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Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 1

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

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  1. Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 1 David R. Brillinger University of California, Berkeley 2 1

  2. 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.

  3. NY Times, 06/08/2009

  4. 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.

  5. 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.

  6. 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

  7. 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

  8. Where to situate holding points? D. Singham

  9. 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)

  10. The early contributors. Tycho Brahe, Danish Astromer 1546 - 1601 Accurate observations of Mars declination

  11. W. Pafko

  12. Johannes Kepler 1571-1630 Used Brahe's results to learn nature of solar system

  13. 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.

  14. Isaac Newton 1642 – 1727 Inferred mechanisms underlying celestial motions, laws

  15. 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

  16. Joseph-Louis Lagrange 1736 – 1813 Lagrangian. Equations of motion found by differentiating a potential/action function.

  17. Gravity. gravitation potential H(r) = –GM0/|r0 – r|, G: constant of gravitation, M0: mass gravitational field F = -grad H, grad = ( /x,  /y)

  18. 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

  19. Tiny mastic grain particles. Perrin collected data to check some predictions of Einstein. Perrin (1913) (Guttorp book)

  20. Przibram (1913)

  21. Canadian - Swiss competition. football - no hockey - no curling - yes

  22. 1996 European Football Championship. Passes between Shearer-Sherrington goals. "Brownian motion" J. Wesson (2002)

  23. 25-pass goal. Argentina vs Serbia-Montenegro, 2006 D. R. Brillinger (2007)

  24. Marine biology. Hawaiian monk seal. 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

  25. Brillinger et al (2008)

  26. 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

  27. Variation of latitude due to nutation predicted by Euler. Chandler discovered period of 428 days. S. Chandler 1846-1913 L. Euler 1707-1783

  28. Rotating solid Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891 Data from International Latitude Observatories setup in 1899

  29. D. R. Brillinger (1973)

  30. 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

  31. 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?

  32. Brillinger et al (2004)

  33. Elephant seal. Were endangered, now 150000 Females: 600-800kg Males: 2300kg Females: live 16-18 yrs: Males: 12-14

  34. 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.

  35. One elephant seal's journey D. R. Brillinger and B. S. Stewart (1998)

  36. Surface of sphere

  37. Popup tag.

  38. Whale shark's tag after release D. R. Brillinger and B. S. Stewart (2009)

  39. 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.

  40. B. P. Kelly

  41. source("plotspin")

  42. Some formalism. Differential equations (t, r(t)) t: time, r: location Deterministic case dr(t)/dt = v(t)ORdr = vdt v: velocity G. Leibniz 1646-1716

  43. 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

  44. 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 = - 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

  45. 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

  46. 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)|)

  47. 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)

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