Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 3

1. Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random trajectories: some theory and applications Lecture 3 David R. Brillinger University of California, Berkeley 2 1

2. Question. Why does time exist? If it didn't, then everything would happen at the same time. Einstein?

3. Lecture 3: Further analyses / special topics (moving) explantories calibration boundaries several objects. particle processes/systems trajectories on surface (?)

4. Rocky Mountain elk and ATV. Starkey Reserve, Oregon, NE pasture April-October 2003

5. Concern: effects of human invaders, e.g. ATV on animals' behavior animals ranged in a confined region 2.4 m high fence 8 GPS equipped t for elk - 5min, t for ATV - 1sec randomization in treatment assignment [SDE drift term depends on location of elk and intruder] Brillinger, Preisler, Ager, Wisdom (2004)

8. Model. SDE dr(t) = m(r)dt + dB(t) m = control, atv Estimated 's

10. Control vs. ATV days.

12. Model. dr(t)= μ(r(t))dt + υ(|r(t)-x(t-τ)|)dt + σdB(t) x(t): location of ATV at time t τ: time lag Plots of |vτ| vs. distance |r - xτ|

14. Discussion and summary. model fit by gam() apparent increase in elk speed at ATV distances up to 1.5km an experiment method useful in assessing animal reactions to recreational uses by humans

15. Whaleshark tagging study. Off Kenya Data collected to study ecology of these fish, e.g. where they travelled, foraged, and when? How to protect?, Brent

16. 29 June - 19 July, 2008. Indian Ocean Locations for tag from instrumented shark Unequally spaced times, about 250 time points Tag released, drifted til batteries expired Our (opportunistic) purpose: to calibrate sea surface current models results

19. Remote sensed data: sea surface heights, zonal and meridional currents Ocean Watch Demonstration Project's Live Access Server http://las.pfeg.noaa.gov/oceanWatch/oceanwatch_safari.php Jason-1, 10 day composite Resolution: .25 deg Study period April-July 2008 5 tags about 200 locations each; Argos for locations Uses gradient+ to get geostrophic current

20. Geostrophic currents for June 29, 2008 (ten day composite) Sri Lanka upper right, Maldives left Backgound bathymetry - yellow is highest

21. Show movie,

22. Brent's interpretation. "Looks like the drifter starts out behaving according to the driving forces of surface current.The odd and interesting event is when it moves south into that small apparently weak gyre towards the end. It then goes back to moving under influence of current heading south when it comes out of gyre, but this is in opposite trajectory that it would have followed if it had followed dominant flow before it had entered gyre. This seems to be a key change in state of expected movement from the null prediction."

23. Functional stochastic differential equation (FSDE) dr(t) = μ(H(t),t)dt + σ(H(t),t)dB H(t): a history based on the past,{r(s), s t} Process is Markov when H(t) = {r(t)} Interpretation r (t) - r(0) = 0tμ(H(s),s)ds + 0t σ(H(s),s)dB(s)

24. Details of SFDE. definition, convergence, ... Approximation r(ti+1)-r(ti)=(H(ti),ti)(ti+1-ti)+(H(ti),ti){ti+1-ti}Zi+1

25. Analysis. Reduced tag data to 46 contiguous 12 hour periods median values Estimated local zonal and meridional velocities graphed versus time and each other Looking for validation of NOAA values

26. Some details of computations. estimated local zonal and meridional velocities by simple differences smoothed/processed these with biweight length 5 interpolated remote sensed values to tag times .....

27. Pre and post running biweight.

31. On same plot

32. Incorporating currents and winds and past locations Regression model, tag velocity (r(ti+1)-r(ti))/(ti+1-ti) =  (H(ti),ti) + + CXC(r(ti),ti) + VXV(r(ti),ti) + σZi+1/√(ti+1-ti) where (H(t),t) =  tt-1r(s)dM(s) M(t) = #{ti t}, counting function

33. Tag currents and residuals from fits

34. regression coeficients, n = 206 zonal case 0.742828 0.051252 14.494 C 0.201452 0.039224 5.136 V -0.009115 0.003862 -2.360 R2 = 0.804 meridional case 0.708062 0.041549 17.042 C 0.240575 0.039707 6.059 V 0.025608 0.005453 4.696 R2 = 0.854

35. Residuals introducing variables successively  = 0,  = 1;  C ;  C ,  V ;  C ,  V , 

36. Discusssion and summary. Use NOAA values with some caution and further processing, if possible Can use SDE result for simulation Residuals to discover things motivations - SDE, FSDE continuous time and then discrete time robust/resistant smoothing

37. The case of bounded regions. Human made fences, islands for seals, ... Suppose the region is D is closed with boundary D . Consider the SDE dr= μ(r)dt + σ(r)dB(t) - dA(r) where A is an adapted process that only increases when r(t) is on the boundary D. Purpose is to reflect particle back to the interior of D. One cannot simply use the Euler scheme throwimg away a point if particle goes outside D. Bias results.

38. Method 1. .Build a sloping wall. That is have a potential rising rapidly at the boundary D when moving to the interior. One might take H(r) = d(r,D) ,  scalars Here grad H dt is an approximation to dA. Method 2. Let D denote the projection operator taking an r to the nearest point of D. Let  0 and (r)={r-D(r)}/. Use the scheme r(tk+1) = r(tk) + (r(tk),tk))(tk+1-tk) + (r(tk),tk) (tk+1-tk)Zk+1 - (r(tk),tk)(tk+1-tk) These points may go outside the boundary, but by taking  small enough gets a point inside

39. Method 3. Consider the scheme r(tk+1) = D(r(tk)+(r(tk)(tk+1-tk) + (r(tk)(tk+1-tk)Zk+1) If a point falls outside D project back to the boundary. These values do lie in D. Brillinger (2003)

40. A crude approximation is provided by the procedure: if generated point goes outside, keep pulling back by half til inside.

41. Second animal. CDA Male juvenile Released La’au Point 4 April 2004 Study ended 27 July n = 754 over 88.4 days I = 144

42. Brillinger, Stewart, Littnan (2005)

43. Potential function employed H(x,y)=β10x+β01y+β20x2+β11xy+β02y2+C/dM(x,y) dM(x,y): distance to Molokai

44. Potential estimate

46. Turing test

47. Discrete markov chain approach, Kushner (1976). Suppose D = {r:(r)  0} with boundary D = {r:(r)=0} Set a(r,t) = ½(r,t)(r,t)' For present convenience suppose aij(r,t) =0 i  j Suppose tk+1 - tk = t Dh refers to lattice points in D with separation h. Suppose r0 in Dh Let ei be unit vector in ith coordinate direction Consider Markov chain with transition probabilities P(rk=r0 eih|rk-1=r0) = (aii(r0,tk-1)+h|i(r0,tk-1)|)/h2 P(rk=r0|rk-1=r0) = 1 -  preceding. For suitable h, 

49. Vector case dri(t) = ji(ri (t)- rj(t))dt + dBi(t) i = 1,...,p for some function  Which ? Are the animals interacting? Difficulties with unequal time spacings Time lags

50. Other topics. Uncertainties - haven't focused on. There are general methods: jackknife and bootstrap Order of approximation Unequal spacings Crossings - trajectori3es heading into regions (eg. football, debris) Moving fronts