c173c273_lec10_w11[1]

c173c273_lec10_w11[1] - University of California, Los...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Ordinary kriging Kriging (Matheron 1963) owes its name to D. G. Krige a South African mining engineer and it was first applied in mining data. Kriging assumes a random field expressed through a variogram or covariance function. It is a very popular method to solve the spatial prediction problem. Let Z = ( Z ( s 1 ) ,Z ( s 2 ) ,...,Z ( s n )) be the vector of the observed data at known spatial locations s 1 ,s 2 ,...,s n . The objective is to estimate the unobserved value Z ( s ) at location s . The model: The model assumption is: Z ( s ) = μ + δ ( s ) where δ ( s ) is a zero mean stochastic term with variogram 2 γ ( · ). The variogram was discussed in previous handouts in detail. The Kriging System The predictor assumption is ˆ Z ( s ) = n X i =1 w i Z ( s i ) i.e. it is a weighted average of the sample values, and ∑ n i =1 w i = 1 to ensure unbiasedness. The w i ’s are the weights that will be estimated. Kriging minimizes the mean squared error of prediction min σ 2 e = E [ Z ( s )- ˆ Z ( s )] 2 or min σ 2 e = E " Z ( s )- n X i =1 w i Z ( s i ) # 2 For intrinsically stationary process the last equation can be written as: σ 2 e = 2 n X i =1 w i γ ( s- s i )- n X i =1 n X j =1 w i w j γ ( s i- s j ) (1) See next page for the proof: 1 Let’s examine ( Z ( s )- ∑ n i =1 w i Z ( s i )) 2 : z ( s )- n X i =1 w i z ( s i ) ! 2 = z 2 ( s )- 2 z ( s ) n X i =1 w i z ( s i ) + n X i =1 n X j =1 w i w j z ( s i ) z ( s j ) = n X i =1 w i z 2 ( s )- 2 n X i =1 w i z ( s ) z ( s i ) + n X i =1 n X j =1 w i w j z ( s i ) z ( s j )- 1 2 n X i =1 w i z 2 ( s i )- 1 2 n X j =1 w j z 2 ( s j ) + n X i =1 w i z 2 ( s i ) =- 1 2 n X i =1 n X j =1 w i w j [ z ( s i )- z ( s j )] 2 + n X i =1 w i [ z ( s )- z ( s i )] 2 If we take expectations on the last expression we have- 1 2 n X i =1 n X j =1 w i w j E [ z ( s i )- z ( s j )] 2 + n X i =1 w i E [ z ( s )- z ( s i )] 2 =- 1 2 n X i =1 n X j =1 w i w j var [ z ( s i )- z ( s j )] + n X i =1 w i var [ z ( s )- z ( s i )] But var [ z ( s i )- z ( s j )] = 2 γ ( · ) is the definition of the variogram, and therefore the previous expression is written as: 2 ∑ n i =1 w i γ ( s- s i )- ∑ n i =1 ∑ n j =1 w i w j γ ( s i- s j ) Therefore kriging minimizes σ 2 e = E [( Z ( s )- n X i =1 w i Z ( s i )] 2 = 2 n X i =1 w i γ ( s- s i )- n X i =1 n X j =1 w i w j γ ( s i- s j ) subject to n X i =1 w i = 1 The minimization is carried out over ( w 1 ,w 2 ,...,w n ), subject to the constraint ∑ n i =1 w i = 1....
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This note was uploaded on 02/11/2012 for the course STATS c173/c273 taught by Professor Nicolaschristou during the Spring '11 term at UCLA.

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c173c273_lec10_w11[1] - University of California, Los...

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