MAXIMUM-LIKELIHOOD ESTIMATION OF MODELS FOR RESIDUAL COVARIANCE IN SPATIAL REGRESSION

2-2. Computational aspects One procedure to maximize Ln is by scoring using the updating equation <f>i = 4>o + B~1 L^. In view of (25) this amounts to computing /?0, for a given 60, using PQ = (X'V-lX)-lX'V-1 Y (2-6)
Some key words: Asymptotic normality; Consistency; Gaussian process; Kriging; Lattice process; Likelihood; Simulation; Spatial covariance.
1. I N T R O D U C T I O N
Biometriica (1984), 71, I, pp. 136-46 Printed in (heal Britain
135
Maximum likelihood estimation of models for residual covariance in spatial regression
i
W
1
(i = l,...,p),
(2-2)
where V, = dV/8d,, V = dV^dd,
derivative matrix of Ln can be written
= - F " Vx F "
1
and W = Y-X0. The second
l0g
M,
LlggJ
(2-3)
+ X'V Y for i = 1, ...,p, and LM (2-4)
2. MAXIMUM LIKELIHOOD
2-1. The likelihood for a Gaussian process
We consider a real valued Gaussian random process { Y(t); t e T} where T is an index set. For example T = Zd describes a d-dimensional lattice process and T = R* is a
In regression analysis the assumption that residuals are uncorrelated is sometimes untenable. This is particularly so in the analysis of spatial data when correlation may exist between neighbouring entities. The nature of the covariance among residuals will usually not be known precisely, but it is often possible to adopt a simple parametric model to describe it. One then has a set of covariance parameters as well as the regression coefficients to estimate. In most applications estimation of the regression coefficients will be of primary importance; however, our interest in this problem arose in the estimation of models of spatial covariance where the nature of the covariance is of particular interest in itself (Mardia, 1980). The approach we adopt, through maximum likelihood, is akin to the earlier studies of Cliff & Ord (1973, Chapter 5), and more recently Cook & Pocock (1983). In § 2 we discuss how the likelihood function may be numerically maximized, giving suitable formulae for the derivatives and information matrix. In §3 we give conditions which ensure that, with dependent observations, estimators are consistent and uniformly asymptotically normal. The relevance of these conditions for the analysis of spatial data is discussed in §§4 and 5. The results of some Monte Carlo simulation experiments for the estimation of spatial covariance is presented in §5. These provide a useful insight into the nature of bias for small samples. In §6 we discuss the computationally straightforward large sample spectral approximation to the likelihood for processes observed on a lattice and we give some illustrative examples.
where Lfifi = -X'V'1
has (i,j)th term
X, L^ has ith column -X'V'Xp
-i{tr(F- x Vtj+VVJ+W'V'JW}, where V,j = d2 V/d9idep Bn = -E(Li2(2-5)
Using ViJ = V~l(Vt V~l Vj+ V} F " 1 V,- F y ) V'1 we obtain the information matrix = diag (B,, Be), Vj) = tr(VVlVVJ). where Bp = X'V~ X and the (i, j)th element of Be is $ttj with
136
K . V. M A R D I A AND R . J .
MARSHALL
continuous parameter process. Or T could be a collection of spatial entities such as regions or counties. We suppose that, for all t e T, E{ Y(t)} = x{t)'fi, where x(t) = {xl(t), ...,xq(t)}' is a g x 1 vector of nonrandom regressors and /? e B is a parameter vector, B being an open subset of Bf. Also let the covariance be defined by a parametric model cov{y(/), Y(s)} = a(t,s; 6), for all t,a e T, where 6 6 0 is a p x l parameter vector, 0 being an open subset of Rp. We assume that <x(t, s; 6) is twice differentiate with respect to 0 a t all points on T2 x 0 , and that it is positive-definite in the sense that for every finite subset Tn = {t1, ...,tn} of T the covariance matrix Vn = {o(tt,ty, 6)}. is positive-definite. Suppose that Y(t) is observed at each point to give the sample vector Yn = { Y ^ ) , . . . , Y(tn)}'. We denote the combined (q+p)xl parameter vector by (j> = (/?', 6')'. The log likelihood for (f> is, if we ignore a constant, Ln(<j>;Yn)=-±\og\Vn\-\(Yn-XnP)'V;1(Yn-XJ), (2-1) where Xn is an n x q regressor matrix with^'th column Xj = {x^t^,..., Xj(tn)}''. We assume Xn to be rank q. By maximizing (21) the maximum likelihood estimates of ft and 0, denoted fin, §n, can be obtained. There are a number of ways this can be done as we indicate in §22. First, however, we give some relevant formulae. For notational convenience, we shall henceforth omit the subscript n on XH, Vn and Yn. The derivative vector of £„, say L^, can be written £</> = (L'^L'g)', where L0 = - X T " 1 I j S + I ' F " 1 Y and the ith element of Lg is