space-time processes nrcse. separability separable covariance structure: cov(z(x,t),z(y,s))=c s...
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Separability
Separable covariance structure:
Cov(Z(x,t),Z(y,s))=CS(x,y)CT(s,t)
Nonseparable alternatives
•Temporally varying spatial covariances
•Fourier approach
•Completely monotone functions
Bruno’s seasonal nonseparability
Nonseparability generated by seasonally changing spatial term
Z1 large-scale feature
Z2 separable field of local features
(Bruno, 2004)
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Y(t,x) = μ(t,x)+ σ t (x)(α xZ1(t)+ Z2 (t,x))+ ε(t,x)
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σt (x )
General stationary space-time covariances
Cressie & Huang (1999): By Bochner’s theorem, a continuous, bounded, symmetric integrable C(h;u) is a space-time covariance function iff
is a covariance function for all .
Usage: Fourier transform of C(u)
Problem: Need to know Fourier pairs
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Cω(u) = e−ihTω∫ C(h,u)dh
C(h,u) = exp(iuτ)C1(h, τ)κ(τ)
f(h; τ)1 24 34 dτ∫
Spectral density
Under stationarity and separability,
If spatially nonstationary, write
Define the spatial coherency as
Under separability this is independent
of frequency τ
f(h;τ) =ϕ (h)κ(τ)
fa,b (τ) =12π
exp(−iuτ)Cov(Z(a, t+u),Z(b, t))du∫
Ra,b (τ) =fa,b (τ)
fa,a (τ)fb,b (τ)⎡⎣ ⎤⎦12
Estimation
Let
where R is estimated using
φa,b (τ) = tanh−1(Ra,b (τ))
fa,b (τ) = gρ (a−s)gρ (b−s)Ia+s ,b+s∗ (τ)ds
R2∫
ANOVA results
Item df rss P-value
Between points
1 0.129 0.68
Between freqs
5 11.14 0.0008
Residual 5 0.346
A class of Matérn-type nonseparable covariances
=1: separable
=0: time is space (at a different rate)
f(,τ) =γ(α2β2 +β2 2 + α2τ2 + 2 τ2 )−ν
scale spatialdecay
temporaldecay
space-timeinteraction
Fuentes model
Prior equal weight on =0 and =1.
Posterior: mass (essentially) 0 for =0 for regions 1, 2, 3, 5; mass 1 for region 4.
Z(s, t) = K (s −s i, t−ti )Zi (s, t)i=1
5
∑
Another approach
Gneiting (2001): A function f is completely monotone if (-1)nf(n)≥0 for all n. Bernstein’s theorem shows that
for some non-decreasing F. In particular, is a spatial covariance function for all dimensions iff f is completely monotone.The idea is now to combine a completely monotone function and a function with completey monotone derivative into a space-time covariance
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f(t) = e−rtdF(r)0
∞∫
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f( h 2)
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C(h,u) =σ2
ψ(u 2)d/ 2ϕ
h 2
ψ(u 2 )
⎛
⎝ ⎜
⎞
⎠ ⎟
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ϕ
Some examples
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ϕ (t) = exp(−ctγ ), c > 0,0 < γ ≤ 1
ϕ(t) =cνtν/ 2
2ν−1Γ(ν)Kν (ct1/ 2 ), c > 0,ν > 0
ϕ(t) = (1+ ctγ )− ν , c,ν > 0,0 < γ ≤ 1
ψ(t) = (atα + 1)β , a > 0,0 < α ≤ 1,0 ≤ β ≤ 1
ψ(t) =ln(atα + b)
ln(b), a > 0,b > 1,0 < α ≤ 1
A particular case
α=1/2,γ=1/2 α=1/2,γ=1
α=1,γ=1/2 α=1,γ=1
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C(h,u) = (u 2α + 1)−1exp −h 2
(u 2α + 1)γ
⎛
⎝ ⎜
⎞
⎠ ⎟
Issues to be developed
•NonstationarityDeformation approach: 3-d spline fit;penalty for roughness in time and space
•AntisymmetryGneiting’s approach has
C(h,u)=C(-h,u)=C(h,-u)=C(-h,-u)Unrealistic when covariance caused by meteorology
•CovariatesNeed to take explicit account of covariates driving the covariance
•Multivariate fieldsMardia & Goodall (1992) use Kronecker product structure (like separability)
Temporally changing spatial deformations
where
(θ, f) a wijk Cijk −C(fk (xi ), fk (xj ); θ)( )i,j,k∑
2
+λkJ (fk ) + ηK (f)
K(f) = (fk (x) −∫k∑ fk−1(x))2dx
Velocity-driven space-time covariances
CS covariance of purely spatial field
V (random) velocity of field
Space-time covariance
Frozen field model: P(V=v)=1 (e.g. prevailing wind)
C(h,u) =EVCS (h−Vu)
C(h,−u) =C0 (h+ vu) ≠C0 (h−vu) =C(h,u)
Taylor’s hypothesis
C(0,u) = C(vu,0) for some v
Relates spatial to temporal covariance
Examples:
Frozen field model
Separable modelsC(h,u) =C0 ((a
2 h2 +b2u2 )
12 )
Irish wind data
Daily average wind speed at 11 stations, 1961-70, transformed to “velocity measures”
Spatial: exponential with nugget
Temporal:
Space-time: mixture of Gneiting model and frozen field
CT (u) =(1+ a u2α )−1