nonstationary covariance structure i: deformations
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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.
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C(x,y) = C( x − y )
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C(x,y) = C( 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.
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C(x,y) = C( 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 J(f) is a penalty for non-smooth transformations, such as the bending energy
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ˆ C ij
(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∫∫
SARMAP
An ozone monitoring exercise in California, summer of 1990, collected data on some 130 sites.
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)
D(x,y) =( f(x) −f(y) ), x,y ∈Rn
RichnessPerrin & Senoussi (2000): Let f and f-1 be differentiable, and let r(x,y) be continuously differentiable. Then
(stationarity) iff
Let f(0)=0,ci ith column of . Then
(isotropy) iff and
r(x,y) =ρ(f(x) −f(y))
(∗)Dxr(x,y)J f−1(x) +Dyr(x,y)J f
−1(y), x ≠y
Jf−1(0)
r(x,y) =ρ( f(x) −f(y) ) (∗)
fi (y)Dxr(0,y)c j =fj (y)Dxr(0,y)ci
The Brownian sheet
Let andThen
So the Brownian sheet can be thought of as a stationary deformation. It is however not an istropic deformation.
r(x,y) =x + y − y−x
2 x y
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.
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)
A Bayesian implementationLikelihood:
Prior:
Linear part: fix two points in the G-D mapping put a (proper) prior on the remaining two parameters
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L(S | Σ) = (2πΣ )−(T−1)/ 2 exp −T2
trΣ−1S ⎧ ⎨ ⎩
⎫ ⎬ ⎭
p(W) ∝ exp −
12τ
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
Other applications
Point process deformation (Jensen & Nielsen, Bernoulli, 2000)
Deformation of brain images (Worseley et al., 1999)
Global processes
Problems such as global warming require modeling of processes that take place on the globe (an oriented sphere). Optimal prediction of quantities such as global mean temperature need models for global covariances.
Note: spherical covariances can take values in [-1,1]–not just imbedded in R3.
Also, stationarity and isotropy are identical concepts on the sphere.
Isotropic covariances on the sphere
Isotropic covariances on a sphere are of the form
where p and q are directions, pq the angle between them, and Pi the Legendre polynomials.
Example: ai=(2i+1)ρi
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C(p,q) = aii= 0
∞
∑ Pi (cosγpq )
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C(p,q) =1− ρ2
1− 2ρcos γpq + ρ2 − 1
Global temperature
Global Historical Climatology Network 7280 stations with at least 10 years of data. Subset with 839 stations with data 1950-1991 selected.
Spherical deformation
Need isotropic covariance model on transformation of sphere/globe
Covariance structure on convex manifolds
Simple option: deform globe into another globe
Alternative: MRF approach
A class of global transformations
Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude).
(Das, 2000)