Selecting an Informative/Discriminating Multivariate Response for Inverse Prediction (original) (raw)

A sensitivity matrix based methodology for inverse problem formulation

Journal of Inverse and Ill-posed Problems, 2009

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate.

Optimal designs for inverse prediction in univariate nonlinear calibration models

Chemometrics and Intelligent Laboratory Systems, 2004

Univariate calibration models are intended to link a quantity of interest X (e.g. the concentration of a chemical compound) to a value Y obtained from a measurement device. In this context, a major concern is to build calibration models that are able to provide precise (inverse) predictions for X from measured responses Y.

Estimation of parameter uncertainty using inverse model sensitivities

Developments in Water Science, 2004

Forward model sensitivities are commonly applied to evaluate the uncertainty in model parameter estimates obtained through inverse analysis. In this case, the forward sensitivity (Jacobian) matrix is applied to compute an approximate representation of the covariance matrix of inverse parameter estimates. However, this approach can produce biased estimates of the covariance matrix because it does not account accurately for correlations between uncertainty of calibration targets and estimates. Typically, these correlations are non-linear and depend on the spatial and temporal structure of inverse targets and estimated parameters. A better but much more computationally intensive method to measure parameter uncertainty, which we call the inverse-sensitivity approach, directly evaluates the sensitivity of inverse estimates of model parameters with respect to the calibration targets. Further, we can evaluate the sensitivity of model predictions based on inverse model parameter estimates with respect to the calibration targets. The proposed methodology can also be applied to problems such as estimation of predictive uncertainty, optimization of data collection strategies, and design of monitoring networks. Its implementation can be performed efficiently through parallelization. Results based on a simple groundwater flow inverse problem are presented to illustrate the basis for the method.

Inverse Parameter Identification Using Bayesian Statistics and Response Surfaces

This paper presents a methodology designed to calibrate a simple Finite Element (FE) model in order to accurately reproduce the mechanical behavior of a material. This desired behavior is based on experimental measurements obtained from a three-point bending test. In order to perform this inverse analysis with a feasible computational effort, the typical FE solution of the forward problem is replaced by a meta-model, based on a response surface approximation. This simplified model still allows for accurate prediction of the output of the model, but at a negligible cost. The inverse problem is solved through a Bayesian perspective, by using the Metropolis-Hastings algorithm. The results presented a good agreement with results obtained by a L2-norm minimization approach, also presented here. The validation of these results was performed by running a FE simulation with the estimated parameters and the results accurately fitted the experimental data.

Inverse Analysis: A Tool for Model Parameter Estimation

This paper reports the development of a generalized inverse analysis formulation for the parameter estimation of four-parameter Burger model that is used to represent the time-dependent deformation of a viscoelastic soil medium such as a consolidating clay stratum. Aided by a suitable optimization technique (Sequential Quadratic Programming, SQP), a mathematical programming has been developed in terms of identification of the design vector, objective function and design constraints. In order to comprehend the efficacy and establish the proper functionality of the developed technique, a synthetic case study accounting only the loading cycle of the Burger model has been considered. Prime issues related to the back-estimation of the parameters namely identification of variable bounds, global optimality and optimal number of data-points required are explored and reported herein. The efficacy of the developed technique is also illustrated with a case-study.

Theory of net analyte signal vectors in inverse regression

The net analyte signal and the net analyte signal vector are useful measures in building and optimizing multivariate calibration models. In this paper a theory for their use in inverse regression is developed. The theory of net analyte signal was originally derived from classical least squares in spectral calibration where the responses of all pure analytes and interferents are assumed to be known. However, in chemometrics, inverse calibration models such as partial least squares regression are more abundant and several tools for calculating the net analyte signal in inverse regression models have been proposed. These methods yield different results and most do not provide results that are in accordance with the chosen calibration model. In this paper a thorough development of a calibration-specific net analyte signal vector is given. This definition turns out to be almost identical to the one recently suggested by Faber (Anal. Chem. 1998; 70: 5108–5110). A required correction of the net analyte signal in situations with negative predicted responses is also discussed.

LBNL-41638 Multiphase Inverse Modeling : An Overview

1998

Inverse modeling is a technique to derive model-related parameters from a variety of observations made on hydrogeologic systems, from small-scale laboratory experiments to field tests to long-term geothermal reservoir responses. If properly chosen, these observations contain information about the system behavior that is relevant to the performance of a geothermal field. Estimating model-related parameters and reducing their uncertainty is an important step in model development, because errors in the parameters constitute a major source of prediction errors. This paper contains an overview of inverse modeling applications using the ITOUGH2 code, demonstrating the possibilities and limitations of a formalized approach to the parameter estimation problem.

The Role of Regression Performance on Multimodel Analysis

Groundwater, 2013

In this work, we provide suggestions for designing experiments where calibration of many models is required and guidance for identifying problematic calibrations. Calibration of many conceptual models which have different representations of the physical processes in the system, as is done in cross-validation studies or multi-model analysis, often uses computationally frugal inversion techniques to achieve tractable execution times. However, because these frugal methods are usually local methods, and the inverse problem is almost always nonlinear, there is no guarantee that the optimal solution will be found. Furthermore, evaluation of each inverse model's performance to identify poor calibrations can be tedious. Results of this study show that if poorly calibrated models are included in the analysis, simulated predictions and measures of prediction uncertainty can be affected in unexpected ways. Guidelines are provided to help identify problematic regressions and correct them.