kriging · 2007-07-07 · Effect of estimated covariance structure The usual geostatistical method...
Transcript of kriging · 2007-07-07 · Effect of estimated covariance structure The usual geostatistical method...
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1.2 Kriging
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Research goals inair quality research
Calculate air pollution fields for healtheffect studiesAssess deterministic air quality modelsagainst dataInterpret and set air quality standardsImproved understanding ofcomplicated systemsPrediction of air quality
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The geostatistical model
Gaussian processµ(s)=EZ(s) Var Z(s) < ∞
Z is strictly stationary if
Z is weakly stationary if
Z is isotropic if weakly stationary and
Z(s),s !D " R2
(Z(s1),...,Z(sk )) =d
(Z(s1 +h),...,Z(sk +h))
µ(s) ! µ Cov(Z(s1),Z(s2 )) = C(s1 " s2)
C(s1 ! s2) = C0( s1 ! s2 )
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The problem
Given observations at n locationsZ(s1),...,Z(sn)estimate
Z(s0) (the process at an unobserved site)
(an average of the process)
In the environmental context often timeseries of observations at the locations.
Z(s)d!(s)A
"or
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Some history
Regression (Galton, Bartlett)Mining engineers (Krige 1951,Matheron, 60s)Spatial models (Whittle, 1954)Forestry (Matérn, 1960)Objective analysis (Grandin, 1961)More recent work Cressie (1993), Stein(1999)
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A Gaussian formula
If
then
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X
Y
!
" #
$
% & ~ N
µX
µY
!
" #
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% & ,
'XX 'XY'YX 'YY
!
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(Y |X) ~ N(µY + !YX!XX"1(X"µX ),
!YY " !YX!XX"1
!XY )
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Simple krigingLet X = (Z(s1),...,Z(sn))T, Y = Z(s0), so that
µX=µ1n, µY=µ,ΣXX=[C(si-sj)], ΣYY=C(0), and
ΣYX=[C(si-s0)].
Then
This is the best unbiased linear predictorwhen µ and C are known (simple kriging).
The prediction variance is
p(X) ! Z(s0 ) = µ + C(si " s0 )[ ]TC(si " sj )#$ %&
"1
X " µ1n( )
m1 = C(0) ! C(si ! s0 )[ ]TC(si ! sj )"# $%
!1
C(si ! s0 )[ ]
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Some variants
Ordinary kriging (unknown µ)
where
Universal kriging (µ (s)=A(s)β for some spatialvariable A)
Still optimal for known C.
p(X) ! Z(s0 ) = µ + C(si " s0 )[ ]TC(si " sj )#$ %&
"1
X " µ1n( )
µ = 1nT C(si ! sj )"# $%
!1
1n( )!1
1nT C(si ! sj )"# $%
!1
X
! = ( A(si )[ ]TC(si " sj )#$ %&
"1A(si )[ ])"1
A(si )[ ]TC(si " sj )#$ %&
"1X
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Universal kriging variance
E Z(s0 ) ! Z(s0 )( )2
= m1 +
A(s0 ) ! [A(si )T[C(si ! sj )]
!1[C(si ! s0 )]( )T
"( A(si )[ ]TC(si ! sj )#$ %&
!1
A(si )[ ])!1
" A(s0 ) ! [A(si )T[C(si ! sj )]
!1[C(si ! s0 )]( )
simple kriging variance
variability due to estimating β
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The (semi)variogram
Intrinsic stationarityWeaker assumption (C(0) needs notexist)Kriging predictions can be expressed interms of the variogram instead of thecovariance.
! ( h ) =1
2Var(Z(s + h) " Z(s)) = C(0) " C( h )
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Ordinary kriging
where
and kriging variance
!
Z(s0 ) = ! iZ(si )i=1
n
"
!T = " + 11# 1T$#1"1T$#11
%
&'(
)*
T
$#1
" = (" (s0 # s1),...," (s0 # sn))T
$ ij = " (si # sj )
m1(s0 ) = 2 ! i" (s0 # si )i=1
n
$ # ! i! j" (si # sj )j=1
n
$i=1
n
$
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Parana data
Built-in geoR data setAverage rainfall over different years forMay-June (dry-season)143 recording stations throughoutParana State, Brazil
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Parana precipitation
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Fitted variogram
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Is it significant?
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Kriging surface
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Kriging standard error
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A better combination
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Spatial trend
Indication of spatial trendFit quadratic in coordinates
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Residual variogram
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Effect of estimatedcovariance structure
The usual geostatistical method is toconsider the covariance known. When it isestimated• the predictor is not linear• nor is it optimal• the “plug-in” estimate of thevariability often has too low mean
Let . Is a goodestimate of m2(θ) ?
p2 (X) = p(X;!(X))
m1(!(X))
m2(!) = E!(p2 (X) " µ)2 m1(!)
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Some results
1. Under Gaussianity, m2(θ) ≥ m1(θ) withequality iff p2(X)=p(X;θ) a.s.2. Under Gaussianity, if is sufficient,and if the covariance is linear in θ, then
3. An unbiased estimator of m2(θ) is
where is an unbiased estimator ofm1(θ).
!
E!m1(!) =m2(!) " 2(m2 (!) "m1(!))
2m !m1(")
m
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Better predictionvariance estimator
(Zimmerman and Cressie, 1992)
(Taylor expansion; often approx. unbiased)
A Bayesian prediction analysis takesaccount of all sources of variability (Le andZidek, 1992; 2006)
Var(Z(s0;!)) "m1(!)
+2 tr cov(!) #cov($Z(s0;!)%&
'(
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Some references
N. Cressie (1993) Statistics for Spatial Data. Rev. ed.Wiley. Pp. 105-112,119-123, 151-157.Zimmerman, D. L. and Cressie, N (1992): Mean squaredprediction error in the spatial linear model withestimated covariance parameters. Annals of theInstitute of Statistical Mathematics 44: 27-43.Le N. D. and Zidek J. V. (1992) Interpolation WithUncertain Spatial Covariances–A Bayesian AlternativeTo Kriging. 43 (2): 351-374.N. D. Le and J. V. Zidek (2006) Statistical Analysis ofEnvironmental Space-Time Processes. Springer-Verlag.