Suppose that observations have been made at a sample of observed y-values at n sites t,...,...

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Suppose that observations have been made at a sample of observed y-values at n sites t,..., tn. Write 2ij var(Y(ti) - Y(t;)) = var(Y(ti) Y(t;)). We want to predict the value of the random variable Y(to) at the site to through (to) where The value of a,... error n (to) = ai (ti) i=1 an can be determined by minimizing the mean square prediction and also to yield unbiased (to) E (Y (to) (to)) - EY (to) EY (to) Show that the solution is the following = where a = F a1 10 720 11 12 721 722 V2n 1 -(0)-(0)-(5) a= An * m* = 1 = : Yn2 Ynn 1 1 1 1 Hint: Use the following relationship between Vij and Cij 2%ij = var(Y(ti) Y(t;)) = var(Y(t;)) + var(Y(t;)) 2cov(Y(t;), Y(t;)) = C'ii + C'jj 2Cij - 20 = var(Y(t;) Y(t;)) = var(Y(t)) + var(Y(t;)) 2cov(Y(ti), Y(t;)) = = Coo + Cii - 2Cro Incorporate these result into 2 n n Coo + asaCy - 2 i=1 j=1 n i=1 on page 5 of the "Kriging_theory" handout, and then carry out the remaining proof. Suppose that observations have been made at a sample of observed y-values at n sites t,..., tn. Write 2ij var(Y(ti) - Y(t;)) = var(Y(ti) Y(t;)). We want to predict the value of the random variable Y(to) at the site to through (to) where The value of a,... error n (to) = ai (ti) i=1 an can be determined by minimizing the mean square prediction and also to yield unbiased (to) E (Y (to) (to)) - EY (to) EY (to) Show that the solution is the following = where a = F a1 10 720 11 12 721 722 V2n 1 -(0)-(0)-(5) a= An * m* = 1 = : Yn2 Ynn 1 1 1 1 Hint: Use the following relationship between Vij and Cij 2%ij = var(Y(ti) Y(t;)) = var(Y(t;)) + var(Y(t;)) 2cov(Y(t;), Y(t;)) = C'ii + C'jj 2Cij - 20 = var(Y(t;) Y(t;)) = var(Y(t)) + var(Y(t;)) 2cov(Y(ti), Y(t;)) = = Coo + Cii - 2Cro Incorporate these result into 2 n n Coo + asaCy - 2 i=1 j=1 n i=1 on page 5 of the "Kriging_theory" handout, and then carry out the remaining proof.