Influence of parameter estimation uncertainty in Kriging

Todini, E.; Ferraresi, M.

doi:10.1016/s0022-1694(96)80024-2

Cited by 26 publications

(9 citation statements)

References 5 publications

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“…(6) (12) If the parameter vector ϑ is no longer considered as a set of a priori fixed values (the common assumption), but rather as θ , an estimated quantity depending both on the model structure and upon the observation vector z * , it is easy to show that, due to the parameter estimation error, both ( ) are small, following Todini and Ferraresi (1996), it is convenient to approximate the expected value and the variance of the Kriging estimate ˆ z 0 , taken as a function of the parameters, by a truncated Taylorseries expansion of ˆ z 0 ϑ ( ) about { } θ E (Benjamin and Cornell, 1970). In addition, since { } θ E is not known, it is common practice to use the estimated parameter value ϑî nstead, to give: (17) have been derived in Todini and Ferraresi (1996).…”

Section: Influence Of Parameter Estimation Uncertaintymentioning

confidence: 99%

“…Following the comprehensive reviews given by Zimmerman and Zimmerman (1991), and by Cressie (1993), these can be divided into least squares based techniques in the space of the semi-variogram (Journel and Huijbregts, 1978;Cressie and Hawkins, 1980;Cressie, 1985); least squares techniques in the space of observations defined in the form of generalised covariance expressed as a linear function of parameters (Delfiner, 1976;Kitanidis, 1983); Maximum Likelihood in the space of residuals from a linear trend (Mardia and Marshall, 1984); Maximum Likelihood in the space of cross-validation errors (Samper and Newman, 1989); Maximum Likelihood in the space of "error contrasts" (Kitanidis, 1983), using what is known as Restricted Maximum Likelihood (REML) (Patterson and Thompson, 1971;1974). All these techniques have pros and cons that will be addressed briefly in the sequel, but this paper will focus only on the ML and REML type estimators, since they allow for the derivation of the covariance matrix of the parameters, which can be computed as the inverse of the Fisher information matrix, and can be used for investigating the effect of parameter uncertainty over the Kriging estimates, as advocated by Kitanidis (1983) and proposed by Todini and Ferraresi (1996).…”

Section: The Maximum Liklihood Estimatormentioning

confidence: 99%

“…Within the frame of previous research aimed at assessing the influence of parameter estimation variance in Kriging, Todini and Ferraresi (1996) (similarly to what was done by Samper and Newman (1989)) made the hypothesis of normal independent cross-validation errors by assuming the covariance matrix ε V to be diagonal. The application of the technique to real data with this independence hypotheses showed that the assumption of independent errors may involve significant distortions in the estimation of the parameters particularly for certain types of variograms (Todini et al, 2001).…”

Section: The Maximum Liklihood Estimatormentioning

confidence: 99%

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Influence of parameter estimation uncertainty in Kriging: Part 1 - Theoretical Development

Todini

2001

Hydrol. Earth Syst. Sci.

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This paper deals with a theoretical approach to assessing the effects of parameter estimation uncertainty both on Kriging estimates and on their estimated error variance. Although a comprehensive treatment of parameter estimation uncertainty is covered by full Bayesian Kriging at the cost of extensive numerical integration, the proposed approach has a wide field of application, given its relative simplicity. The approach is based upon a truncated Taylor expansion approximation and, within the limits of the proposed approximation, the conventional Kriging estimates are shown to be biased for all variograms, the bias depending upon the second order derivatives with respect to the parameters times the variance-covariance matrix of the parameter estimates. A new Maximum Likelihood (ML) estimator for semi-variogram parameters in ordinary Kriging, based upon the assumption of a multi-normal distribution of the Kriging cross-validation errors, is introduced as a mean for the estimation of the parameter variance-covariance matrix.

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Section: Influence Of Parameter Estimation Uncertaintymentioning

confidence: 99%

Section: The Maximum Liklihood Estimatormentioning

confidence: 99%

Section: The Maximum Liklihood Estimatormentioning

confidence: 99%

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Influence of parameter estimation uncertainty in Kriging: Part 1 - Theoretical Development

Todini

2001

Hydrol. Earth Syst. Sci.

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“…(3), (4) and (5) was made by using the ML estimator under the assumption of spatially independent cross-validation errors, as suggested by Samper and Newman (1989) and by Todini and Ferraresi (1996).…”

Section: Spherical (6)mentioning

confidence: 99%

Influence of parameter estimation uncertainty in Kriging: Part 2 - Test and case study applications

Todini

Pellegrini

Mazzetti

2001

Hydrol. Earth Syst. Sci.

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The theoretical approach introduced in Part 1 is applied to a numerical example and to the case of yearly average precipitation estimation over the Veneto Region in Italy. The proposed methodology was used to assess the effects of parameter estimation uncertainty on Kriging estimates and on their estimated error variance. The Maximum Likelihood (ML) estimator proposed in Part 1, was applied to the zero mean deviations from yearly average precipitation over the Veneto Region in Italy, obtained after the elimination of a non-linear drift with elevation. Three different semi-variogram models were used, namely the exponential, the Gaussian and the modified spherical, and the relevant biases as well as the increases in variance have been assessed. A numerical example was also conducted to demonstrate how the procedure leads to unbiased estimates of the random functions. One hundred sets of 82 observations were generated by means of the exponential model on the basis of the parameter values identified for the Veneto Region rainfall problem and taken as characterising the true underlining process. The values of parameter and the consequent cross-validation errors, were estimated from each sample. The cross-validation errors were first computed in the classical way and then corrected with the procedure derived in Part 1. Both sets, original and corrected, were then tested, by means of the Likelihood ratio test, against the null hypothesis of deriving from a zero mean process with unknown covariance. The results of the experiment clearly show the effectiveness of the proposed approach.

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“…The complication of required information would increase the difficulty of analyses, particularly when the data are not enough. In the Kriging method, statistical hypotheses are made in evaluating and identifying the multidimensional spatial structure of the hydrological process of interest (Todini and Ferraresi, 1996;Dirks et al, 1998a, b). If rainfall stations are not sufficient, the statistical data will be limited, and the Kriging method will not be suitable for estimating precipitation.…”

Section: Introductionmentioning

confidence: 99%

The Parameter Optimization in the Inverse Distance Method by Genetic Algorithm for Estimating Precipitation

Chang

2006

Environ Monit Assess

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The inverse distance method, one of the commonly used methods for analyzing spatial variation of rainfall, is flexible if the order of distances in the method is adjustable. By applying the genetic algorithm (GA), the optimal order of distances can be found to minimize the difference between estimated and measured precipitation data. A case study of the Feitsui reservoir watershed in Taiwan is described in the present paper. The results show that the variability of the order of distances is small when the topography of rainfall stations is uniform. Moreover, when rainfall characteristic is uniform, the horizontal distance between rainfall stations and interpolated locations is the major factor influencing the order of distances. The results also verify that the variable-order inverse distance method is more suitable than the arithmetic average method and the Thiessen Polygons method in describing the spatial variation of rainfall. The efficiency and reliability of hydrologic modeling and hence of general water resource management can be significantly improved by more accurate rainfall data interpolated by the variable-order inverse distance method.

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Influence of parameter estimation uncertainty in Kriging

Cited by 26 publications

References 5 publications

Influence of parameter estimation uncertainty in Kriging: Part 1 - Theoretical Development

Influence of parameter estimation uncertainty in Kriging: Part 1 - Theoretical Development

Influence of parameter estimation uncertainty in Kriging: Part 2 - Test and case study applications

The Parameter Optimization in the Inverse Distance Method by Genetic Algorithm for Estimating Precipitation

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