Optimal Experimental Design Problem Research Articles

Optimal experimental design criterion for discriminating semiparametric models

This paper studies the optimal experimental design problem to discriminate two regression models. Recently, López-Fidalgo et al. [2007. An optimal experimental design criterion for discriminating between non-normal models. J. Roy. Statist. Soc. B 69, 231–242] extended the conventional T-optimality criterion by Atkinson and Fedorov [1975a. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70; 1975b. Optimal design: experiments for discriminating between several models. Biometrika 62, 289–303] to deal with non-normal parametric regression models, and proposed a new optimal experimental design criterion based on the Kullback–Leibler information divergence. In this paper, we extend their parametric optimality criterion to a semiparametric setup, where we only need to specify some moment conditions for the null or alternative regression model. Our criteria, called the semiparametric Kullback–Leibler optimality criteria, can be implemented by applying a convex duality result of partially finite convex programming. The proposed method is illustrated by a simple numerical example.

COMPUTING OPTIMAL DYNAMIC EXPERIMENTS FOR MODEL CALIBRATION IN PREDICTIVE MICROBIOLOGY

ABSTRACT The potential of mathematical models describing the microbial behavior during food processing and storage largely depends on their predictive capabilities and, in this concern, model calibration plays a crucial role. Unfortunately, model calibration may only be performed successfully if the sources of information are sufficiently rich. Therefore, a careful experimental design is required. This contribution formulated the optimal experimental design (OED) problem as a general dynamic optimization problem where the objective was to optimize a certain criterion depending on the Fisher information matrix. This formulation allows for more flexibility in the experimental design, including initial conditions, sampling times, experimental durations, time‐dependent manipulable variables and number of experiments as degrees of freedom. Moreover, the use of robust confidence regions for the parameter estimates was suggested as an alternative to evaluate the quality of the proposed experimental schemes. The OED for the calibration of the thermal death time and Ratkowsky‐type secondary models was considered for illustrative purposes, showing how the usually disregarded E‐optimality criterion results in the experimental schemes offering the best compromise precision/decorrelation among the parameters.PRACTICAL APPLICATIONSThis work addresses a general methodology for designing optimal dynamic experiments for the purpose of model calibration. This methodology is general in the sense that it may be applied to any type of food processing model, being particularly relevant for predictive microbiology and quality assessment as the experimentation is both time consuming and expensive.The main advantages of the proposed technique are twofold: on one hand, it is able to significantly reduce the overall experimental burden, contributing not only to simplify the experimental planning, devising the most adequate experiments, but also minimizing the number of experiments, and on the other hand, the resultant experiments provide the maximum quantity and quality of information to improve the predictive capabilities of the models under consideration, of key importance for process design, optimization and control.

Optimal current patterns in dynamical electrical impedance tomography imaging

In this paper, the topic of optimal experiment design in electrical impedance tomography (EIT) is studied. More specifically, we consider determination of optimal current patterns in EIT in cases of time-varying targets. The reconstruction problem associated with EIT imaging is known to be an ill-posed inverse problem. Statistical inversion methods have been shown to be advantageous in many cases in EIT. In Kaipio et al (2004 Inverse Problems 20 919–36), we considered the problem of optimal experiment design in statistical framework and we proposed an approach for determining optimal current patterns in cases of imaging of time-invariant targets. The approach was based on the statistical interpretation of the reconstruction problem and optimal current patterns were obtained by minimizing the trace of an approximate posterior covariance matrix. In this paper, we utilize a similar approach to determining optimal current patterns in cases of time-varying targets. The image reconstruction problem of EIT is formulated as a state estimation problem. As in the time-invariant case, the optimality criterion is based on the posterior covariances but instead of considering one specific time instant we minimize the time-averaged mean posterior variance. It is shown in a numerical study that the uncertainties of the estimates obtained with optimized current patterns are smaller than those obtained with conventional current patterns. In addition, the results indicate that in time-varying problems a single optimized current pattern may be sufficient to achieve good accuracy, i.e., multiple optimized current patterns do not provide substantial further information on the target. We also demonstrate that in some cases the increase in the number of current patterns can even decrease the reliability of the estimates. This is one of the reasons for the topic of optimal current patterns being quite important in the case of imaging time-varying targets.

Optimal design of dynamic experiments for improved estimation of kinetic parameters of thermal degradation

Thermal processing is widely used for ensuring food safety and extended shelf life. However, standard methods of thermal processing have a significant impact on food quality due to thermal degradation of nutrients and other quality factors. Model-based methods can be successfully used for thermal process design, optimization and control. However, building sound models requires suitable estimation of the unknown kinetic parameters. Further, the accuracy of these estimates will largely depend on the quality and quantity of the available experimental data. Optimal experimental design (OED) of dynamic experiments allows for the calculation of the scheme of controls and measurements which improve the estimation of model parameters. In this contribution, the OED problem is formulated as a general dynamic optimization problem where the objective is to find those experimental conditions which result in maximum information content, as measured by the Fisher information matrix. The numerical solution of this problem is then approached using a combination of the control vector parameterization approach with a non-linear global optimization solver. As an illustrative application, we consider the optimal experimental design for the parameter estimation of the thiamine degradation kinetic parameters during the thermal processing of canned tuna. Results confirm that the use of optimal dynamic experiments not only improves identifiability but also results in reduced confidence regions for the parameters (a maximum error of the 2% in the parameter estimates), substantially decreasing the experimental effort (up to a 50%). Particularly the use of six optimally designed experiments results in a 30% reduction of the confidence regions with respect to previously published results using 10 typical experiments.

ON OPTIMAL INPUT DESIGN IN SYSTEM IDENTIFICATION1

System identification concerns the construction and validation of mathematical models of dynamical systems from experimental data. The objective of this contribution is to discuss new research directions in experiment design , e.g. how to design informative experiments which satisfy specifications on the resulting model quality and practical limitations such as constraints on input and output signals, but also experimental time. In particular, we discuss how input design is instrumental for alleviating the problem of modelling complex systems. Many optimal experiment design problems can be formulated as optimal control problems, but with nonstandard cost functions. A difficulty is that the solution often depends on the true system. To solve optimal control problems, we can exploit recent advances in numerical optimization for control design, including convex optimization and relaxation methods. As a more concrete example, we study how to estimate the ℋ∞ norm of a system.

IDENTIFICATION OF QUANTUM SYSTEMS: MAXIMUM LIKELIHOOD AND OPTIMAL EXPERIMENT DESIGN FOR STATE TOMOGRAPHY

An optimal experiment design problem invoked by the Cramér-Rao Inequality is applied to the problem of quantum state tomography. The optimization problem is integer-combinatorial and we use an established relaxation which results in a convex programming problem whose solution can be used to guide a more efficient experiment.

Experimental design for outflow experiments based on a multi-level identificationmethod for material laws

A multi-level method for the determination of nonlinear coefficient functions ofparabolic partial differential equations is presented as an example of thereconstruction of the hydraulic properties of porous media from soil columnoutflow experiments. A linear stability and error analysis is used to characterizethe ill-posedness of the inverse problem. Founded on this analysis, optimalexperimental design problems can be formulated. Numerical examples arepresented which show that the method described leads to a stable solution of theinverse problem.

Optimal experimental design for parameter estimation in column outflow experiments

Inverse modeling is increasingly used for water flow and reactive solute transport in the unsaturated zone to determine unknown parameters, e.g., soil hydraulic parameters, degradation rates or dispersion coefficients. However, experimental data are frequently not suitable for parameter estimation leading instead to practically singular estimation problems. Optimal experimental design aims at identifying experimental conditions and sampling schemes that deliver measurement data which are most sensitive to unknown parameters. A new numerical approach to optimize experimental designs for parameter estimation in coupled partial differential equations (PDEs) and ordinary differential equation (ODEs) arising from transport and degradation processes is presented. The optimal experimental design problem is embedded in the framework of optimal control theory. This results in an intricate, nonlinear, constrained optimization problem that is solved by a direct approach based on a structured SQP method. These methods are used to investigate several scenarios for a column outflow experiment in a simulation study. Simultaneously, experimental conditions, such as irrigation schemes, substance concentrations, and sampling schemes are optimized. Result is an enormous reduction of parameter uncertainty. Additionally, it is demonstrated that it is possible to estimate both soil hydraulic and solute transport parameters together in one experiment on the basis of leachate data only. The usefulness of the numerical approach for practical applications is discussed.

Computation of Optimal Identification Experiments for Nonlinear Dynamic Process Models: a Stochastic Global Optimization Approach

The problem of optimal experimental design (OED) for parameter estimation of nonlinear dynamic systems is considered. It is shown how this problem can be formulated as a dynamic optimization (optimal control) problem where the performance index is usually a scalar function of the Fisher information matrix. Numerical solutions can be obtained using direct methods, which transform the original problem into a nonlinear programming (NLP) problem via parametrizations. However, because of the frequent nonsmoothness of the cost functions, the use of gradient-based methods to solve this NLP might lead to local solutions. Stochastic methods of global optimization are suggested as robust alternatives. A case study considering the OED for parameter estimation in a fed-batch bioreactor is used to illustrate the performance and advantages of two selected stochastic algorithms.

Optimal experiment design for regression polynomial models identification

In this paper the problem of optimal experimental design for parameter identification of static non-linear blocks is addressed. Non-linearities are assumed to be polynomial and represented according to the Vandermonde base. The optimality problem is formulated in a set membership context and the cost functions to be minimized are the worst case parameter uncertainties. Closed form optimal input sequences are derived when the input u is allowed to vary on a given interval [ u a, u b ]. Since optimal input sequences are, in general, not invariant to base changes, results and criteria for representing polymomials with different bases, still preserving the optimal set of input levels derived from the Vandermonde parameterization, are introduced as well. Finally numerical results are reported showing the effectiveness of using optimal input sequences especially when identifying some block described dynamic models that include in their structure static non-linearities (such as Hammerstein and LPV models). In such cases the improvement achieved in the confidence of the estimates can add up to a factor of several hundreds with respect to the case of random generated inputs.

Optimal adaptive control for estimation of parameters of ARX models

The objective of this paper is to find an adaptive control strategy which would enable us to estimate the parameters of the ARX model as accurately as possible while keeping the output of the system below a specified level of variability. It turns out that this optimal experiment design problem can be solved by using the Astrom-Wittenmark self-tuning tracker (1973) with a specific choice of the reference signal.

D-optimal experiment design for static non linear systems described by 2-D polynomial models

In this paper the problem of optimal experiment design for the identification of non linear static system described by 2-D polynomial models is of concern. The optimal sampling schedule minimizing the volume of the uncertainty of the estimated parameters has been analytically derived when the available measurements are corrupted by an error that is described both in statistical and set membership terms. General conditions about the choice of the sampling points making the model identifiable have been found.

Experimental design for estimating the optimum point in a response surface

The problem of optimal experimental design for response optimization is considered. The optimal point (control)x* of a response surface is to be determined by estimating the response parametersθ from measurements performed at design pointsxi,i=1,...,N. Classical sequential approaches for choosing thexi's are recalled. A loss function related to the issue of response optimization is used to define control-oriented design criteria. The design policies differ depending on whether least-squares or minimum risk estimation is used to estimateθ. Connections between various criteria suggested in the literature are exhibited. Special attention is given to quadratic model responses. Most approaches presented assume that the response is correctly described by a given parametric function over the region of interest. Possible deterministic departures from this function raise the problem of model robustness, and the literature on the subject is briefly surveyed.

Systems Analysis in Ground‐Water Planning and Management

The objective of this paper is to review the state of the art of systems analysis and optimization techniques developed in the field of water resources for the planning and management of a ground‐water system. The areas reviewed include the following: ground‐water management models, inverse solution techniques for parameter identification, and optimal experimental design methods. Emphasis is placed upon ground‐water supply management models, as opposed to models used for ground‐water quality management. The techniques that have been used in the optimization of ground‐water management include: linear programming, mixed‐integer and quadratic programming, differential dynamic programming, nonlinear programming, and simulation. The inverse problem of parameter identification pertains the optimal determination of model parameters using historical input and output observations. Because of data limitation in both quantity and quality, the inverse problem is inherently ill posed. This paper summarizes recent advances made in the inverse procedures and methods developed to alleviate the problems of instability ard nonuniqueness of the identified parameters. The optimal experimental design problem addresses the issue of data requirements and optimal sampling strategies for the purpose of parameter identification. A criterion must be established for the optimal design of a pumping test. The fundamental concept of optimal experimental design and various criteria used for optimization are reviewed.

A method to find the minimax of functionals dependent on probability measures

The paper suggests a method for finding the minimax of functionals dependent on probability measures. Almost sure convergence of the method is proved. The method is applied to optimal experimental design problems.

Experiment design in a bounded-error context: Comparison with D-optimality

The problem of optimal experiment design for linear regression models is addressed. The measurement errors are supposed to be bounded, with no other hypothesis on their distribution, contrary to the classical statistical approach. An experiment is considered here as optimal if it minimizes the volume of the set of all parameters that are consistent with the data, the model structure, and the noise bounds. This new design policy is compared to the classical D-optimal design used in a statistical context. When the number of measurements is equal to the number of parameters to be estimated, the new optimality criterion, although relying on quite different hypotheses on the noise distribution, is shown to lead to the same optimal policy as a D-optimality criterion. This is no longer true when more measurements are to be made. Depending on the shape of the admissible experimental domain, it is sometimes possible to design optimal experiments without any calculations. An algorithmic procedure is suggested, to be used when this is not so.

An extended identifiability in aquifer parameter identification and optimal pumping test design

An extended identifiability, called “δ identifiability”, is developed for groundwater modeling and management. Some fundamental concepts are developed for establishing a criterion in connection with the problem of optimal experimental design, such as the design of an optimum pumping test to assist aquifer parameter identification. In practice, the quantity and quality of data prohibits a unique solution of the inverse problem of parameter identification. The proposed “δ identifiability” is based upon the concept of weak uniqueness. The determination of the admissibility of a given design is formulated as a nonlinear programming problem. The original constrained problem is transformed into solving a sequence of unconstrained problems by a penalty function method. Numerical experiments are conducted to illustrate the proposed concept and algorithm.

General Differential and Lagrangian Theory for Optimal Experimental Design

The problem of optimal experimental design for estimating parameters in linear regression models is placed in a general convex analysis setting. Duality results are obtained using two approaches, one based on subgradients and the other on Lagrangian theory. The subgradient concept is also used to derive a potentially useful equivalence theorm for establishing the optimality of a singular design and, finally, general versions of the original equivalence theorems of Kiefer and Wolfowitz (1960) are obtained.

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.

Optimal experiment design for autoregressive model with output power constraints

The output power constraint problem of optimal experiment design for an autoregressive model is considered. It is shown that besides the known minimum variance control law design, a D - optimal experiment consisting of sinusoidai input test signals without feedback can be obtained by solving a set of nonlinear equations. Two examples are also given to illustrate this design methodology.