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By Wilsky A.S.

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Ch (xn , xn ) Cg (xn , x1 ) · · · Cg (xn , xn ) 0 ··· .. ··· ⎤ 0 .. ⎥ . ⎦ σA2 = K h + Bww T B T + diag σA2 , σA2 , . . , σA2 . 25) We note that under our model formulations, typically we would select ζi2 , ζ j2 and ζi,2 j all to zero, since we already have a baseline covariance function given by the independent spatial GP h(·). We may now reinterpret the model covariance as a form of covariance regression which admits a representation as a random effects model, making it an extension of the framework proposed in [17].

This is quite a generic structure since many possible choices may be made for what would go into w. 9) is a valid Mercer kernel and will construct a covariance matrix which will be symmetric and positive definite so long as c1 , c2 > 0 and kernels k1 and k2 are Mercer kernels. The construction of the covariance kernel in this manner admits two different types of interpretation of the resulting spatial model. The first is based on a linear combination of two GPs, the second is based on a hybrid model which involves a linear combination of a GP and a Gaussian graphical model (GMM) of [18].

It follows from f g and (CYY + εn I)−1 ≤ 1/εn that ξ yx = (CYY + εn I)−1 k(·, y) k(·, x) ≤ 1 k(·, y) εn f ⊗g = k(·, x) ≤ a2 , εn and E ξYX 2 = E {(CYY + εn I)−1 k(·, Y )} ⊗ k(·, X) = E k(·, X) 2 (CYY + εn I)−1 k(·, Y ) ≤ a 2 E (CYY + εn I)−1 k(·, Y ) 2 2 2 = a 2 E (CYY + εn I)−2 k(·, Y ), k(·, Y ) = a 2 ETr (CYY + εn I)−2 (k(·, Y ) ⊗ k(·, Y )∗ ) = a 2 Tr (CYY + εn I)−2 CYY ≤ a2 a2 Tr (CYY + εn I)−1 CYY = N(εn ). εn εn Here k(·, Y )∗ is the dual element of k(·, Y ) and k(·, Y ) ⊗ k(·, Y )∗ is regarded as an operator on HY .

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