Optimal Experimental Design For Parameter Estimation Research Articles

Tractable global solutions to chance-constrained Bayesian optimal experiment design for arbitrary prior and noise distributions

Optimal experiment design (OED) aims to optimize the information content of experimental observations by designing the experimental conditions. In Bayesian OED for parameter estimation, the design selection is based on an expected utility metric that accounts for the joint probability distribution of the uncertain parameters and the observations. This work presents solution methods for two approximate formulations of the Bayesian OED problem based on Kullback–Leibler divergence for the particular case of Gaussian prior and observation noise distributions and the general case of arbitrary prior distributions and arbitrary observation noise distributions when the observation noise corresponds to arbitrary functions of the states and random variables with an arbitrary multivariate distribution. The proposed methods also allow satisfying path constraints with a specified probability. The solution approach relies on the reformulation of the approximate Bayesian OED problem as an optimal control problem (OCP), for which a parsimonious input parameterization is adopted to reduce the number of decision variables. An efficient global solution method for OCPs via sum-of-squares polynomials and parallel computing is then applied, which is based on approximating the cost of the OCP by a polynomial function of the decision variables and solving the resulting polynomial optimization problem to global optimality in a tractable way via semidefinite programming. It is established that the difference between the cost obtained by solving the polynomial optimization problem and the globally optimal cost of the OCP is bounded and depends on the polynomial approximation error.

Practical Identifiability Analysis and Optimal Experimental Design for the Parameter Estimation of the ASM2d-Based EBPR Anaerobic Submodel

Identifiability analysis is a precondition for reliable parameter estimation. Building on previous work on structural identifiability, this paper focuses on the practical identifiability and optimal experimental design (OED) of the EBPR anaerobic submodel. The nonnegative determinant of the Fisher information matrix (FIM) found in this study clearly demonstrates that the parametersYPO4,KA,qPHA, andXPAOin the submodel are practically identifiable usingSAandSPO4as the measured variables and fixingKPPas the default value. Furthermore, fixingKPPto study the practical identifiability of the other parameters and to estimate their values is shown to be valid. Subsequently, a modeling-based procedure for the OED for parameter estimation was proposed and applied successfully to anaerobic phosphorus release experiments. According to the FIMD-criterion, the optimal experimental condition was determined to be an initialSAconcentration of 300 mg/L. Under the optimal experimental condition, errors in the values ofYPO4,KA,qPHA, andXPAOare all below 20%, and the estimated values were 0.35 ± 0.02 mg P/mg COD, 3.88 ± 0.41 mg COD/L, 3.35 ± 0.27 mg P/(mgCOD⁎d-1), and 1500 ± 72 mg COD/L, respectively. Compared to the results from the nonoptimal experimental condition, the practical identifiability and the estimation precision of the four parameters were improved.

Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature

Optimal experiment design for parameter estimation (OED/PE) has become a popular tool for efficient and accurate estimation of kinetic model parameters. When the kinetic model under study encloses multiple parameters, different optimization strategies can be constructed. The most straightforward approach is to estimate all parameters simultaneously from one optimal experiment ( single OED/PE strategy). However, due to the complexity of the optimization problem or the stringent limitations on the system's dynamics, the experimental information can be limited and parameter estimation convergence problems can arise. As an alternative, we propose to reduce the optimization problem to a series of two-parameter estimation problems, i.e., an optimal experiment is designed for a combination of two parameters while presuming the other parameters known. Two different approaches can be followed: (i) all two-parameter optimal experiments are designed based on identical initial parameter estimates and parameters are estimated simultaneously from all resulting experimental data ( global OED/PE strategy), and (ii) optimal experiments are calculated and implemented sequentially whereby the parameter values are updated intermediately ( sequential OED/PE strategy). This work exploits OED/PE for the identification of the Cardinal Temperature Model with Inflection (CTMI) ( Rosso et al., 1993). This kinetic model describes the effect of temperature on the microbial growth rate and encloses four parameters. The three OED/PE strategies are considered and the impact of the OED/PE design strategy on the accuracy of the CTMI parameter estimation is evaluated. Based on a simulation study, it is observed that the parameter values derived from the sequential approach deviate more from the true parameters than the single and global strategy estimates. The single and global OED/PE strategies are further compared based on experimental data obtained from design implementation in a bioreactor. Comparable estimates are obtained, but global OED/PE estimates are, in general, more accurate and reliable.

Accurate estimation of cardinal growth temperatures of Escherichia coli from optimal dynamic experiments

Prediction of the microbial growth rate as a response to changing temperatures is an important aspect in the control of food safety and food spoilage. Accurate model predictions of the microbial evolution ask for correct model structures and reliable parameter values with good statistical quality. Given the widely accepted validity of the Cardinal Temperature Model with Inflection (CTMI) [Rosso, L., Lobry, J. R., Bajard, S. and Flandrois, J. P., 1995. Convenient model to describe the combined effects of temperature and pH on microbial growth, Applied and Environmental Microbiology, 61: 610–616], this paper focuses on the accurate estimation of its four parameters ( T min, T opt, T max and µ opt) by applying the technique of optimal experiment design for parameter estimation (OED/PE). This secondary model describes the influence of temperature on the microbial specific growth rate from the minimum to the maximum temperature for growth. Dynamic temperature profiles are optimized within two temperature regions ([15 °C, 43 °C] and [15 °C, 45 °C]), focusing on the minimization of the parameter estimation (co)variance (D-optimal design). The optimal temperature profiles are implemented in a computer controlled bioreactor, and the CTMI parameters are identified from the resulting experimental data. Approximately equal CTMI parameter values were derived irrespective of the temperature region, except for T max. The latter could only be estimated accurately from the optimal experiments within [15 °C, 45 °C]. This observation underlines the importance of selecting the upper temperature constraint for OED/PE as close as possible to the true T max. Cardinal temperature estimates resulting from designs within [15 °C, 45 °C] correspond with values found in literature, are characterized by a small uncertainty error and yield a good result during validation. As compared to estimates from non-optimized dynamic experiments, more reliable CTMI parameter values were obtained from the optimal experiments within [15 °C, 45 °C].

Practical application of dynamic temperature profiles to estimate the parameters of the square root model

Optimal experimental design for parameter estimation (OED/PE) is a promising method to improve parameter estimation accuracy and minimise experimental effort in the field of predictive microbiology. In this paper, the OED/PE methodology was applied on two practical examples: the growth of Bacillus cereus and Enterobacter cloacae in liquid whole egg product. Both strains were recovered from samples of a commercial product. The goal of the modelling exercise was to quantify the influence of temperature on bacterial growth. The Baranyi-model for bacterial growth combined with the Ratkowsky square root model to describe temperature dependence was used. Using this model, a temperature step profile was calculated based on the optimal D-criterion. The model was then fitted against the experimental bacterial growth curve measured under the dynamic temperature conditions. This process was repeated until the parameters could be estimated with sufficient accuracy, apparent by the model prediction errors. For B. cereus, prior information could be extracted from the literature, allowing calculating a dynamic temperature profile directly. Two-step profiles were sufficient to obtain a good estimation for the model parameters. No prior information could be found for E. cloacae. Therefore, a limited series of static experiments had to be conducted to obtain usable prior model parameters estimates. Only one dynamic experiment was then needed to achieve a good estimation.

Optimal experiment design for cardinal values estimation: guidelines for data collection

Optimal experiment design for parameter estimation (OED/PE) is an interesting technique for modelling practices when aiming for maximum parameter estimation accuracy. Nowadays, experimental designs for secondary modelling within the field of predictive microbiology are mostly arbitrary or based on factorial design. The latter type of design is common practice in response surface modelling approaches. A number of levels of the factor(s) under study are selected and all possible treatment combinations are performed. It is however not always clear which levels and treatment combinations are most relevant. An answer to this question can be obtained from optimal experiment design for—in this particular case—parameter estimation. This technique is based on the extremisation of a scalar function of the Fisher information matrix. The type of scalar function determines the final focus of the optimised design. In this paper, optimal experiment designs are computed for the cardinal temperature model with inflection point (CTMI) and the cardinal pH model (CPM). A model output sensitivity analysis (depicting the sensitivity of the model output to a small change in the model parameters) yields a first indication of relevant temperature or pH treatments. Performed designs are: D-optimal design aiming for a maximum global parameter estimation accuracy (by minimising the determinant of the Fisher information matrix), and E-optimal design improving the confidence in the most uncertain model parameter (by maximising the smallest eigenvalue of the Fisher information matrix). Although lowering the information content of a set of experiments, boundary values on the design region need to be imposed during optimisation to exclude unworkable experiments and partly account for incorrect nominal parameter values. Opposed to the frequently applied equidistant or arbitrary treatment placement, optimal design results show that typically four informative temperature or pH levels are selected and replicate experiments are to be performed at these points. Informative experiments are typically placed at points with an extreme model output sensitivity.

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.