Spatial prediction Stat 498A, Winter 2008. The geostatistical model Gaussian process  (s)=EZ(s) Var Z(s) < ∞ Z is strictly stationary if Z is weakly.

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Presentation transcript:

Spatial prediction Stat 498A, Winter 2008

The geostatistical model Gaussian process  (s)=EZ(s) Var Z(s) < ∞ Z is strictly stationary if Z is weakly stationary if Z is isotropic if weakly stationary and

The spatial prediction problem Given observations at n locations Z(s 1 ),...,Z(s n ) estimate Z(s 0 ) (the process at an unobserved site) or (an average of the process) In the environmental context often time series of observations at the locations.

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)

A Gaussian formula If then

Simple kriging Let X = (Z(s 1 ),...,Z(s n )) T, Y = Z(s 0 ), so that  X =  1 n,  Y = ,  XX =[C(s i -s j )],  YY =C(0), and  YX =[C(s i -s 0 )]. Then This is the best unbiased linear predictor when  and C are known (simple kriging). The prediction variance is

Some variants Ordinary kriging (unknown  ) where Universal kriging (  (s)=A(s)  for some spatial variable A) Still optimal for known C.

The (semi)variogram Intrinsic stationarity Weaker assumption (C(0) needs not exist) Kriging predictions can be expressed in terms of the variogram instead of the covariance.