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7 Command file: Specifying variance structures 128New⎡⎤σ 11⎢⎥⎣ σ 21σ 22 ⎦ ,σ 31σ 32σ 33that is, initial values are given in the order, 1 = σ 11, 2 = σ 21, 3 = σ 22, . . .• the US model is associated with several special features of ASReml. Whenused in the R structure for multivariate data, ASReml automatically recognisespatterns of missing values in the responses (see Chapter 8). Also, there is anoption to update its values by EM rather than AI when its AI updates makethe matrix non positive definite.• The Matérn class of isotropic covariance models is now described. ASRemluses an extended Matérn class which accomodates geometric anisotropy and achoice of metrics for random fields observed in two dimensions. This extension,described in detail in Haskard (2006), is given byρ(h; φ) = ρ M (d(h; δ, α, λ); φ, ν)where h = (h x , h y ) T is the spatial separation vector, (δ, α) governs geometricanisotropy, (λ) specifies the choice of metric and (φ, ν) are the parameters ofthe Matérn correlation function. The function isρ M (d; φ, ν) ={2 ν−1 Γ(ν)} −1( dφ) νK ν( dφ), (7.1)where φ > 0 is a range parameter, ν > 0 is a smoothness parameter, Γ(·) is thegamma function, K ν (.) is the modified Bessel function of the third kind of orderν (Abramowitz and Stegun, 1965, section 9.6) and d is the distance defined interms of X and Y axes: h x = x i − x j ; h y = y i − y j ; s x = cos(α)h x + sin(α)h y ;s y = cos(α)h x − sin(α)h y ; d = (δ|s x | λ + |s y | λ /δ) 1/λ .For a given ν, the range parameter φ affects the rate of decay of ρ(·) withincreasing d. The parameter ν > 0 controls the analytic smoothness of theunderlying process u s , the process being ⌈ν⌉ − 1 times mean-square differentiable,where ⌈ν⌉ is the smallest integer greater than or equal to ν (Stein, 1999,page 31). Larger ν correspond to smoother processes. ASReml uses numericalderivatives for ν when its current value is outside the interval [0.2,5].When ν = m + 1 2with m a non-negative integer, ρ M(·) is the product ofexp(−d/φ) and a polynomial of degree m in d. Thus ν = 1 2yields the exponentialcorrelation function, ρ M (d; φ, 1 2) = exp(−d/φ), and ν = 1 yields Whittle’selementary correlation function, ρ M (d; φ, 1) = (d/φ)K 1 (d/φ) (Webster andOliver, 2001).When ν = 1.5 thenρ M (d; φ, 1.5) = exp(−d/φ)(1 + d/φ)