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

2001 ◽  
Vol 5 (2) ◽  
pp. 215-223 ◽  
Author(s):  
E. Todini
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Covariance Matrix ◽  
Theoretical Development ◽  
Comprehensive Treatment ◽  
Parameter Estimates ◽  
Wide Field ◽  
Estimation Uncertainty ◽  
Parameter Estimation Uncertainty ◽  
Variance Covariance Matrix

Abstract. 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. Keywords: Kriging, maximum likelihood, parameter estimation, uncertainty

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

2001 ◽  
Vol 5 (2) ◽  
pp. 225-232 ◽  
Author(s):  
E. Todini ◽  
F. Pellegrini ◽  
C. Mazzetti
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Cross Validation ◽  
Ratio Test ◽  
Veneto Region ◽  
Numerical Example ◽  
Estimation Uncertainty ◽  
Yearly Average ◽  
Average Precipitation ◽  
Parameter Estimation Uncertainty

Abstract. 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. Keywords: yearly rainfall, maximum likelihood, Kriging, parameter estimation uncertainty

scholarly journals Neglect Of Parameter Estimation Uncertainty Can Significantly Overestimate Structural Reliability

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Árpád Rózsás ◽  
Miroslav Sýkora
Keyword(s):  
Parameter Estimation ◽  
Bayesian Statistics ◽  
Failure Probability ◽  
Structural Reliability ◽  
Probabilistic Models ◽  
Estimation Uncertainty ◽  
Point Estimates ◽  
Distribution Parameters ◽  
Order Of Magnitude ◽  
Parameter Estimation Uncertainty

Abstract Parameter estimation uncertainty is often neglected in reliability studies, i.e. point estimates of distribution parameters are used for representative fractiles, and in probabilistic models. A numerical example examines the effect of this uncertainty on structural reliability using Bayesian statistics. The study reveals that the neglect of parameter estimation uncertainty might lead to an order of magnitude underestimation of failure probability.

scholarly journals Conditional Maximum Likelihood Estimation in Probability-Branched Multistage Designs

2021 ◽  
Author(s):  
Jan Steinfeld ◽  
Alexander Robitzsch
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Parameter Estimate ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Conditional Maximum Likelihood ◽  
Item Parameter Estimates ◽  
Item Parameter Estimation ◽  
Conditional Maximum

This article describes the conditional maximum likelihood-based item parameter estimation in probabilistic multistage designs. In probabilistic multistage designs, the routing is not solely based on a raw score j and a cut score c as well as a rule for routing into a module such as j < c or j ≤ c but is based on a probability p(j) for each raw score j. It can be shown that the use of a conventional conditional maximum likelihood parameter estimate in multistage designs leads to severely biased item parameter estimates. Zwitser and Maris (2013) were able to show that with deterministic routing, the integration of the design into the item parameter estimation leads to unbiased estimates. This article extends this approach to probabilistic routing and, at the same time, represents a generalization. In a simulation study, it is shown that the item parameter estimation in probabilistic designs leads to unbiased item parameter estimates.

Analysis of errors in parameter estimation with application to physiological systems

1980 ◽  
Vol 239 (5) ◽  
pp. R390-R400
Author(s):  
T. M. Grove ◽  
G. A. Bekey ◽  
L. J. Haywood
Keyword(s):  
Parameter Estimation ◽  
Covariance Matrix ◽  
Measurement Errors ◽  
Estimation Error ◽  
Nonlinear Dynamic System ◽  
Parameter Estimates ◽  
Taylor Series Expansions ◽  
System Output ◽  
Model Equations ◽  
Physiological Systems

The accuracy of parameter estimation applied to physiological systems is analyzed. The method of analysis is applicable to procedures utilizing minimization of squared output error and a nonlinear dynamic system model. Three major sources of estimation error are described: 1) measurement error, 2) modeling error, and 3) optimization error. Measurement errors affect values used for the system output, the model input, and nonestimated parameters of the model. Modeling errors are due to failure to adequately describe the structure of the system and to numerical errors that occur in the digital computer solution of the model equations. Linearization by use of Taylor series expansions in the region of the nominal solution is used to obtain an expression for the covariance matrix of the parameter estimates in terms of the covariance matrix of each error source. The analysis is applied to the example of cardiac output estimation from respiratory measurements. The results demonstrate that an analysis of system identifiability is not sufficient to ensure usable estimates and that systematic error analysis is essential for assessing the usefulness of parameter estimation techniques.

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