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Cycle Romand de Statistique, 2009 September 2009 Ovronnaz, Switzerland Random trajectories: some theory and appl

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

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Cycle Romand de Statistique, 2009 September 2009 Ovronnaz, Switzerland Random trajectories: some theory and appl

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

  2. Lecture 2: Inference methods and some results Lecture 1 provided motivating examples This lecture presents analyses EDA and CDA (Stefan)

  3. 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 coordinates Data would provide information on the interior structure of the Earth

  4. Monthly 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()|2

  5. periodogram - 1972 graphics!

  6. Model. Arato, Kolmogorov, Sinai, (1962) set down the SDE dX = - Xdt -  Ydt +  dB dY =  Xdt -  Ydt +  dC Z = X + iY  = B + iC General 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 mo

  7. If 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 errors Gaussian estimation, Whittle method

  8. Discussion. Perhaps nonlinearity Looked for association with earthquakes, atmospheric pressure by filtering at Chandler frequency. None apparent Mystery "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.

  9. "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 sec

  10. Perrin (1913)

  11. Potential function. Quadratic. H(x,y) = γ1x + γ2y + γ11x2 + γ12xy + γ22y2 real-valued drift. μ = - grad H = - (γ1 + 2γ11x + γ12y , γ2 + γ12x + 2γ22y ) stack (r(ti+1)-r(ti))/ (ti+1-ti) =μ(r(ti)) + σZi+1/√(ti+1-ti) WLS martingale differences asymptotic normality +, Lai and Wei (1982)

  12. Estimate of H

  13. Estimate of μ

  14. Discussion. Ornstein-Uhlenbeck like Potential function for O-U H = (a - r)'A(a-r)/2 A symmetric  0 quadratic

  15. Bezerkeley football

  16. 25-pass goal. 2006 Argentina vs. Serbia-Montenegro

  17. H(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)) + σZi+1/√(ti+1-ti) μ = -grad H, stack, WLS

  18. Estimate of H, image plot

  19. Vector field

  20. Discussion. Modelled path, not score Asymmetry, down one side of the field Ball speed, slow, then quick

  21. Hawaiian Monk seal. 2.2 m, 250 kg, life span 30 yr Endangered – environmental change, habitat modification, reduction in prey, humans, random fluctuations ~ 1200 remain

  22. Location 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, km

  23. Female, 4-5 years old Released La’au Point 13 April 2004 Study period til 27 July n = 573 over 87.4 days (ti ,r(ti), LCi), i=1,…,I unequally spaced in time well-determined: LC = 3, 2, 1 I = 189 Spatial feature: Molokai

  24. Brillinger, Stewart and Littnan (2008)

  25. 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/resistant

  26. Penguin Bank!

  27. Journeys? - distance from La’au Point - foraging?

  28. Modelling. H(r,t) - two points of attraction, one offshore, one atshore Potential function ½σ2log |r-a| - δ|r-a| a(t) changes

  29. Parametric μ= -grad H Approximate likelihood from (r(ti+1)-r(ti))/ (ti+1-ti) =μ(r(ti)) + σZi+1/√(ti+1-ti) Robust/resistant WLS Estimate σ2 from mean squared error

  30. Discussion and summary. Time spent foraging in Penguin Bank appeared constrained by a terrestrial atractor (haulout spot – safety?). Seal spent more time offshore than thought previously EDA robust/resistant methods basic

  31. 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 Data via optical instrumentation

  32. Kinesin motor attached to microtubule Malik, Brillinger and Vale (1994)

  33. Location (X(t),Y(t)) Rotate via svd to get parallel displacement, Z(t) 2 D becomes 1 D Model Step function, N(t)? Z(t) =  + N(t) + E(t)

  34. As stationary increment process fZZ = 2 fNN + fEE If N(t) renewal fNN = p(1 - ||2) / (2  |1 - |2), p rate,  characteristic function Interjump, time j+1 - j constant, v velocity of movement power spectrum j(/v - 2j/) periodic spikes

  35. Prewhitened 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 microtubules To assess simulated various gamma distributions

  36. For some l set Y(t) = t + klklk (t/T) + noise with lk(x) = 2l/2(2lx - k) Haar scaling wavelet (x) = 1 0  x < 1 = 0 otherwise Fit by least squares Shrink: replace estimate alk by w(|alk|/slk)alk w(u) = (1-u2)+

  37. D. R. Brillinger (1996)

  38. Discussion and summary. "Malik et al (1994) were able to rule out large, regular (that is, strictly periodic) steps for kinesin movement along microtubules (for  > 12 nm) and argued they would not have been able to detect smaller steps (say 8 nm or less) unless the motions were highly regular (quasi periodic), with the step-to-step interval varying by less than 20%" Since then Brownian motion has been reduced revealing steps

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