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Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 1 David R. Brillinger University of California, Berkeley 2   1. Hieronymus Brillinger. 30.9.1469 à Bâle, apr. le 10.1.1537 à Fribourg-en-Brisgau

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Cycle Romand de Statistique, 2009

Ovronnaz, Switzerland

Random trajectories: some theory and applications

Lecture 1

David R. Brillinger

University of California, Berkeley

2 1


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Hieronymus Brillinger. 30.9.1469 à Bâle, apr. le 10.1.1537 à Fribourg-en-Brisgau

Fils de Kaspar, procureur au tribunal épiscopal, et de Clara.

Diacre en 1482, immatriculé à l'université de Bâle en 1485, proviseur de l'école de la cathédrale en 1487.

Chapelain de Saint-Pierre (1492) et du chapitre (1502).

Recteur de l'université en 1505.

En 1510 il fouilla la tombe de la reine Anne, première épouse de l'empereur Rodolphe Ier, enterrée dans la cathédrale, et transféra sa couronne dans le trésor de l'église.



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Lecture 1: Some history and some background

Meant to be a succession of motivating examples, questions and methods

Data sets and their analysis

"trajectories" and "trajectoires" are old words for "processes"

E.g. Loève (1955), p. 500: "The values Xt() at  of a random function Xt will be called sample functions or trajectories or paths of the random function; ..."

Here there is a moving particle and t is physical time.


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Trajectoire. La ligne décrite par n'importe quel point d'un objet en mouvement, et notamment par son centre de gravité.

Astronomie. La courbe que décrit le centre de gravité d'une planète accomplissant sa révolution autour du soleil, ou d'un satellite autour d'une planète.

Physique des particules. Le trajet d'une particule élémentaire, ou d'un élément émis à partir d'une source de rayonnement.

Ingénierie. En balistiqu la trajectoire est la courbe que décrit le centre de gravité d'un projectile pendant son trajet dans l'espace.

Ecologie. On parle de trajectométrie pour signifier l'étude des déplacements des animaux. Ceux-ci peuvent être suivis directement ou équipés d'émetteurs / récepteur GPS ou d'émetteurs VHF.


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  • Mathématiques. L'ensemble des positions successives occupées par ce point au cours du temps.

  • On introduit le formalisme des arcs paramétrés pour décrire d'une part la trajectoire, d'autre part la façon dont elle est parcourue, ou paramétrage.

  • Des résultats mathématiques établissent des différences fondamentales entre les trajectoires possibles d'une masse ponctuelle sur différentes surface:

  • le long d'une ligne, où par exemple une marche aléatoire repasse presque partout presque surement

  • sur une surface (en deux dimensions), et plus spécifiquement sur un plan, une sphère, un tore

  • dans un volume


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1 D trajectories.

Rapid Bus going north on San Pablo Avenue, Berkeley

weekdays October 2008 6:10 to 19:30, approx every 20 min

velocity in seconds

distance = velocity  time, dx = vdt



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Some physics history. Vector-valued trajectories

Planets, latitude (n = 12) vs. longitude (n=30)

Eleventh century, (H. P. Lattin, Isis (1948))

Coordinates employed by N. Oresme (d. 1382)


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The early contributors.

Tycho Brahe, Danish Astromer 1546 - 1601

Accurate observations of Mars declination



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Johannes Kepler 1571-1630

Used Brahe's results to learn nature of solar system


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1. Planets move in ellipses with the Sun at one focus.

2. The radius vector describes equal areas in equal times.

3. The squares of the periodic times are to each other as the cubes of the mean distances.


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Isaac Newton 1642 – 1727

Inferred mechanisms underlying celestial motions, laws


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  • Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it.

  • The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

  • F: force, m: mass, a: acceleration F = ma = mÿ

  • 3. For every action force, there is an equal and opposite reaction force.

  • Understanding motion required the development of calculus


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Joseph-Louis Lagrange 1736 – 1813 motion in a right line, unless it is compelled to change that state by forces impressed on it.

Lagrangian. Equations of motion found by differentiating a potential/action function.


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Gravity motion in a right line, unless it is compelled to change that state by forces impressed on it..

gravitation potential

H(r) = –GM0/|r0 – r|, G: constant of gravitation, M0: mass

gravitational field

F = -grad H, grad = ( /x,  /y)


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2 D trajectories motion in a right line, unless it is compelled to change that state by forces impressed on it..

Brownian motion. Observed phenomenon

Robert Brown (1828). “A brief account of microscopal observations made in the months of June, July, and August, 1827, on the particles contained in the pollen of plants; …”

Bachelier (1900). “Théorie de la speculation”.

Einstein (1905), Smoluchowsky (1905)

Langevin (1908). Worked to verify Einstein

1773-1858


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Tiny mastic grain particles. Perrin collected data to check some predictions of Einstein.

Perrin (1913) (Guttorp book)


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Przibram (1913) some predictions of Einstein.


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Canadian - Swiss competition. some predictions of Einstein.

football - no

hockey - no

curling - yes


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1996 European Football Championship some predictions of Einstein..

Passes between Shearer-Sherrington goals. "Brownian motion"

J. Wesson (2002)


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25-pass goal some predictions of Einstein.. Argentina vs Serbia-Montenegro, 2006

D. R. Brillinger (2007)


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Marine biology. Hawaiian monk seal some predictions of Einstein..

Most endangered marine mammal in US waters, 1300

Live 30 yrs. Male 230 kg, female 270 kg

Motivation: management purposes, to learn where they forage geographically and vertically


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Brillinger et al (2008) some predictions of Einstein.


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Brownian motor some predictions of Einstein..

Kinesin: a two-headed motor protein that powers organelle transport along microtubules.

Biophycist's question. "Do motor proteins actually make steps?"

Hunt for the periodic positions at which a motor might dwell

Biophycist's goal. "To formulate and test hypotheses relating motor structure to function"

Data via optical instrumentation


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Variation of latitude due to nutation predicted by Euler. Chandler discovered period of 428 days.

S. Chandler

1846-1913

L. Euler

1707-1783


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Rotating solid Chandler discovered period of 428 days.

Euler predicted free nutation of the rotating Earth in 1755

Discovered by Chandler in 1891

Data from International Latitude Observatories setup in 1899


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D. R. Brillinger (1973) Chandler discovered period of 428 days.


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Whale shark Chandler discovered period of 428 days..

Slow moving filter feeder.

Largest living fish species.

Can grow up to 60 ft in length and can weigh up to 15 tons

Brent


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Starkey Reserve, Oregon Chandler discovered period of 428 days.

Designed to answer management questions, ...

Can elk, deer, cows, bikers, hikers, riders, hunters coexist?

Foraging strategies, habitat preferences, dynamics of population densities?


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Brillinger et al (2004) Chandler discovered period of 428 days.


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Elephant seal Chandler discovered period of 428 days..

Were endangered, now 150000

Females: 600-800kg Males: 2300kg

Females: live 16-18 yrs: Males: 12-14


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Elephant seals range over vast areas of the Eastern North Pacific between California rookeries and distant foraging areas.

How do they navigate?

Perhaps they follow great circle paths.


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One elephant seal's journey Pacific between California rookeries and distant foraging areas.

D. R. Brillinger and B. S. Stewart (1998)


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Surface of sphere Pacific between California rookeries and distant foraging areas.


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Popup tag Pacific between California rookeries and distant foraging areas..


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Whale shark's tag after release Pacific between California rookeries and distant foraging areas.

D. R. Brillinger and B. S. Stewart (2009)


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3 D trajectory Pacific between California rookeries and distant foraging areas..

Ringed seal. Litle is known about their behavior or activity patterns - much of the time underwater and surface activities hidden by snow

Primitive among the phocid seal group, therefore, of particular interest in comparative behavioral studies.


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B. P. Kelly Pacific between California rookeries and distant foraging areas.


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source("plotspin") Pacific between California rookeries and distant foraging areas.


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Some formalism Pacific between California rookeries and distant foraging areas..

Differential equations

(t, r(t)) t: time, r: location

Deterministic case

dr(t)/dt = v(t)ORdr = vdt v: velocity

G. Leibniz

1646-1716


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Newtonian mechanics. F:force, m: mass, d Pacific between California rookeries and distant foraging areas.v/dt: accel

F = mdv/dt

Block on incline. : elevation, g: accel gravity, x: horiz dist, : coeff friction

d2x/dt2 = g(sin  - cos )

I. Newton, 1689

1643-1727


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Newton’s second law, F = ma Pacific between California rookeries and distant foraging areas.

Scalar-valued potential function, H

Planar case, location r = (x,y)’, time t

An example dr(t)/dt = v(t)

dv(t)/dt = - βv(t) – β grad H(r(t),t)

v: velocity β: damping (friction)

becomes dr/dt = - grad H(r,t) = μ(r,t), for β large


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Potential functions. Pacific between California rookeries and distant foraging areas.

Attraction

To point a, H(r) = α|r-a|2,

½σ2log |r-a| - δ|r-a| bird motion

To region, a nearest point

Repulsion

From point, H(r) = |r-a|-2

From region, a nearest point


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Attraction and repulsion Pacific between California rookeries and distant foraging areas.

H(r) = α(1/r12 – 1/r6)

Quadratic

H(r) = β10x + β01y + β20x2 + β11xy + β02y2

Nonparametric, β(.), smooth

e.g. wavelets, local regression, spline expansion

Moving attractor/repellor

H(r,t) = β(|r-a(t)|)


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Basic concepts of probability. Pacific between California rookeries and distant foraging areas.

Probability space, (, F, P)

Sample space, 

-field F, subsets of 

Probability measure, P

Random variable X,

{ in :X()  x} in F for x in R

Vector-valued case - on same probability space

Filtration {Fn}, sequence of increasing -fields each in F

{Yn} adapted to F, Yn is Fn measureable for all n

Grimmett and Stirzaker (2001)


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Stochastic calculus, Ito integral Pacific between California rookeries and distant foraging areas.

W. Doeblin 1915-1940

K. Ito, 1967 1915-2008


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Random function, { Pacific between California rookeries and distant foraging areas.B(t;)}  a r.v

Brownian motion, B(t), values in Rp

position of particle at time t

continous time - form of random walk

Disjoint increments are independent and such that B(t+s)-B(t) is Np(0,sIp).

dB(t)=B(t+dt)-B(t) is Np(0,dtIp), dt small

Almost surely: continuous, nowhere differentiable, unbounded variation


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The It Pacific between California rookeries and distant foraging areas.ô stochastic integral.

I() = 0T(u) dB(u), t  0

Assumptions. There is a "filtration" Ft , t  0 with the properties,

1. s  t implies every set in Fs is also in Ft

2. B(t) is Ft measureable for all t

3. For t t1  ...  tn, the increments ... are independent of Ft

Concerning (t),

1. (t) is Ft -measureable for all t

2. E 0T (t)2dt <  for all T


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Construction of I( Pacific between California rookeries and distant foraging areas.).

Simple case.

={t0,t1,...,tn} partition of [0,T]

Elementary process, (t) constant on each [tk,tk+1]

I() = j=0k-1 (tj)[B(tj+1)-B(tj)]+(tk)[B(t)-B(tk)]


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General case. Pacific between California rookeries and distant foraging areas.

Assume (t) is Ft -measureable and

E 0T(t)2dt < 

There is sequence of predictable step functions {n} with

limn E 0T|n(t)-(t)||2dt = 0

I(n) = 0Tn(t)dB(t)

I() = 0T(t)dB(t) = limn I(n)

( {I(n)} is a Cauchy sequence in L2(P))

I() may be approximated by I(n)


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Stochastic differential equations (SDEs). Pacific between California rookeries and distant foraging areas.

dr(t) = μ(r(t),t)dt + σ(r(t),t)dB(t) (*)

To be interpreted as

r(t) - r(0) = 0t dr(s) = 0tμ(r(s),s)dt + 0tσ(r(s),s)dB(s)

μ: drift, σ: diffusion, {B(t)}: Brownian

There are conditions for a unique solution to (*), e.g.

|μ(u,t) - μ(v,t)|2 + |σ(u,t) - σ(v,t)|2 C|u - v|2

|μ(u,t)|2 + |σ(u,t)|2 D(1+u2)


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 denotes length of the largest interval in  Pacific between California rookeries and distant foraging areas.n and r the associated approximate solution. Suppose

E|r(0)|2 < 

E|r(0) - r(0)|  C1 

|μ(u,t) - μ(v,t)| + |σ(u,t) - σ(v,t)|  C2|u - v|

|μ(u,t)| + |σ(u,t)|  C3(1+|u|)

|μ(u,s) - μ(u,t)| + |σ(u,s) - σ(u,t)|  C4(1+|u|)|s-t|

then uniformly for 0 t T

E|r(t) - r(t)|  C5 

Kloeden and Platen


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Interpretations Pacific between California rookeries and distant foraging areas.

E{dr(t)|r(u), u  t} = (r(t),t)dt

Var{dr(t)|r(u), u  t} = (r(t))(r(t))'dt

Surprises

0t B(s)dB(s) = ½(B(t)2 - 1)

If X(t) = g(B(t)), then dX(t) = g'(B(t))dB(t) + ½ g"(B(t))dt

"Brownian-based developments have had an incredible impact on physics, probability, ..."


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Particular cases. Pacific between California rookeries and distant foraging areas.

Langevin equation for motion of a particle.

x: position, a: radius, m: mass, : viscosity

m d2x/dt2 = -6a dx/dt + X

X: "complimentary force", dB/dt

Paul Langevin

1872-1946


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d Pacific between California rookeries and distant foraging areas.r(t) = μ(r(t),t)dt + σ(r(t),t)dB(t)

(Vector) Ornstein-Uhlenbeck.

drift, diffusion terms

μ(r,t) = A(a - r), σ(r,t) = σ

gradient process

μ(r,t) = -grad H(r,t) for some scalar-valued H

grad = ( /x,  /y)

O-U potential function

H = (a - r)TA(a - r)/2, A symmetric 0


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Modelling and data analysis Pacific between California rookeries and distant foraging areas..

Available data, {r(tj),tj)}

The Euler scheme approximate solution to the SDE is

(r(ti+1)-r(ti))/(ti+1-ti) =μ(r(ti),ti) + σ(r(ti),ti)Zi+1/√(ti+1-ti)

Zi: independent standard vector normals, with the values for t between ti and ti+1 obtained by simple interpolation.


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( Pacific between California rookeries and distant foraging areas.r(ti+1)-r(ti))/(ti+1-ti) =μ(r(ti),ti) + σ(r(ti),ti)Zi+1/√(ti+1-ti)

will not be used as an approximate solution.

It will be used to suggest a likelihood function for use in estimation.

It leads , ignoring an initial term, to the log likelihood,

-½ i (log 2 + log |ii'| + tr{(ri+1 - ri - i)(ii')-1(ri+1-ri-i)'}

Maximize to estimate parameters


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Discussion Pacific between California rookeries and distant foraging areas..

Interpretations result from using SDE approach.

Conceptual models can arise directly.

Results are extendable, analytic expressions are available, predictions can be set down, and the process is Markov.

Advantages of potential function approach include:

the function is real-valued-valued

both parametric and nonparametric estimates are available

Literature exists for the case where there is a boundary


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Show videos Pacific between California rookeries and distant foraging areas.

ringed seal, goal


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