PowerPoint PresentationSlide 2Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Cycle Romand de Statistique, 2009September 2009Ovronnaz, SwitzerlandRandom trajectories: some theory and applicationsLecture 2David R. BrillingerUniversity of California, Berkeley2  1Lecture 2: Inference methods and some results Lecture 1 provided motivating examples This lecture presents analyses EDA and CDA (Stefan)The Chandler wobble.Chandler inferred the presence of 12 and approx 14 months components in the wobble.Serious concern to scientists and at the end of the 1800s Network of stations set up to collect North Star coordinatesData would provide information on the interior structure of the EarthMonthly data, t = 1 month.Work with complex-values, Z(t) = X(t) + iY(t).Compute the location differences, Z(t), and then the finite FT dZT() = t=0T-1 exp {-it}[Z(t+1)-Z(t)]Periodogram IZZT() = (2T)-1|dZT()|2periodogram - 1972 graphics!Model.Arato, Kolmogorov, Sinai, (1962) set down the SDE dX = - Xdt -  Ydt +  dB dY =  Xdt -  Ydt +  dCZ = X + iY  = B + iCGeneral stimulus dZ = -  Zdt +  d  =  - i Adding measurement noise, the power spectrum is |i + |-2f()+2|1-exp{-i}|2/2But what is the source of ? Source of 12 mo, 14 moIf series stationary, mixing periodograms, Is at  = 2s/T approximately independent exponentials parameter fs Suggesting estimation criterion (quasi-likelihood) L = s fs-1 exp{-Is/fs}and approximate standard errorsGaussian estimation, Whittle methodDiscussion. Perhaps nonlinearity Looked for association with earthquakes, atmospheric pressure by filtering at Chandler frequency.None apparentMystery "solved" by modern data and models. Using 1985 to 1996 data, R. S. Gross (NASA) concluded two thirds of wobble caused by changes in ocean-bottom water pressure, one-third by changes in atmospheric pressure. NASA interested. One of the biggest sources of uncertainty in navigating interplanetary spacecraft is not knowing Earth's rotation changes."Brownian-like" data. Perrin's mastic grain particles Viscosity, so can't be exactly Brownian Perrin checking on Einstein and Smoluchowsky n = 48, t = 30 secPerrin (1913)Potential function.Quadratic. H(x,y) = γ1x + γ2y + γ11x2 + γ12xy + γ22y2 real-valueddrift. μ = - grad H = - (γ1 + 2γ11x + γ12y , γ2 + γ12x + 2γ22y ) stack (r(ti+1)-r(ti))/ (ti+1-ti) = μ(r(ti)) + σ Zi+1/√(ti+1-ti)WLSmartingale differencesasymptotic normality +, Lai and Wei (1982)Estimate of HEstimate of μDiscussion. Ornstein-Uhlenbeck like Potential function for O-U H = (a - r)'A(a-r)/2 A symmetric  0quadraticBezerkeley football25-pass goal. 2006 Argentina vs. Serbia-MontenegroH(r) =  log |r| +  |r| + γ1x + γ2y + γ11x2 + γ12xy + γ22y2 r = (x,y) attraction (goalmouth) plus smooth |r – a0|, a0 closest point of goalmouth (r(ti+1)-r(ti))/ (ti+1-ti) = μ(r(ti)) + σ Zi+1/√(ti+1-ti)μ = -grad H, stack, WLSEstimate of H, image plotVector fieldDiscussion. Modelled path, not score Asymmetry, down one side of the field Ball speed, slow, then quickHawaiian Monk seal. 2.2 m, 250 kg, life span 30 yrEndangered – environmental change, habitat modification, reduction in prey, humans, random fluctuations ~ 1200 remainLocation data. Satellite-linked time depth recorder + radio transmitter Argos Data Collection & Location Service Location estimate + index (Location class (LC) = 3,2,1,0,A,B,Z) UTM coordinates – better projection, euclidian geometry, kmFemale, 4-5 years oldReleased La’au Point 13 April 2004Study period til 27 Julyn = 573 over 87.4 days (ti ,r(ti), LCi), i=1,…,I unequally spaced in time well-determined: LC = 3, 2, 1 I = 189Spatial feature: MolokaiBrillinger, Stewart and Littnan (2008)Bagplot. Multi-d generalization of boxplot Center is multi-d median Bag contains 50% of observations with greatest depth (based on halfspaces) Fence separates inliers from outliers – inflates bag by factor of 3 Equivariant under affine transforms Robust/resistantPenguin Bank!Journeys? - distance from La’au Point - foraging?Modelling. H(r,t) - two points of attraction, one offshore, one atshore Potential function ½σ2log |r-a| - δ|r-a| a(t) changesParametric μ = -grad H Approximate likelihood from (r(ti+1)-r(ti))/ (ti+1-ti) = μ(r(ti)) + σ Zi+1/√(ti+1-ti) Robust/resistant WLS Estimate σ2 from mean squared errorDiscussion and summary. Time spent foraging in Penguin Bank appeared constrained by a terrestrial atractor (haulout spot – safety?).Seal spent more time offshore than thought previouslyEDArobust/resistant methods basicBrownian motor. KinesinA 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 dwellData via optical instrumentationKinesin motor attached to microtubuleMalik, Brillinger and Vale (1994)Location (X(t),Y(t))Rotate via svd to get parallel displacement, Z(t) 2 D becomes 1 DModel Step function, N(t)? Z(t) =  + N(t) + E(t)As stationary increment process fZZ = 2 fNN + fEEIf N(t) renewal fNN = p(1 - ||2) / (2  |1 - |2), p rate,  characteristic functionInterjump, time j+1 - j constant, v velocity of movement power spectrum  j (/v - 2j/)periodic spikesPrewhitened for greater sensitivity. Robust line fitted to Z(t) Periodogram of residuals Robust line fit to log(periodogram) at low frequencies and subtracted Averaged results for several microtubulesTo assess simulated various gamma distributionsFor