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2.1 Spatial covariances
NRCSE
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Valid covariance functionsBochner’s theorem: The class ofcovariance functions is the class ofpositive definite functions C:
Why?
aij!
i! ajC(si,sj ) " 0
aij
!i
! ajC(si,sj ) = Var( ai! Z(si ))
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Spectral representation
By the spectral representation any isotropiccontinuous correlation on Rd is of the form
By isotropy, the expectation depends only onthe distribution G of . Let Y be uniform on theunit sphere. Then
�
!(v) = E eiuTX"
#
$
% , v = u ,X &Rd
�
X
�
!(v) = Eeiv X Y
= E"Y (v X )
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Isotropic correlation
Jv(u) is a Bessel function of the firstkind and order v.Hence
and in the case d=2
(Hankel transform)
�
!(v) = "Y (sv)0
#
$ dG(s)
�
!Y(u) =2
u
"
#
$
%
d
2&1
'd
2
"
#
$
% Jd2&1(u)
!(v) = J0 (sv)dG(s)0
"
#
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The Bessel function J0
0 200 400 600 800 1000 1200 1400 1600-0.5
0
0.5
1
–0.403
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The exponential correlation
A commonly used correlation functionis ρ(v) = e–v/φ. Corresponds to aGaussian process with continuous butnot differentiable sample paths.More generally, ρ(v) = c(v=0) + (1-c)e–v/φ
has a nugget c, corresponding tomeasurement error and spatialcorrelation at small distances.All isotropic correlations are a mixtureof a nugget and a continuous isotropiccorrelation.
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The squared exponential
Using yields
corresponding to an underlyingGaussian field with analytic paths.This is sometimes called the Gaussiancovariance, for no really good reason.A generalization is the power(ed)exponential correlation function,
G'(x) =2x
!2e"4x2 /!2
!(v) = e" v
#( )2
!(v) = exp " v#$% &'
(( ), 0 < ( ) 2
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The spherical
Corresponding variogram
!(v) =1" 1.5v + 0.5 v
#( )3
; h < #
0, otherwise
$%&
'&
!2+"2
23 t
# + (t# )3( ); 0 $ t $ #
!2+ "
2; t > #
nugget
sill range
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The Matérn class
where is a modified Bessel function ofthe third kind and order κ. It corresponds toa spatial field with κ–1 continuousderivativesκ=1/2 is exponential;κ=1 is Whittle’s spatial correlation; yields squared exponential.
G'(x) =2!
"2!
x
(x2 + "#2 )1+!
!(v) =1
2"#1$(")v
%&'(
)*+
"
K"
v
%&'(
)*+
K!
! " #
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Some othercovariance/variogram
families
VariogramCovarianceName
NonePower law
NoneLinear
Rationalquadratic
Wave!2 sin("t)
"t
!2 (1"
t2
1+ #t2)
!2+ "
2 (1#sin($t)
$t)
!2+
"2t2
1+ #t2
!2+ "
2t
!2+ "
2t#
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RecallMethod of moments: square of all pairwisedifferences, smoothed over lag bins
Problems: Not necessarily a valid variogramNot very robust
Estimation ofvariograms
! (h) =1
N(h)(Z(si ) " Z(sj ))
2
i,j#N(h)
$
N(h) = (i, j) : h!"h
2# si ! sj # h+
"h
2
$%&
'()
! (v) = "2 (1# $(v))
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A robust empiricalvariogram estimator
(Z(x)-Z(y))2 is chi-squared for GaussiandataFourth root is variance stabilizingCressie and Hawkins:
!! (h) =
1
N(h)Z(si ) " Z(sj )
12#
$%&
'()
4
0.457 +0.494
N(h)
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Least squares
Minimize
Alternatives:•fourth root transformation•weighting by 1/γ2
•generalized least squares
�
!! ([ Z(si ) " Z(sj)]2" # ( si " sj ;!)( )
j
$i
$2
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Maximum likelihood
Z~Nn(µ,Σ) Σ= α[ρ(si-sj;θ)] = α V(θ)Maximize
and θ maximizes the profile likelihood
�
!(µ ,! ,") = #n
2log(2$! ) #
1
2logdetV(")
+1
2!(Z# µ )TV(")#1(Z# µ )
�
ˆ µ = 1TZ / n ˆ ! = G(ˆ " ) / n G(") = (Z# ˆ µ )TV(")#1(Z# ˆ µ )
�
! * (!) = "n
2log
G2(!)
n"1
2logdetV(!)
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Parana data
ml
ls
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A peculiar ml fit
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Some more fits
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All together now...
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Bayesian kriging
Instead of estimating the parameters,we put a prior distribution on them, andupdate the distribution using the data.Model:Prior:
Posterior:
(Z !) ~ N(",#2 C($) + %2I)
f(!) = f(")f(#2 )f($)f(%2 )
f(! Z = z) " f(!) f(z #)f($2 )f(%)f(&2 )d''' $2d%d&2
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geoR
Prior is assigned to φ and . Thelatter assumed zero unless specified.The distributions are discretized.Default prior on mean β is flat (if notspecified, assumed constant).(Lots of different assignments arepossible)
! "
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Prior/posterior of φ
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Variogram estimates
meanmedianWLS
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Bayes vs universal kriging
Bayes predictive mean Universal kriging
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Spectral representationStationary processes
Spectral process Y has stationaryincrements
If F has a density f, it is called thespectral density.
Z(s) = exp(isT!)dY(!)Rd
"
E dY(!)2= dF(!)
Cov(Z(s1),Z(s2 )) = ei(s1-s2 )T!f(!)d!R2
"
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Estimating the spectrum
For process observed on nxn grid,estimate spectrum by periodogram
Equivalent to DFT of sample covariance
In,n (!) =1
(2"n)2z(j)ei!
Tj
j#J
$
2
! =2"j
n; J = (n# 1) / 2$% &' ,...,n# (n# 1) / 2$% &'{ }
2
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Properties of theperiodogram
Periodogram values at Fourierfrequencies (j,k)π/Δ are•uncorrelated•asymptotically unbiased•not consistentTo get a consistent estimate of thespectrum, smooth over nearbyfrequencies
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Some commonisotropic spectra
Squared exponential
Matérn
f(!)="2
2#$exp(% !
2/ 4$)
C(r) = "2 exp(%$ r
2)
f(!) = "(#2+ !
2)$%$1
C(r) =&"(# r )%K
%(# r )
2%$1'(% + 1)#2%
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A simulated process
Z(s) = gjk cos 2!js1m
+ks2n
"#$
%&'+Ujk
()*
+,-k=.15
15
/j=0
15
/
gjk = exp(! j+ 6 ! k tan(20°) )
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Thetford canopy heights
39-year thinned commercialplantation of Scots pine inThetford Forest, UKDensity 1000 trees/ha36m x 120m area surveyed forcrown heightFocus on 32 x 32 subset
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Spectrum of canopy heights
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Whittle likelihood
Approximation to Gaussian likelihoodusing periodogram:
where the sum is over Fourier frequencies,avoiding 0, and f is the spectral densityTakes O(N logN) operations to calculateinstead of O(N3).
!(!) = logf(";!) +IN,N (")
f(";!)
#$%
&'("
)
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Using non-gridded data
Consider
where
Then Y is stationary with spectraldensity
Viewing Y as a lattice process, it hasspectral density
Y(x) = !"2 h(x " s)# Z(s)ds
h(x) = 1( xi ! " / 2, i = 1,2)
fY (!) =1
"2H(!)
2fZ (!)
f! ,Y (") = H(" +
2#q
!)
2
fZq$Z2% (" +
2#q
!)
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Estimation
Let where Jx is the grid square with centerx and nx is the number of sites in thesquare. Define the tapered periodogram
where . The Whittlelikelihood is approximately
Yn2(x) =
1
nxh(si ! x)Z(si )
i"Jx
#
Ig1Yn2(!) =
1
g12 (x)"
g1(x)Yn2 (x)e# ixT!
"2
g1(x) = nx / n
!Y
=n2
2!( )2
logf" ,Y (2!j / n) +Ig1,Yn2
(2!j / n)
f" ,Y (2!j / n)
#$%
&%
'(%
)%j
*
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A simulated example
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Estimated variogram
trueexact mleapprox mle
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Thetford revisited
Features depend on spatial location
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Some references
Bertil Matern; Spatial Variation. Medd. StatensSkogsforskningsinst.49: 5, Ch 2-3.(Reprinted in Springer Lecture Notes In Statistics, vol. 36)
Cressie: ch. 2.3.1, 2.4, 2.6.
P. Guttorp, M. Fuentes and P. D. Sampson (2006): Usingtransforms to analyze space-time processes.http://www.nrcse.washington.edu/pdf/trs80.pdf (toappear,SemStat 2004 proceedings, Chapman & Hall)
Banerjee, S., Carlin, B. P. and Gelfand, A. E.: HierarchicalModeling and Analysis for Spatial Data, Chapman andHall, 2004. pp. 129-135.