Continuity and covariance

Recall that for stationary processes

so if C is continuous then Z is mean square continuous.

To get results about sample paths need stronger conditions. Isotropic case: if

then fractal dimension is 3-/2

2m times differentiable at origin yields paths having m derivatives

γ(h) = 12 E(Z(u + h) − Z(u))2 = C(0) − C(h)

1−(t) : t

Nugget and smoothness

Matérn covariance, =3/2,=1. Nugget 2 = 0 (left); small (right)

Simulated using RandomFields in R.

Nonstationary covariance structure I: Deformations

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Evidence of anisotropy15o red60o green105o blue150o brown

Another view of anisotropy

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σe2 = 127.1(259)

σs2 = 68.8 (255)

θ = 10.7 (45)

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σe2 = 154.6 (134)

σs2 = 141.0 (127)

θ = 29.5 (35)

General setup

Z(x,t) = (x,t) + (x)1/2E(x,t) + (x,t)

trend + smooth + error

We shall assume that is known or constant

t = 1,...,T indexes temporal replications

E is L2-continuous, mean 0, variance 1, independent of the error C(x,y) = Cor(E(x,t),E(y,t))

D(x,y) = Var(E(x,t)-E(y,t)) (dispersion)

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Cov(Z(x,t),Z(y,t)) =ν(x)ν(y)C(x,y) x ≠ y

ν(x) + σε2 x = y

⎧ ⎨ ⎩

Geometric anisotropy

Recall that if we have an isotropic covariance (circular isocorrelation curves).

If for a linear transformation A, we have geometric anisotropy (elliptical isocorrelation curves).

General nonstationary correlation structures are typically locally geometrically anisotropic.

C(x,y) =( x−y )

C(x,y) =( Ax−Ay )

The deformation idea

In the geometric anisotropic case, write

where f(x) = Ax. This suggests using a general nonlinear transformation

. Usually d=2 or 3. G-plane D-space

We do not want f to fold.

C(x,y) =( f(x) −f(y) )

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f:R2 → Rd

Implementation

Consider observations at sites x1, ...,xn. Let be the empirical covariance between sites xi and xj. Minimize

where P(f) is a penalty for non-smooth transformations, such as the bending energy

Solution: pair of thin-plate splines

(θ, f) a wij Cij −C(f(xi ), f(xj ); θ)( )

i,j∑

2+ λTP(f)

P(f)i =∂2fi∂x2

⎛

⎝⎜⎞

⎠⎟

2

+ 2∂2fi∂x∂y

⎛

⎝⎜⎞

⎠⎟

2

+∂2fi∂y2

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥dxdy∫∫

Cij

SARMAP

An ozone monitoring exercise in California, summer of 1990, collected data on some 130 sites.

Transformation

This is for hr. 16 in the afternoon

Identifiability

Perrin and Meiring (1999): Let

If (1) f and f-1 are differentiable in Rn

(2) (u) is differentiable for u>0

then (f,) is unique up to a scaling for and a homothetic transformation for f (scaling, rotation, reflection)

C(x,y) =( f(x) −f(y) ), x,y ∈Rn

RichnessPerrin & Senoussi (2000): Let f and f-1 be differentiable, and let C(x,y) be continuously differentiable. Then

(stationarity) iff

Let f(0)=0,ci ith column of . Then

(isotropy) iff and

C(x,y) =(f(x) −f(y))

(∗)DxC(x,y)J f−1(x) =DyC(x,y)J f

−1(y), x ≠y

Jf−1(0)

C(x,y) =( f(x) −f(y) ) (∗)

fi (y)DxC(0,y)c j =fj (y)DxC(0,y)ci

Levy’s spatial Brownian motion

C(x,y) =x + y − y−x

2 x y

Deformation of Brownian process

Let and

Then

So this Gaussian process can be thought of as a stationary deformation. It is however not an istropic deformation.

f1(x) =ln( x ) f2 (x) =arctan(x1 / x2 )

(s) = 12 {exp(s1 / 2 + exp(−s1 / 2)

− exp(s1 / 2 + exp(−s1 / 2) − 2cos(s2 )}

Estimating variability

Resample time slices with replacement from the original data (to maintain spatial structure).

Re-estimate deformation based on each bootstrap sample.

Kriging estimates can be made based on each of the bootstrap estimates, to get a better sense of the variability.

French rainfall data

Altitude-adjusted 10-day aggregated rainfall data Nov-Dec 1975-1992 for 39 sites from Languedoc-Rousillon region of France.

Estimated deformation

(a)

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G-plane equicorrelation contours

D-plane Equicorrelation Contours

Uncertainty in deformationModel refitted using 24 of the 36 sites.

The smoothing parameter λ

Cross-validation: Leave out sampling station i, estimate(θ,f) from remaining n-1 stations. Minimize the prediction error for site i, summed over i.

Together with bootstrap estimate of variability, very computer intensive.

Thin-plate splines

f(s) =c + As

linear part1 2 3 + WT %σ(s)

%σ(s) = σ(s − x1),...,σ(s − xn )( )T

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σ(h) = h 2 log( h )

1T W =0 XTW=0

P(f) =tr(WT %SW)

%S = %σ(x1)L %σ(xn)( )

A Bayesian implementationLikelihood:

(Integrated) likelihood:

S Σ⎡⎣ ⎤⎦= Z,S m, Σ⎡⎣ ⎤⎦ m[ ]dm∫∝ Σ −(T−1) 2

exp −T2

trΣ−1S⎧⎨⎩

⎫⎬⎭

Z1,K ,ZN m,θ, f, ,σ 2⎡⎣ ⎤⎦

= Z,S m, Σ⎡⎣ ⎤⎦

=2πΣ −T 2exp −

T2

trΣ−1S −T2

Z−m( )TΣ−1 Z−m( )

⎧⎨⎩

⎫⎬⎭

[m] ∝ 1

Prior on deformation

Linear part: fix two points in the G-D mapping put a (proper) prior on the remaining two parameters

p(W) ∝ exp −

12

WiT %SWi

i=1

2

∑⎛⎝⎜

⎞⎠⎟

smoothing parameter

Computation

Metropolis-Hastings algorithm for sampling from highly multidimensional posterior.

Given estimates of D-plane locations, f(xi), the transformation is extrapolated to the whole domain using thin plate splines. Predictive distributions for

(a) temporal variance at unobserved sites,

(b) the spatial covariance for pairs of observed and/or unobserved sites,

(c) the observation process at unobserved sites.

California ozone

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63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 00:12:36 PST 2003

Posterior samples

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N=63, S. Calif: 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 22:18:29 PST 2003

Application to US presidential elections

NY Times election map 2004

Other applications

Point process deformation (Jensen & Nielsen, Bernoulli, 2000)

Deformation of brain images (Worseley et al., 1999)