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Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 3 David R. Brillinger University of California, Berkeley 2   1. Question. Why does time exist? . If it didn't, then everything would happen at the same time.

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slide1
Cycle Romand de Statistique, 2009

Ovronnaz, Switzerland

Random trajectories: some theory and applications

Lecture 3

David R. Brillinger

University of California, Berkeley

2 1

slide2
Question. Why does time exist?

If it didn't, then everything would happen at the same time.

Einstein?

slide3
Lecture 3: Further analyses / special topics

(moving) explantories

calibration

boundaries

several objects. particle processes/systems

trajectories on surface (?)

slide4
Rocky Mountain elk and ATV.

Starkey Reserve, Oregon, NE pasture

April-October 2003

slide5
Concern: effects of human invaders, e.g. ATV on animals' behavior

animals ranged in a confined region

2.4 m high fence

8 GPS equipped

t for elk - 5min, t for ATV - 1sec

randomization in treatment assignment

[SDE drift term depends on location of elk and intruder]

Brillinger, Preisler, Ager, Wisdom (2004)

slide8
Model.

SDE

dr(t) = m(r)dt + dB(t)

m = control, atv

Estimated 's

slide12
Model.

dr(t)= μ(r(t))dt + υ(|r(t)-x(t-τ)|)dt + σdB(t)

x(t): location of ATV at time t

τ: time lag

Plots of |vτ| vs. distance |r - xτ|

slide14
Discussion and summary.

model fit by gam()

apparent increase in elk speed at ATV distances up to 1.5km

an experiment

method useful in assessing animal reactions to recreational uses by humans

slide15
Whaleshark tagging study. Off Kenya

Data collected to study ecology of these fish, e.g. where they travelled, foraged, and when?

How to protect?,

Brent

slide16
29 June - 19 July, 2008.

Indian Ocean

Locations for tag from instrumented shark

Unequally spaced times, about 250 time points

Tag released, drifted til batteries expired

Our (opportunistic) purpose: to calibrate sea surface current models results

slide19
Remote sensed data: sea surface heights, zonal and meridional currents

Ocean Watch Demonstration Project's Live Access Server

http://las.pfeg.noaa.gov/oceanWatch/oceanwatch_safari.php

Jason-1, 10 day composite

Resolution: .25 deg

Study period April-July 2008

5 tags about 200 locations each; Argos for locations

Uses gradient+ to get geostrophic current

slide20
Geostrophic currents for June 29, 2008 (ten day composite)

Sri Lanka upper right, Maldives left

Backgound bathymetry - yellow is highest

slide22
Brent's interpretation.

"Looks like the drifter starts out behaving according to the driving forces of surface current.The odd and interesting event is when it moves south into that small apparently weak gyre towards the end. It then goes back to moving under influence of current heading south when it comes out of gyre, but this is in opposite trajectory that it would have followed if it had followed dominant flow before it had entered gyre. This seems to be a key change in state of expected movement from the null prediction."

slide23
Functional stochastic differential equation (FSDE)

dr(t) = μ(H(t),t)dt + σ(H(t),t)dB

H(t): a history based on the past,{r(s), s t}

Process is Markov when H(t) = {r(t)}

Interpretation

r (t) - r(0) = 0tμ(H(s),s)ds + 0t σ(H(s),s)dB(s)

slide24
Details of SFDE. definition, convergence, ...

Approximation

r(ti+1)-r(ti)=(H(ti),ti)(ti+1-ti)+(H(ti),ti){ti+1-ti}Zi+1

slide25
Analysis.

Reduced tag data to 46 contiguous 12 hour periods

median values

Estimated local zonal and meridional velocities

graphed versus time and each other

Looking for validation of NOAA values

slide26
Some details of computations.

estimated local zonal and meridional velocities by simple differences

smoothed/processed these with biweight length 5

interpolated remote sensed values to tag times

.....

slide32
Incorporating currents and winds and past locations

Regression model, tag velocity

(r(ti+1)-r(ti))/(ti+1-ti) =  (H(ti),ti) + + CXC(r(ti),ti) + VXV(r(ti),ti) + σZi+1/√(ti+1-ti)

where

(H(t),t) =  tt-1r(s)dM(s)

M(t) = #{ti t}, counting function

slide34
regression coeficients, n = 206

zonal case

0.742828 0.051252 14.494

C 0.201452 0.039224 5.136

V -0.009115 0.003862 -2.360

R2 = 0.804

meridional case

0.708062 0.041549 17.042

C 0.240575 0.039707 6.059

V 0.025608 0.005453 4.696

R2 = 0.854

slide35
Residuals introducing variables successively

 = 0,  = 1;  C ;  C ,  V ;  C ,  V , 

slide36
Discusssion and summary.

Use NOAA values with some caution and further processing, if possible

Can use SDE result for simulation

Residuals to discover things

motivations - SDE, FSDE

continuous time and then discrete time

robust/resistant smoothing

slide37
The case of bounded regions.

Human made fences, islands for seals, ...

Suppose the region is D is closed with boundary D .

Consider the SDE

dr= μ(r)dt + σ(r)dB(t) - dA(r)

where A is an adapted process that only increases when r(t) is on the boundary D. Purpose is to reflect particle back to the interior of D.

One cannot simply use the Euler scheme throwimg away a point if particle goes outside D. Bias results.

slide38
Method 1. .Build a sloping wall. That is have a potential rising rapidly at the boundary D when moving to the interior. One might take H(r) = d(r,D) ,  scalars

Here grad H dt is an approximation to dA.

Method 2. Let D denote the projection operator taking an r to the nearest point of D. Let  0 and (r)={r-D(r)}/.

Use the scheme

r(tk+1) = r(tk) + (r(tk),tk))(tk+1-tk) + (r(tk),tk) (tk+1-tk)Zk+1 -

(r(tk),tk)(tk+1-tk)

These points may go outside the boundary, but by taking  small enough gets a point inside

slide39
Method 3. Consider the scheme

r(tk+1) = D(r(tk)+(r(tk)(tk+1-tk) + (r(tk)(tk+1-tk)Zk+1)

If a point falls outside D project back to the boundary. These values do lie in D.

Brillinger (2003)

slide40
A crude approximation is provided by the procedure: if generated point goes outside, keep pulling back by half til inside.
slide41
Second animal. CDA

Male juvenile

Released La’au Point 4 April 2004

Study ended 27 July

n = 754 over 88.4 days

I = 144

slide43
Potential function employed

H(x,y)=β10x+β01y+β20x2+β11xy+β02y2+C/dM(x,y)

dM(x,y): distance to Molokai

slide47
Discrete markov chain approach, Kushner (1976).

Suppose D = {r:(r)  0} with boundary D = {r:(r)=0}

Set a(r,t) = ½(r,t)(r,t)'

For present convenience suppose aij(r,t) =0 i  j

Suppose tk+1 - tk = t

Dh refers to lattice points in D with separation h. Suppose r0 in Dh

Let ei be unit vector in ith coordinate direction

Consider Markov chain with transition probabilities

P(rk=r0 eih|rk-1=r0) = (aii(r0,tk-1)+h|i(r0,tk-1)|)/h2

P(rk=r0|rk-1=r0) = 1 -  preceding. For suitable h, 

slide49
Vector case

dri(t) = ji(ri (t)- rj(t))dt + dBi(t)

i = 1,...,p for some function 

Which ?

Are the animals interacting?

Difficulties with unequal time spacings

Time lags

slide50
Other topics.

Uncertainties - haven't focused on. There are general methods: jackknife and bootstrap

Order of approximation

Unequal spacings

Crossings - trajectori3es heading into regions (eg. football, debris)

Moving fronts

slide52
Acknowledgements. Data/background providers, collaborators

Aager, Guckenheimer, Guttorp, Kie, Oster, Preisler, Stewart, Wisdom, Littnan, Mendolssohn, Foley, Dewitt

Lovett, Spector

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