Matching objective and subjective information in groundwater inverse analysis by Akaike's Bayesian information criterion

Honjo, Yusuke; Kashiwagi, Nobuhisa

doi:10.1029/98wr02365

Cited by 16 publications

(8 citation statements)

References 21 publications

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“…For linear models, a number of these estimates are available such as Akakie information criterion (AIC), Bayes information criterion (BIC), minimum description length, etc. Among them, AIC and BIC are very popular in hydrologic literature (Xu and Li 2002;Gyasi-Agyei 2001;Honjo and Kashiwagi 1999;Knotters and De Gooijer 1999;Mutua 1994;Gregory et al 1992). The general form of the classical estimate of prediction risk for a linear model can be written as…”

Section: Asymptotic Analysismentioning

confidence: 99%

On selection of kernel parametes in relevance vector machines for hydrologic applications

Tripathi

Govindaraju

2006

Stoch Environ Res Risk Assess

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Recent advances in statistical learning theory have yielded tools that are improving our capabilities for analyzing large and complex datasets. Among such tools, relevance vector machines (RVMs) are finding increasing applications in hydrology because of (1) their excellent generalization properties, and (2) the probabilistic interpretation associated with this technique that yields prediction uncertainty. RVMs combine the strengths of kernel-based methods and Bayesian theory to establish relationships between a set of input vectors and a desired output. However, a bias-variance analysis of RVM estimates revealed that a careful selection of kernel parameters is of paramount importance for achieving good performance from RVMs. In this study, several analytic methods are presented for selection of kernel parameters. These methods rely on structural properties of the data rather than expensive re-sampling approaches commonly used in RVM applications. An analytical expression for prediction risk in leave-one-out cross validation is derived. For brevity, the effectiveness of the proposed methods is assessed first by data generated from the benchmark sinc function, followed by an example involving estimation of hydraulic conductivity values over a field based on observations. It is shown that a straightforward maximization of likelihood function can lead to misleading results. The proposed methods are found to yield robust estimates of parameters for kernel functions.

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Section: Asymptotic Analysismentioning

confidence: 99%

On selection of kernel parametes in relevance vector machines for hydrologic applications

Tripathi

Govindaraju

2006

Stoch Environ Res Risk Assess

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“…The ABIC minimization method has been applied to various geophysical inversions, and it has given successful results: for example, analysis of earth tide data (Tamura et al . 1991), estimation of surficial density from gravity data (Murata 1993), deconvolution of palaeomagnetic remanence data (Oda and Shibuya 1994), groundwater inverse analysis (Honjo and Kashiwagi 1999), 2D inversion of magnetotelluric data (Uchida 1993; Ogawa and Uchida 1996), seismic waveform inversion (Yoshida 1989), the Fourier transform (Mitsuhata et al . 2001) and 2.5D inversion of controlled‐source electromagnetic data (Mitsuhata, Uchida and Amano 2002).…”

Section: Introductionmentioning

confidence: 99%

Adjustment of regularization in ill‐posed linear inverse problems by the empirical Bayes approach

Mitsuhata¹

2004

Geophysical Prospecting

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A B S T R A C TRegularization is the most popular technique to overcome the null space of model parameters in geophysical inverse problems, and is implemented by including a constraint term as well as the data-misfit term in the objective function being minimized. The weighting of the constraint term relative to the data-fitting term is controlled by a regularization parameter, and its adjustment to obtain the best model has received much attention. The empirical Bayes approach discussed in this paper determines the optimum value of the regularization parameter from a given data set. The regularization term can be regarded as representing a priori information about the model parameters. The empirical Bayes approach and its more practical variant, Akaike's Bayesian Information Criterion, adjust the regularization parameter automatically in response to the level of data noise and to the suitability of the assumed a priori model information for the given data. When the noise level is high, the regularization parameter is made large, which means that the a priori information is emphasized. If the assumed a priori information is not suitable for the given data, the regularization parameter is made small. Both these behaviours are desirable characteristics for the regularized solutions of practical inverse problems. Four simple examples are presented to illustrate these characteristics for an underdetermined problem, a problem adopting an improper prior constraint and a problem having an unknown data variance, all frequently encountered geophysical inverse problems. Numerical experiments using Akaike's Bayesian Information Criterion for synthetic data provide results consistent with these characteristics. In addition, concerning the selection of an appropriate type of a priori model information, a comparison between four types of difference-operator model -the zeroth-, first-, second-and third-order difference-operator models -suggests that the automatic determination of the optimum regularization parameter becomes more difficult with increasing order of the difference operators. Accordingly, taking the effect of data noise into account, it is better to employ the lower-order difference-operator models for inversions of noisy data.

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“…There are yet more methods for model discrimination that are not considered here. Such methods are the minimum description length criterion [Rissanen, 1978] and the modified versions of AIC [e.g., Hurvich and Tsai, 1993;Honjo and Kashiwagi, 1999].…”

Section: Review Of Methods On Model Selectionmentioning

confidence: 99%

A Bayesian Approach to the quantification of the effect of model error on the predictions of groundwater models

Gaganis¹,

Smith²

2001

Water Resources Research

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Abstract. Errors arising from the imperfect mathematical representation of the structure of a hydrologic system (model error) are not random but systematic. Their effect on model predictions varies in space and time and differs for the flow and solute transport components of a groundwater model. Such errors do not necessarily have any probabilistic properties that can be easily exploited in the construction of a model performance criterion. A Bayesian approach is presented for quantifying model error in the presence of parameter uncertainty. Insight gained in updating the prior information on the model parameters is used to assess the correctness of the model structure, which is defined relative to the accuracy required of the model predictions. Model error is evaluated for each measurement of the dependent variable through an examination of the correctness of the model structure for different accuracy levels. The effect of model error on each dependent variable, which is quantified as a function of location and time, represents a measure of the reliability of the model in terms of each model prediction. This method can be used in identifying possible causes of model error and in discriminating among models in terms of the correctness of the model structure. It also offers an improved description of the uncertainties associated with a modeling exercise that may be useful in risk assessments and decision analyses. IntroductionA mathematical model is a set of equations or functions that contain adjustable parameters and constants. These functions or equations are mathematical descriptions of hypotheses regarding properties and processes of a more complex physical system. Groundwater models are always simplifications of reality, as we are unable to either characterize or describe mathematically the true complexity of a hydrologic system. We will refer to structural errors in a model, which can arise from incorrect hypotheses, unmodeled processes, or unknown correlations between processes, as model error. The parameters of a groundwater model, such as hydraulic conductivity, are always uncertain because of measurement error, heterogeneity, and scaling issues [Beckie, 1996]. The error produced from uncertainties in parameter values is called parameter error. In a stochastic modeling approach, a subset of the parameters is assigned error probability distributions to quantify the effects of parameter error. Error in the deterministically specified components of these probability distributions, such as in the form of the pdf, can also be considered a type of model error.Simulation models have two main uses in hydrogeology.

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Matching objective and subjective information in groundwater inverse analysis by Akaike's Bayesian information criterion

Cited by 16 publications

References 21 publications

On selection of kernel parametes in relevance vector machines for hydrologic applications

On selection of kernel parametes in relevance vector machines for hydrologic applications

Adjustment of regularization in ill‐posed linear inverse problems by the empirical Bayes approach

A Bayesian Approach to the quantification of the effect of model error on the predictions of groundwater models

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