K. V. MARDIA AND R. J. MARSHALL, Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika 71 (1984), pp. 289-295.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....j Kj Gamma t Gamma log 2; 5) where K = K oe I and j Kj denotes the determinant of K. It is also possible to express analytically the partial derivatives of the log likelihood with respect to the parameters l = Gamma 2 tr 1 T K t; 6) see, e.g. [11]) Given l and its derivatives with respect to it is straightforward to feed this information to an optimization package in order to obtain a local maximum of the likelihood. In general one may be concerned about making point estimates when the number of parameters is large relative to the ....

K. V. Mardia and R. J. Marshall. Maximum likelihood estimation for models of residual covariance in spatial regression. Biometrika, 71(1):135-- 146, 1984.

....Stein (1999) raises concerns about variogram estimation. In keeping with the recommendations of O Connell and Wolfinger (1997) we propose to use a mixed model approach with residual maximum likelihood for estimation of = 2 x . Precedents of this likelihood approach to kriging include Mardia and Marshall (1984) and Zimmerman (1989) However, one is still faced with an n n matrix inversion. The Upper Cape 8 Cod reproductive data involves n = 1630 observations, rendering (6) infeasible. An attractive solution is to use reduced knot or low rank kriging as proposed by Nychka et al. 1997) Let f 1 ; ....

Mardia, K.V. and Marshall, R.J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika, 72, 135--146.

....nonstationarity of Z. 4.3.2 Spectral likelihood function If we assume that we are sampling from a Gaussian process, then it is straightforward in principle to write down the exact likelihood function and hence to maximize it numerically with respect to the unknown parameters. Kitanidis (1983) and Mardia and Marshall (1984) were the first to advocate estimating spatial processes in this way. The evaluation of the likelihood function requires computing the inverse and determinant of the model covariance matrix. In general, environmental datasets are very large and calculating the determinants that we have in the ....

Mardia, K. V. and Marshall, R. J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika, 71, 135-146.

....information about # is lost by quantization. We do this by comparing the variance of the estimators of # obtained from the original Gaussian data Z and from the clipped version X. Estimation from the Gaussian data has been studied by Kitanidis [37] Kitanidis and Lane [38] Mardia and Marshall [41], Warnes and Ripley [62] and Mardia and Watkins [42] 5.2 Estimation from the original (Gaussian) data. 5.2.1 Maximum likelihood estimator Suppose we have an observation Z of a stationary Gaussian field with mean zero and the covariance function r # over a grid S. As usual, Z is a n = n 1 n 2 ....

.... normal (as the grid expands) The properties of the maximum likelihood estimator for dependent data in the general case are discussed among other places in Billingsley [4] Bhat [3] Crowder [16] and Sweeting [58] 77 Our starting point is the following theorem (Theorem 2 in Mardia and Marshall [41]) Theorem 3 Suppose that the n vector Z is a single observation from N(0, T # ) The length of # is p.Let# 1 # # 2 ### n denote the eigenvalues of T # , let # i 1 # # i 2 ### i n denote the eigenvalues of #T # ## i , and let # ij 1 # # ij 2 # ## ij n denote the eigenvalues ....

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Mardia, K. V., and Marshall, R. J. (1984), "Maximum likelihood estimation of models for residual covariance in spatial regression," Biometrika, 71, 135146.

....is that there is no easy way to obtain standard errors for the estimators, or tests of hypotheses about the parameters. Most of the alternatives to this method are based on some form of likelihood procedure, assuming a Gaussian process: 476 Smith Maximum likelihood estimation (Kitanidis 1983, Mardia and Marshall 1984). This is more complicated than Cressie s method because the evaluation of the exact likelihood is appreciably harder than (18) but it is computationally feasible for reasonably sized problems. If there are n data points, likelihood evaluation requires storage and inversion of a n n covariance ....

Mardia, K.V. and Marshall, R.J. (1984), Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135-146.

....is that the operator L V is homogeneous over the interior sites in V o . This allows for the calculation of the eigenfunctions and eigenvalues of the operator associated with Z k , corresponding to complex exponentials and Fourier transforms of the operator. Such expressions appear in [32] [33] and particularly in the early work of Whittle [34] and recent work by Kent and Mardia [35] Our extension of these ideas relies on the fact that the normalized finite volume partition function has dependence on the boundary which goes as O( #V V ) This provides a handle on the asymptotic ....

K.V. Mardia and R.J. Marsall, "Maximum likelihood estimation of models for residual covariance in spatial regression," Biometrika, vol. 71, no. 1, pp. 135--46, 1984.

....for S = diag( s s 1 2 35 2 spatial correlation among the measurement errors could be used. Under this model, the diagonal S would be replaced with a general covariance matrix S=Var(e) The model parameters #, #, and # are estimated by maximum likelihood; see Cressie (1993) and Mardia and Marshall (1984) for details about fitting and asymptotic properties of ML estimators. We estimate the trend Z(s o ) at any location s o using the best linear unbiased predictor 13 1 ( R Z s ds R (8) # ( # ( # ) Z s x Z X o = o o b t b S 1 (6) s t t t t a 2 1 1 1 1 1 ( ....

Mardia, K. V. and Marshall, R. J. (1984). `Maximum likelihood estimation of models for residual covariance in spatial regression'. Biometrika 71, 135-146.

....variograms. 3 The central theme of the present paper is to propose an alternative, likelihood based, approach to fitting the model (1. 2) There are a number of reasons why this might be considered desirable: i) The maximum likelihood approach to spatial processes was first proposed by Mardia and Marshall (1984) and has been shown to be computationally feasible for data sets consisting of as many as several hundred sites. This has not been unchallenged, since Warnes and Ripley (1987) argued that the likelihood may be highly multimodal even in the case of a very simple homogeneous model for the spatial ....

Mardia, K.V. and Marshall, R.J. (1984), Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135-146.

....r and ff is based on the distribution of z of (2.3) the components of which are not independent. So the standard results on maximum likelihood estimators cannot be applied directly. However, we can use a similar method used by Stein in Stein (1989) and Stein (1990) In Stein (1989) a result from Mardia and Marshall (1984) is used to investigate the properties of maximum likelihood estimators in a linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model like our case here. In order to use the result in Mardia and Marshall (1984) we first calculate ....

....(1989) a result from Mardia and Marshall (1984) is used to investigate the properties of maximum likelihood estimators in a linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model like our case here. In order to use the result in Mardia and Marshall (1984), we first calculate the Fisher information of model (2.3) We have the following: Lemma 3.1.1 Let I be the expected Fisher information matrix for ( r; ff) in the distribution of z in (2.3) and let I = 0 B B B B I I r I ff I r I rr I rff I ff I rff I ffff 1 C C C C A : Then we ....

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Mardia, K. V. and Marshall, R. J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika 71: 135--146.

....optimization routine using Equation (5) as the function to be optimized. For a given value of , estimates of fi and oe 2 are calculated using Equations (6) and (7) respectively. Next, Equation (8) is used in calculating the partial derivatives of the objective function. See Mardia and Marshall [34] for an overview of the maximum likelihood procedure. 4.5 Estimating and using derivatives In the manufacturing sciences, deterministic simulators help describe the relationships between product design, and the manufacturing process to the product s final characteristics. This allows the product ....

Mardia, K.V. and Marshall, R.J. Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika, 71(1):135--146, 1984.

....I and j Kj denotes the determinant of K. It is also possible to express analytically the partial derivatives of the log likelihood with respect to the parameters l i = Gamma 1 2 tr K Gamma1 K i 1 2 t T K Gamma1 K i K Gamma1 t; 6) see, e.g. [11]) Given l and its derivatives with respect to it is straightforward to feed this information to an optimization package in order to obtain a local maximum of the likelihood. In general one may be concerned about making point estimates when the number of parameters is large relative to the ....

K. V. Mardia and R. J. Marshall. Maximum likelihood estimation for models of residual covariance in spatial regression. Biometrika, 71(1):135--146, 1984.

....Sm ( 0, with probability tending to one. ii) p m( m Gamma ) d Gamma N(0; F Gamma1 ) 3.13) A more general asymptotic theory for the maximum likelihood estimation for the model in (3.1) can be developed, as N 1, where N = nmk, denotes the total number of observations. See Mardia and Marshall (1984) for the regularity conditions needed for the general case. Further Remarks on Estimation Returning to the important special case, L given in (3.3) suppose first that Sigma(j) is known. The mle of fi is then given by fi(j) m X j=1 (X 1j Sigma Gamma1 (j)X 0 1j ) Gamma1 ....

Mardia, K. V. and Marshall, R. J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135-146.

....the empirical and theoretical variograms, using curve fitting methods; see, for example, Cressie (1991) However, a growing body of work on inference for spatial stochastic processes has resulted in a move towards the adoption of likelihood based methods of parameter estimation. See, for example, Mardia and Marshall (1984), Warnes and Ripley (1987) Vecchia (1988, 1992) Mardia and Watkins (1989) Zimmerman (1989) and Laslett (1994) A potentially serious limitation of conventional geostatistical methodology is that it uses the prediction variance as an estimate of precision without taking account of the ....

MARDIA, K. V. and MARSHALL, R. J. (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika, 72, 135--146.

....easily be controlled by writing the covariance function in terms of hyperparameters . One approach to adapting these hyperparameters to the observed data is to estimate them by maximum likelihood (or maximum penalized likelihood) as has long been done in the context of spatial statistics (eg, Mardia and Marshall 1984). In a fully Bayesian approach, the hyperparameters are given prior distributions. Predictions are then made by averaging over the posterior distribution for the hyperparameters, which can be done using Markov chain Monte Carlo methods. These two approaches often give similar results (Williams and ....

....above, this is easily done using the Cholesky decomposition of C. For some of the Markov chain sampling methods, the derivatives of L with respect to the various hyperparameters are also required. The derivative of the log likelihood with respect to a hyperparameter can be written as follows (Mardia and Marshall 1984): L = Gamma 1 2 tr C Gamma1 C 1 2 t T C Gamma1 C C Gamma1 t (21) The trace of the product in the first term can be computed in time proportional to n 2 , assuming that C Gamma1 has already been computed. The second term can also be computed in time ....

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Mardia, K. V. and Marshall, R. J. (1984) "Maximum likelihood estimation of models for residual covariance in spatial regression", Biometrika, vol. 71, pp. 135-146.

.... missing data and is similar in flavor to back fitting algorithms of additive models, see Hastie and Tibshirani, 1990 and Green and Silverman 1994) and also the Cochrane Orcutt procedure, which iterates between estimation of mean and covariance parameters (Cochrane and Orcutt, 1949; see also Mardia and Marshall, 1984 and Neter, Wasserman and Kutner, 1990, p. 496) First, based on rough data analysis, choose initial values for all unknown parameters. Call these values, the current estimates. Next, iterate the following procedure until convergence: 1. Imputation: Estimate the missing data Z u 1 and Z u 2 ....

Mardia, K.V. and Marshall, R.J. (1984), "Maximum likelihood estimation of models for residual covariance in spatial regression," Biometrika, 71(1), 135-146.

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K. V. MARDIA AND R. J. MARSHALL, Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika 71 (1984), pp. 289-295.

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Mardia, K.V. and Marshall, R.J. (1984) Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression. Biometrika, 71, 135-146.

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