c173c273_lec10a_w11[1]

c173c273_lec10a_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 in terms of the covariance function 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 Kriging System The predictor assumption is ˆ Z ( s ) = n X i =1 w i Z ( s i ) 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 second order stationary process the last equation can be written as: σ 2 e = C (0)- 2 n X i =1 w i C ( s ,s i ) + n X i =1 n X j =1 w i w j C ( 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 ) + μ- μ ! 2 = ( [ z ( s )- μ ]- n X i =1 w i [ z ( s i )- μ ] ) 2 = [ z ( s )- μ ] 2- 2 n X i =1 w i [ z ( s i )- μ ][ z ( s )- μ ] + n X i =1 n X j =1 w i w j [ z ( s i )- μ ][ z ( s j )- μ ] ....
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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_lec10a_w11[1] - University of California, Los...

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