Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland
description
Transcript of Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland
![Page 1: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/1.jpg)
Cycle Romand de Statistique, 2009
Ovronnaz, Switzerland
Random trajectories: some theory and applications
Lecture 1
David R. Brillinger
University of California, Berkeley
2 1
![Page 2: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/2.jpg)
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.
![Page 3: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/3.jpg)
NY Times, 06/08/2009
![Page 4: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/4.jpg)
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.
![Page 5: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/5.jpg)
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.
![Page 6: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/6.jpg)
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
![Page 7: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/7.jpg)
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
![Page 8: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/8.jpg)
Where to situate holding points? D. Singham
![Page 9: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/9.jpg)
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
![Page 10: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/10.jpg)
Tycho Brahe, Danish Astromer 1546 - 1601
Accurate observations of Mars declination
The early contributors.
![Page 11: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/11.jpg)
W. Pafko
![Page 12: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/12.jpg)
Used Brahe's results to learn nature of solar system
Johannes Kepler 1571-1630
![Page 13: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/13.jpg)
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.
![Page 14: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/14.jpg)
Isaac Newton 1642 – 1727
Inferred mechanisms underlying celestial motions, laws
![Page 15: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/15.jpg)
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
![Page 16: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/16.jpg)
Joseph-Louis Lagrange 1736 – 1813
Lagrangian. Equations of motion found by differentiating a potential/action function.
![Page 17: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/17.jpg)
Gravity.
gravitation potential
H(r) = –GM0/|r0 – r|, G: constant of gravitation, M0: mass
gravitational field
F = -grad H, grad = ( /x, /y)
![Page 18: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/18.jpg)
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
![Page 19: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/19.jpg)
Perrin (1913) (Guttorp book)
Tiny mastic grain particles. Perrin collected data to check some predictions of Einstein.
![Page 20: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/20.jpg)
Przibram (1913)
![Page 21: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/21.jpg)
Canadian - Swiss competition.
football - no
hockey - no
curling - yes
![Page 22: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/22.jpg)
1996 European Football Championship.
Passes between Shearer-Sherrington goals. "Brownian motion"
J. Wesson (2002)
![Page 23: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/23.jpg)
![Page 24: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/24.jpg)
25-pass goal. Argentina vs Serbia-Montenegro, 2006
D. R. Brillinger (2007)
![Page 25: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/25.jpg)
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
![Page 26: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/26.jpg)
Brillinger et al (2008)
![Page 27: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/27.jpg)
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
![Page 28: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/28.jpg)
![Page 29: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/29.jpg)
![Page 30: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/30.jpg)
L. Euler
1707-1783
Variation of latitude due to nutation predicted by Euler. Chandler discovered period of 428 days.
S. Chandler
1846-1913
![Page 31: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/31.jpg)
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
![Page 32: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/32.jpg)
D. R. Brillinger (1973)
![Page 33: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/33.jpg)
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
![Page 34: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/34.jpg)
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?
![Page 35: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/35.jpg)
Brillinger et al (2004)
![Page 36: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/36.jpg)
Elephant seal.
Were endangered, now 150000
Females: 600-800kg Males: 2300kg
Females: live 16-18 yrs: Males: 12-14
![Page 37: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/37.jpg)
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.
![Page 38: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/38.jpg)
One elephant seal's journey
D. R. Brillinger and B. S. Stewart (1998)
![Page 39: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/39.jpg)
td
Surface of sphere
![Page 40: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/40.jpg)
Popup tag.
![Page 41: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/41.jpg)
Whale shark's tag after release
D. R. Brillinger and B. S. Stewart (2009)
![Page 42: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/42.jpg)
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.
![Page 43: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/43.jpg)
B. P. Kelly
![Page 44: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/44.jpg)
source("plotspin")
![Page 45: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/45.jpg)
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
![Page 46: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/46.jpg)
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
![Page 47: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/47.jpg)
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
![Page 48: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/48.jpg)
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
![Page 49: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/49.jpg)
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)|)
![Page 50: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/50.jpg)
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)
![Page 51: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/51.jpg)
Stochastic calculus, Ito integral
K. Ito, 1967 1915-2008 W. Doeblin 1915-1940
![Page 52: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/52.jpg)
![Page 53: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/53.jpg)
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
![Page 54: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/54.jpg)
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
![Page 55: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/55.jpg)
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)]
![Page 56: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/56.jpg)
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)
![Page 57: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/57.jpg)
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)
![Page 58: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/58.jpg)
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
![Page 59: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/59.jpg)
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, ..."
![Page 60: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/60.jpg)
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
![Page 61: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/61.jpg)
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
![Page 62: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/62.jpg)
![Page 63: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/63.jpg)
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.
![Page 64: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/64.jpg)
(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
![Page 65: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/65.jpg)
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
![Page 66: Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland](https://reader037.fdocument.pub/reader037/viewer/2022102809/56813adb550346895da321cb/html5/thumbnails/66.jpg)
Show videos
ringed seal, goal