Download - Variograms/Covariances and their estimation
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Variograms/Covariancesand their estimation
STAT 498B
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The exponential correlation
A commonly used correlation function is (v) = e–v/. Corresponds to a Gaussian process with continuous but not differentiable sample paths.More generally, (v) = c(v=0) + (1-c)e–v/ has a nugget c, corresponding to measurement error and spatial correlation at small distances.All isotropic correlations are a mixture of a nugget and a continuous isotropic correlation.
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The squared exponential
Usingyields
corresponding to an underlying Gaussian field with analytic paths.This is sometimes called the Gaussian covariance, for no really good reason. A generalization is the power(ed) exponential correlation function,
G'(x) =2xφ2 e −4x2 /φ2
(v) = e− v
φ( )2
(v) = exp − vφ⎡⎣ ⎤⎦
κ( ), 0 < κ ≤ 2
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The spherical
Corresponding variogram
(v) = 1− 1.5v + 0.5 vφ( )
3; h < φ
0, otherwise
⎧⎨⎪⎩⎪
τ2 + σ2
23 t
φ + ( tφ)3( ); 0 ≤ t ≤ φ
τ2 + σ2; t > φ
nugget
sill range
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The Matérn class
where is a modified Bessel function of the third kind and order . It corresponds to a spatial field with –1 continuous derivatives=1/2 is exponential; =1 is Whittle’s spatial correlation; yields squared exponential.
G'(x) = 2φ2
x(x2 + φ−2 )1+
(v) = 12κ−1Γ(κ)
vφ
⎛⎝⎜
⎞⎠⎟
κ
Kκvφ
⎛⎝⎜
⎞⎠⎟
K
→ ∞
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Some other covariance/variogram
families
Name Covariance Variogram
Wave
Rational quadratic
Linear None
Power law None
σ2 sin(φt)φt
σ2 (1− t2
1+ φt2 )
τ2 + σ2 (1− sin(φt)φt
)
τ2 + σ2t2
1+ φt2
τ2 + σ2t
τ2 + σ2tφ
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Recall Method of moments: square of all pairwise differences, smoothed over lag bins
Problems: Not necessarily a valid variogram
Not very robust
Estimation of variograms
γ(h) = 1N(h)
(Z(si ) − Z(s j ))2
i,j∈N(h)∑
N(h) = (i,φ): h−Δh2
≤σi −σφ ≤h+Δh2
⎧⎨⎩⎫⎬⎭
γ(v) = σ2 (1− ρ(v))
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A robust empirical variogram estimator
(Z(x)-Z(y))2 is chi-squared for Gaussian dataFourth root is variance stabilizingCressie and Hawkins:
%γ(h) =
1N(h)
Z(si ) − Z(s j )12∑⎧
⎨⎩
⎫⎬⎭
4
0.457 + 0.494N(h)
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Least squares
Minimize
Alternatives: • fourth root transformation• weighting by 1/γ2
• generalized least squares
€
θ a ([ Z(si ) − Z(s j)]2 − γ( si − s j ;θ)( )
j∑
i∑
2
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Maximum likelihood
Z~Nn(,) = a[(si-sj;θ)] = a V(θ)Maximize
and θ maximizes the profile likelihood
€
l(μ,α,θ) = − n2
log(2πα ) − 12
logdetV(θ)
+ 12α
(Z − μ)TV(θ)−1(Z − μ)
€
ˆ μ = 1TZ / n ˆ α = G(ˆ θ ) / n G(θ) = (Z − ˆ μ )TV(θ)−1(Z − ˆ μ )
€
l * (θ) = − n2
log G2(θ)n
− 12
logdetV(θ)
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A peculiar ml fit
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Some more fits
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All together now...