c173c273_lec10_w11[1]

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

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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 )) 0 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 0 ) at location s 0 . 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 0 ) = 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 0 ) - ˆ Z ( s 0 )] 2 or min σ 2 e = E " Z ( s 0 ) - 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 0 - 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
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Let’s examine ( Z ( s 0 ) - n i =1 w i Z ( s i )) 2 : z ( s 0 ) - n X i =1 w i z ( s i ) ! 2 = z 2 ( s 0 ) - 2 z ( s 0 ) 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 0 ) - 2 n X i =1 w i z ( s 0 ) 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 0 ) - 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 0 ) - 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 0 ) - 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 0 - 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 0 ) - n X i =1 w i Z ( s i )] 2 = 2 n X i =1 w i γ ( s 0 - 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. Therefore the minimization problem can be written as: min 2 n X i =1 w i γ ( s 0 - s i ) - n X i =1 n X j =1 w i w j γ ( s i - s j ) - 2 λ ( n X i =1 w i - 1) (2) where λ is the Lagrange multiplier. After differentiating (2) with respect to w 1 , w 2 , ..., w n , and λ and set the derivatives equal to zero we find that - n X j =1 w j γ ( s i - s j ) + γ ( s 0 - s i ) - λ = 0 , i = 1 , ..., n and n X i =1 w i = 1 2
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Using matrix notation the previous system of equations can be written as ΓW = γ Therefore the weights w 1 , w 2 , ..., w n and the Lagrange multiplier
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