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

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.

NY Times, 06/08/2009

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.

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.

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

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

Where to situate holding points? D. Singham

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

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

Coordinates employed by N. Oresme (d. 1382)

Some physics history. Vector-valued trajectories

Tycho Brahe, Danish Astromer 1546 - 1601

Accurate observations of Mars declination

The early contributors.

W. Pafko

Used Brahe's results to learn nature of solar system

Johannes Kepler 1571-1630

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.

Isaac Newton 1642 – 1727

Inferred mechanisms underlying celestial motions, laws

1. 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.

2. 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

Joseph-Louis Lagrange 1736 – 1813

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

Gravity.

gravitation potential

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

gravitational field

F = -grad H, grad = ( /x, /y)

2 D trajectories.

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

Perrin (1913) (Guttorp book)

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

Przibram (1913)

Canadian - Swiss competition.

football - no

hockey - no

curling - yes

1996 European Football Championship.

Passes between Shearer-Sherrington goals. "Brownian motion"

J. Wesson (2002)

25-pass goal. Argentina vs Serbia-Montenegro, 2006

D. R. Brillinger (2007)

Marine biology. Hawaiian monk seal.

Most endangered marine mammal in US waters, 1300Live 30 yrs. Male 230 kg, female 270 kgMotivation: management purposes, to learn where they forage geographically and vertically

Brillinger et al (2008)

Brownian motor.

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

L. Euler

1707-1783

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

S. Chandler

1846-1913

Euler predicted free nutation of the rotating Earth in 1755

Discovered by Chandler in 1891

Data from International Latitude Observatories setup in 1899

Rotating solid

D. R. Brillinger (1973)

Whale shark.

Slow moving filter feeder.

Largest living fish species.

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

Brent

Starkey Reserve, Oregon

Designed to answer management questions, ...

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

Foraging strategies, habitat preferences, dynamics of population densities?

Brillinger et al (2004)

Elephant seal.

Were endangered, now 150000

Females: 600-800kg Males: 2300kg

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

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.

One elephant seal's journey

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

td

Surface of sphere

Popup tag.

Whale shark's tag after release

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

3 D trajectory.

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.

B. P. Kelly

source("plotspin")

Some formalism.

Differential equations

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

Deterministic case

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

G. Leibniz

1646-1716

Newtonian mechanics. F:force, m: mass, dv/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

Newton’s second law, F = ma

Scalar-valued potential function, H

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

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

dv(t)/dt = - β v(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

Potential functions.

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

Attraction and repulsion

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)|)

Basic concepts of probability.

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)

Stochastic calculus, Ito integral

K. Ito, 1967 1915-2008 W. Doeblin 1915-1940

Random function, {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

The Itô 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

Construction of I().

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)]

General case.

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)

Stochastic differential equations (SDEs).

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

To be interpreted as

r(t) - r(0) = 0t dr(s) = 0

tμ(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)

denotes length of the largest interval in 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

Interpretations

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, ..."

Particular cases.

Langevin equation for motion of a particle.

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

m d2x/dt2 = -6a dx/dt + X

X: "complimentary force", dB/dt

Paul Langevin

1872-1946

dr(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

Modelling and data analysis.

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.

(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

Discussion.

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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ringed seal, goal