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
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 coordinates
Data would provide information on the interior structure of the Earth
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=0
T-1 exp {-it}[Z(t+1)-Z(t)]
Periodogram
IZZT() = (2T)-1|dZ
T()|2
periodogram - 1972 graphics!
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
If series stationary, mixing periodograms, Is at = 2s/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
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.
"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
Perrin (1913)
Potential function.
Quadratic. H(x,y) = γ1x + γ2y + γ11x2 + γ12xy + γ22y
2
real-valued
drift.
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)
WLS
martingale differencesasymptotic normality +, Lai and Wei (1982)
Estimate of H
Estimate of μ
Discussion.
Ornstein-Uhlenbeck like
Potential function for O-U
H = (a - r)'A(a-r)/2 A symmetric 0
quadratic
Bezerkeley football
25-pass goal. 2006 Argentina vs. Serbia-Montenegro
H(r) = log |r| + |r| + γ1x + γ2y + γ11x2 + γ12xy + γ22y
2
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, WLS
Estimate of H, image plot
Vector field
Discussion. Modelled path, not score
Asymmetry, down one side of the field
Ball speed, slow, then quick
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
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
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
Brillinger, 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/resistant
Penguin 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) changes
Parametric 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 error
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
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
Kinesin motor attached to microtubule
Malik, Brillinger and Vale (1994)
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)
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 - 2j/)
periodic spikes
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
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)+
D. R. Brillinger (1996)
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
Starkey.
Kernel density estimate
based on the locations r(tj)
Relation with potential function in stationary case
(r) = c exp{-2H(r)/2} H(r) = -½ 2 log (r)/c
Brillinger, Preisler, Ager, Kie, Stewart (2002)
Vector field model.
(r(ti+1)-r(ti))/ (ti+1-ti) = μ(r(ti),ti) + σ Zi+1/√(ti+1-ti)
Robust fit via generalized additive model for smooth μ
Discussion.
The elk appear to be more active at 0600 and 1800 hours, but staying iin a local area at 1200 and 2400 hours.
There are hot spots
The time of day effect does not appear additive.
Elephant seal journey.
Were endangered
Formulas for: great circle, SDE
D. R. Brillinger and B. S. Stewart (1998)
"Particle" heading towards North Pole
speed
bivariate Brownian disturbance (U,V). s.d.
(,): longitude, colatitude
dt = dUt + (2/2tan t - )dt
dt = (/sin t) dVt
Brownian with drift on a sphere
Parameter estimation.
Discrete approximation
t+1 - t = 2/(2 tan t) - + t+1
t+1 - t = / (sin t )t+1
Measurement error
t' = t + t'
t' = t + / (sin t')t
Results. Likelihood by simulation
No measurement error
parameter estimate s.e.
.0112 rad .001
.00805
Measurement error
out .0126 .0001
in .0109 .0001
.000489 .0000004
.0175 .0011
Discussion and summary.
Mostly location measurement error (done by light levels)
Model appraisal.
Synthetic plots (Neyman and Scott)
Simulate realization of fitted model
Put real and synthetic side by side Assessment
Turing test
Compute same quantity for each?
Distance from La’au Point in synthetic journey
Ringed seal.
Might wish to simulate other such paths
Suppose seal travels in segments with a constant velocity then,
dr = vdt
i.e. the segments are straight.
It may be that the speed |v| is approximately constant for all
Discussion and summary.
Acoustic tracking - attached pinger
Dives forages and surfaces
Finds its way back to breathing hole - need to navigate back to and between holes
straight line segments?
running biweight?
Some formal aspects.
Consider,for example,
H(x,y) = ∑ βkl gk(x)hl(y) with gk , hl differentiable
Hx(x,y) = ∑ βkl g(1)
k(x)hl(y), Hy(x,y) = ∑ βkl gk(x)h(1)l(y)
with = -(Hx,Hy)
(r(ti+1) – r(ti))/(ti+1 – ti) = (r(ti)) + σZi+1
stack the data
linear model
Martingale difference.
E{Xn+1|{X0,...,Xn}) = 0
E{Xn} = 0
CLT
Martingale.
E{Sn+1|{S0,...,Sn}) = Sn
CLT
Theorem A.1. [Lai and Wei (1982)]. Suppose yi = xiTβ + εi, i=1,2,…
with {εi} martingale differences wrt increasing sequence of σ-fields
{Fn}. Suppose further that
supn E(||εn||α|Fn-1) < ∞ a.s.
for some α > 2 and that limn→∞ var{εn|Fn-1) = σ2 a.s. for some
nonstochastic σ.
Assume that xn Fn-1-measurable r.v. and existance of non-random
positive definite symmetric L by L matrix Bn for which
Bn-1(Xn
TXn)1/2→I and sup1≤i≤n|| Bn
-1xn||→0 in probability.Then
(XnTXn)
1/2(b-β) →N(0, σ2I), in distribution as n→∞.
Note that 0 mean independent observations like the σZi+1 of
the basic model
(r(ti+1)-r(ti))/ (ti+1-ti) = μ(r(ti),ti) + σ Zi+1/√(ti+1-ti)
form a martingale difference sequence with respect to the σ-field Fi generated by {r(t1),…,r(ti)}.
CI for φ(r)T β
Theorem A.2. Under the assumptions of Theorem A.1 and lim log λmax(Xn
TXn)/n→0 almost surely, one has
((φ(r)(XnTXn)
-1φ(r)T)-1/2φ(r)T (b-β)/sn → N(0,1)
in distribution as n→∞.
sn = ((n-1)p)-1RSS
Chang and Chin (1995) least squares when
var{εn|Fn-1) = g(zn; )
zn, is observable and Fn-1 measureable
The estimate of from
min t=1n (êt
2 - g(zn; ))2
the êt residuals from the OLS fit
Asymptotic normality results
Summary and discussion.
Array of biological and physical mechanisms control how animals move, particularly on large landscapes.
Models of movement is one useful tool to study ecology of animal behavior and to test ideas concerning foraging strategies, habitat preferences, and dynamics of population densities
Cleaning the data robust/resistant
simulation
Constrained trajectories by shrinking
Potential function - effective approach
SDE motivated parameter estimate
New stochastic models result
SDE benefits.
conceptual, extendable, simulation, analytic results, prediction, effective
Potential function benefit
Motivates parametric and nonparametric estimates
difficulties: enforcing boundary
Aager, Guckenheimer, Guttorp, Kie, Oster, Preisler, Stewart, Wisdom, Littnan, Roy Mendolssohn, Dave Foley, ?
Lovett, Spector
Acknowledgements. Data/background providers and collaborators