.1 and lim log ?max(XnTXn)/n?0 almost
surely, one has ((f(r)(XnTXn)-1f(r)T)-1/2f(r)T
(b-ß)/sn ? N(0,1) in distribution as n?8. sn
((n-1)p)-1RSS
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Assessment. How to check for gradient system?
?Hy/?x ?Hx/?y ? Model µ of dr(t) µ(r(t),t)dt
s(r(t),t)dB(t) then estimate partials µ1y , µ2x
and compare
How to check full model? Generate synthetic
plots. Compare to the data. Neyman and Scott
(1956)
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1.5 Difficulties that can arise. Excessive
gaps in data Outliers Location error
Memory Nonlinearity ...
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1.6 Results for the empirical examples. 1.6.2
Perrins data, H(r) quadratic OLS estimate of H,
ignoring r1
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Tricky though, the Langevin model,
doesn't fit in directly
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1.6.3 Monk seal. H(r,t) - two points of
attraction, one offshore, one onshore Potential
function quadratic C/dM(x,y) Attractor a
changes, write a(t) C keeps the animal off Molokai
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Synthetic journey.
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1.6.4 Elk Experiment. Explanatory, x(t) 8
animals ATV days, ?t 5min for elk
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dr(t) µ(r(t))dt ?(r(t)-x(t-t))dt sdB(t)
x(t) location of ATV at time t t time lag
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1.6.5 World Cup. H(r) ? log r ? r ?1x
?2y ?11x2 ?12xy ?22y2 Use r a0, a0
closest point of goalmouth
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Vector field
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1.6.6 Whale shark SST first -day
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SSH
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Ocean surface Chlorophyll-a
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With H(x,y) temperature, current, chlorophyl
surface or all fit (r(ti1) r(ti))/(ti1 ti)
-(?H/?xi,?H/?yi)T sZi1
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1.7 Other aspects/approaches models.
Stochastic functional differential equations
memory beyond most recent allowed Rt
denotes history r(u), -8 lt u t dr(t)
µ(Rt)dt s(Rt)dB(t) For inference
(r(ti1) - r(ti)) µ(Rti)(ti1 - ti) s v(ti1
- ti)Zi1
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Consider
When the ti are equispaced ...
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weak convergence of approximations Levy
processes Markov chain approximation
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Summary. Flexble approach - H is scalar-valued
Argentina went down one side of the field
Perrin particle attracted to a region Monk
seal between food and safety
- inside Penguin Bank -
synthetic reasonable Elk - ATV had an effect
Whale shark temperature, current,chlorophyl
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Acknowledgements. Ager, Brown, Kie, Littnan,
Mendelssohn, Panaretos, Preisler, Spector,
Stewart , Wisdom
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Further examples and extensions Several
particles Risk of collision - space debris
tracking Control Approximation errors
Markov chain approximations Comparison
Simulation Prediction Model appraisal
Handling explanatories