c173c273_lec15_w11[1]

c173c273_lec15_w11[1] - University of California, Los...

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University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Co-kriging Suppose that at each spatial location s i ,i = 1 ,...,n we observe k variables as follows: Z 1 ( s 1 ) Z 1 ( s 2 ) ... Z 1 ( s n ) Z 2 ( s 1 ) Z 2 ( s 2 ) Z 2 ( s n ) . . . . . . . . . . . . Z k ( s 1 ) Z k ( s 2 ) Z k ( s n ) We want to predict Z 1 ( s 0 ), i.e. the value of variable Z 1 at location s 0 . This situation that the variable under consideration (the target variable) occurs with other vari- ables (co-located variables) arises many times in practice and we want to explore the possibility of improving the prediction of variable Z 1 by taking into account the correlation of Z 1 with these other variables. For example, the prediction of lead can be improved if we also know the values of zinc at each spatial location. The value of lead will be predicted by the observed values of lead but also by the observed values of zinc. The predictor assumption: ˆ Z 1 ( s 0 ) = k X j =1 n X i =1 w ji Z j ( s i ) = w 11 z 1 ( s 1 ) + w 12 z 1 ( s 2 ) + + w 1 n z 1 ( s n ) + w 21 z 2 ( s 1 ) + w 22 z 2 ( s 2 ) + + w 2 n z 2 ( s n ) + . . . + . . . + + . . . + w k 1 z k ( s 1 ) + w k 2 z k ( s 2 ) + + w kn z k ( s n ) We see that there are weights associated with variable Z 1 but also with each one of the other variables. We will examine ordinary co-kriging, which means that E ( Z j ( s i )) = μ j for all j and i . In vector form: E ( Z ( s )) = E ( Z 1 ( s )) E ( Z 2 ( s )) . . . . . . E ( Z k ( s ) = μ 1 μ 2 . . . . . . μ k = μ . 1
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We want the predictor ˆ Z 1 ( s 0 ) to be unbiased, that is E ± ˆ Z 1 ( s 0 ) ² = μ 1 : E ± ˆ Z 1 ( s 0 ) ² = k X j =1 n X i =1 w ji E ( Z j ( s i )) = w 11 E ( z 1 ( s 1 )) + w 12 E ( z 1 ( s 2 )) + ... + w 1 n E ( z 1 ( s n )) + w 21 E ( z 2 ( s 1 )) + w 22 E ( z 2 ( s 2 )) + + w 2 n E ( z 2 ( s n )) + . . . + . . . + + . . . + w k 1 E ( z k ( s 1 )) + w k 2 E ( z k ( s 2 )) + + w kn E ( z k ( s n )) = n X i =1 w 1 i μ 1 + n X i =1 w 2 i μ 2 + + n X i =1 w ki μ k = μ 1 . Therefore, we must have the following set of constraints: n X i =1 w 1 i = 1 n X i =1 w 2 i = 0 . . . . . . . . . n X i =1 w ki = 0 As with the other forms of kriging, co-kriging minimizes the mean squared error of prediction (MSE): min σ 2 e = E [ Z ( s 0 ) - ˆ Z ( s 0 )] 2 or min σ 2 e = E Z ( s 0 ) - k X j =1 n X i =1 w ji Z j ( s i ) 2 subject to the constraints: n X i =1 w 1 i = 1 n X i =1 w 2 i = 0 . . . . . . . . . n X i =1 w ki = 0 2
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For smplicity, lets assume k = 2, in other words, we observe variables Z 1 and Z 2 and we want to predict Z 1 . Therefore, min σ 2 e = E " Z ( s 0 ) - n X i =1 w 1 i Z 1 ( s i ) - n X i =1 w 2 i Z 2 ( s i ) # 2 Let’s add the following quantities: - μ 1 + μ 1 + n i =1 w 2 i μ 2 : min σ 2 e = E " Z ( s 0 ) - n X i =1 w 1 i Z 1 ( s i ) - n X i =1 w 2 i Z 2 ( s i ) - μ 1 + μ 1 + n X i =1 w 2 i μ 2 # 2 or min σ 2 e = E " ( Z ( s 0 ) - μ 1 ) - n X i =1 w 1 i [ Z 1 ( s i ) - μ 1 ] - n X i =1 w 2 i [ Z 2 ( s i ) - μ 2 ] # 2 We complete the square above to get: [ Z ( s 0 ) - μ 1 ] 2 - 2 n X i =1 w 1 i [ Z 1 ( s 0 ) - μ 1 ][ Z 1 ( s i ) - μ 1 ] - 2 n X i =1 w 2 i [ Z 1 ( s 0 ) - μ 1 ][ Z 2 ( s i ) - μ 2 ] + n X i =1 n X j =1 w 1 i w 1 j [ Z 1 ( s i ) - μ 1 ][ Z 1 ( s j ) - μ 1 ] + n X i =1 n X j =1 w 2 i w 2 j [ Z 2 ( s i ) - μ 2 ][ Z 2 ( s j ) - μ 2 ] + 2 " n X i =1 w 1 i [ Z 1 ( s i ) - μ 1 ] #" n X i =1 w 2 i [ Z 2 ( s i ) - μ 2 ] #
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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_lec15_w11[1] - University of California, Los...

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