# Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Arbitrary directions: some hypothesis and applications Addre.

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Rough Mountain elk and ATV. Starkey Reserve, Oregon, NE field. April-October 2003 ... evident increment in elk speed at ATV separations up to 1.5km ...
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﻿Cycle Romand de Statistique, 2009 Ovronnaz, Switzerland Random directions: some hypothesis and applications Lecture 3 David R. Brillinger University of California, Berkeley 2   1

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Question. Why does time exist? In the event that it didn\'t, then everything would happen in the meantime. Einstein?

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Lecture 3: Further investigations/extraordinary themes (moving) explantories alignment limits a few items. molecule forms/frameworks directions on surface (?)

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

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Concern: impacts of human trespassers, e.g. ATV on creatures\' conduct creatures ran in a kept area 2.4 m high fence 8 GPS prepared  t for elk - 5min,  t for ATV - 1sec randomization in treatment task [SDE float term relies on upon area of elk and intruder] Brillinger, Preisler, Ager, Wisdom (2004)

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Model. SDE d r (t) =  m ( r )dt +  d B (t) m = control, atv Estimated  \'s

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Control versus ATV days.

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Model. d r (t)= μ ( r (t))dt + υ (| r (t)- x (t-τ )|)dt + σ d B (t) x (t): area of ATV at time t τ : time slack Plots of | v τ | versus remove |r - x τ |

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Discussion and outline. show fit by gam() obvious increment in elk speed at ATV removes up to 1.5km an examination strategy helpful in surveying creature responses to recreational uses by people

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Whaleshark labeling study. Off Kenya Data gathered to study nature of these fish, e.g. where they voyaged, scrounged, and when? Step by step instructions to secure?, Brent

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29 June - 19 July, 2008. Indian Ocean Locations for tag from instrumented shark Unequally dispersed times, around 250 time focuses Tag discharged, floated til batteries terminated Our (shrewd) reason: to adjust ocean surface ebb and flow models comes about

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Remote detected information: ocean surface statures, zonal and meridional streams 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 labels around 200 areas every; Argos for areas Uses gradient+ to get geostrophic ebb and flow

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Geostrophic ebbs and flows for June 29, 2008 (ten day composite) Sri Lanka upper right, Maldives left Backgound bathymetry - yellow is most noteworthy

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Show motion picture,

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Brent\'s translation. "Looks like the stray begins carrying on as indicated by the main thrusts of surface current.The odd and intriguing occasion is the point at which it moves south into that little clearly frail gyre towards the end. It then does a reversal to moving under impact of current traveling south when it leaves gyre, yet this is in inverse direction that it would have taken after on the off chance that it had taken after prevailing stream before it had entered gyre. This is by all accounts a key change in condition of expected development from the invalid prediction."

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Functional stochastic differential condition (FSDE) d r(t) = μ (H(t),t)dt + σ (H(t),t)d B H(t): a history in light of the past,{ r (s), s  t} Process is Markov when H(t) = { r (t)} Interpretation r (t) - r (0) =  0 t μ (H(s),s)ds +  0 t σ (H(s),s)d B (s)

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Details of SFDE. definition, merging, ... Estimate r (t i+1 )- r (t i )=  (H(t i ),t i )(t i+1 - t i )+  (H(t i ),t i )  {t i+1 - t i } Z i+1

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Analysis. Diminished label information to 46 adjacent 12 hour terms middle qualities Estimated nearby zonal and meridional speeds diagramed versus time and each other Looking for approval of NOAA qualities

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Some subtle elements of calculations. assessed neighborhood zonal and meridional speeds by straightforward contrasts smoothed/handled these with biweight length 5 interjected remote detected qualities to label times .....

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Pre and post running biweight.

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On same plot

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Incorporating streams and winds and past areas Regression display, label speed ( r (t i+1 )- r (t i ))/(t i+1 - t i ) =  (H(t i ),t i ) +  +  C X C ( r (t i ),t i ) +  V X V ( r (t i ),t i ) + σ Z i+1/√ ( t i+1 - t i ) where  (H(t),t) =   t-1 r (s)dM(s) M(t) = #{ t i  t}, checking capacity

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Tag ebbs and flows and residuals from fits

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relapse 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 R 2 = 0.804 m e ridional case  0.708062 0.041549 17.042  C 0.240575 0.039707 6.059  V 0.025608 0.005453 4.696 R 2 = 0.854

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Residuals presenting factors progressively  = 0,  = 1;  C ;  C ,  V ;  C ,  V , 

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Discusssion and rundown. Utilize NOAA values with some alert and further handling, if conceivable Can utilize SDE result for reenactment Residuals to find things inspirations - SDE, FSDE ceaseless time and afterward discrete time strong/safe smoothing

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The instance of limited areas . Human made wall, islands for seals, ... Assume the district is D is shut with limit  D . Consider the SDE d r = μ ( r )dt + σ ( r )d B (t) - d A ( r ) where An is an adjusted procedure that lone increments when r (t) is on the limit  D. Reason for existing is to reflect molecule back to the inside of D. One can\'t just utilize the Euler plot throwimg away a point if molecule goes outside D. Predisposition comes about.

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Method 1. .Fabricate an inclining divider. That is have a potential rising quickly at the limit  D when moving to the inside. One may take H( r ) =  d( r ,  D)  ,  scalars Here graduate H dt is an estimate to d A . Technique 2. Let  D signify the projection administrator taking a r to the closest purpose of D. Let   0 and  ( r )={ r -  D (r)}/ . Utilize the plan r (t k+1 ) = r (t k ) +  ( r (t k ),t k ))(t k+1 - t k ) +  ( r (t k ),t k )  (t k+1 - t k ) Z k+1 -  ( r (t k ),t k )(t k+1 - t k ) These focuses may go outside the limit, yet by taking  sufficiently little gets a point inside

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Method 3. Consider the plan r (t k+1 ) =  D ( r (t k )+  (r( t k )(t k+1 - t k ) +  ( r (t k )  (t k+1 - t k ) Z k+1 ) If a point falls outside D extend back to the limit. These qualities do lie in D. Brillinger (2003)

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An unrefined estimate is given by the strategy: if created point goes outside, continue pulling back significantly til inside.

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Second creature. CDA Male adolescent Released La\'au Point 4 April 2004 Study finished 27 July n = 754 more than 88.4 days I = 144

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Brillinger, Stewart, Littnan (2005)

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Potential capacity utilized H(x,y)= β 10 x+β 01 y+β 20 x 2 +β 11 xy+β 02 y 2 +C/d M (x,y) d M (x,y): separation to Molokai

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Potential gauge

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Turing test

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Discrete markov chain approach, Kushner (1976). Assume D = {r:  (r)  0} with limit  D = {r:  (r)=0} Set a(r,t) = ½  (r,t)  (r,t)\' For present accommodation assume an ij (r,t) =0 i  j Suppose t k+1 - t k =  t D h alludes to cross section focuses in D with partition h. Assume r 0 in D h Let e i be unit vector in ith arrange heading Consider Markov chain with move probabilities P(r k =r 0  e ih |r k-1 =r 0 ) =  (an ii (r 0 ,t k-1 )+h|  i (r 0 ,t k-1 )|  )/h 2 P(r k =r 0 |r k-1 =r 0 ) = 1 -  going before. For reasonable h, 

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Vector case d r i (t) =  j i  ( r i (t)- r j (t))dt + d B i (t) i = 1,...,p for some capacity  Which  ? Are the creatures communicating? Troubles with unequal time spacings Time slacks

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Other themes. Vulnerabilities - haven\'t concentrated on. There are general strategies: folding blade and bootstrap Order of estimation Unequal spacings Crossings - trajectori3es heading into districts (eg. football, flotsam and jetsam) Moving fronts

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Discussion and rundown.

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Acknowledgments. Information/foundation suppliers, teammates Aager, Guckenheimer, Guttorp, Kie, Oster, Preisler, Stewart, Wisdom, Littnan, Mendolssohn, Foley, Dewitt Lovett, Spector

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