Chemical Engineering Science 65 (2010) 5482–5494
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Sequential experimental design based on multiobjective optimization procedures Andre´ L. Alberton a, Marcio Schwaab b, Evaristo Chalbaud Biscaia Jr.a, Jose´ Carlos Pinto a,n a b
´ria—CP: 68502, Rio de Janeiro, RJ 21941-972, Brazil Programa de Engenharia Quı´mica/COPPE, Universidade Federal do Rio de Janeiro, Cidade Universita ´ria, Santa Maria, RS 97105-900, Brazil Departamento de Engenharia Quı´mica, Universidade Federal de Santa Maria, Av. Roraima, 1000, Cidade Universita
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 October 2009 Received in revised form 8 July 2010 Accepted 20 July 2010 Available online 24 July 2010
Model-based sequential experimental designs are frequently applied for discrimination of rival models and/or estimation of precise model parameters. Although the development and use of a single design criterion to perform the simultaneous model discrimination and precise parameter estimation seem appealing, published material indicates that previous attempts to develop such a single design criterion have not been successful. Despite that, this problem has rarely been analyzed with the help of multiobjective optimization procedures. In this work, a multiobjective optimization method based on the particle swarm optimization procedure is used to build the Pareto fronts in experimental design problems where distinct design criteria used for discrimination of rival models and/or estimation of precise model parameters are considered simultaneously. It is shown through the rigorous analysis of the Pareto sets that both design objectives are frequently conflicting, which means that optimum discrimination of rival models and estimation of precise model parameters cannot be performed simultaneously in many cases. However, it is also shown that the use of the posterior covariance matrix of estimated model parameters for model discrimination makes the design of experiments for the simultaneous optimum model discrimination and estimation of model parameters possible in many experimental design problems. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Parameter identification Kinetics Mathematical modeling Optimization Sequential experimental design Model discrimination
1. Introduction The development of mathematical models is an important and widespread activity for analysis of chemical and biological processes. Mathematical models are needed for design, optimization and control purposes. However, a mathematical model should only be used after the appropriate validation step (performed by comparing model simulations with available experimental data), in order to guarantee the reliability of the obtained simulation results. As experimental values are always corrupted to some extent with errors, the use of statistically based methods for analysis of experimental data and model performances is of fundamental importance. The use of statistically based methods for design of experiments is particularly important, as these techniques allow for the development of models with optimum performances (as defined by the user) with the fewest number of experiments. However, as the pursued objectives can constitute a very broad set of desired performance indexes, distinct experimental design criteria can be proposed. For instance, during the initial model building stages,
n
Corresponding author. Tel.: +55 21 25628337; fax: +55 21 25628300. E-mail address:
[email protected] (J. Carlos Pinto).
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.07.010
different models with distinct mathematical structures can be capable of representing the available experimental data with similar performances. Consequently, the analyst must design experiments in order to discriminate the best model among the available alternatives. After selecting the best suited model, it may be necessary to design experiments to improve the precision of the parameter estimates, which can also lead to improved model predictions (and performance). Experimental designs are almost always performed iteratively. Based on previous experimental observations (or in previous experience), new experiments are designed (and carried out) in order to optimize the performance index defined by the analyst. Then, the new observations are included in the experimental data set and the desired objectives are analyzed. If the obtained results do not meet the required performance indexes, new experiments are designed and realized. The use of such sequential experimental design procedures can improve the reliability of the obtained results, as new experimental observations are incorporated when needed in order to improve the specific objectives defined by the analyst. The sequential experimental design for precise parameter estimation is usually performed through the optimization of a norm of the posterior covariance matrix of parameter estimates, described as the expected covariance matrix of parameter
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estimates after the inclusion of the new set of observations in the experimental data set. The posterior covariance matrix of parameter estimates can be defined as (Bard, 1974) follows: " ^ h, m ¼ V
N þK X
#1 1 BTi,m V1 i Bi,m þ V h,m
ð1Þ
i ¼ N þ1
where Vh,m is the current covariance matrix of parameter estimates of model m (obtained with the available N experi^ h, m is the posterior covariance matrix of parameter ments), V estimates (after inclusion of the new K observations), Vi is the covariance matrix of experimental uncertainties at experimental condition i and Bi,m is the sensitivity matrix, which contains the first derivatives of the responses of model m with respect to the model parameters at the ith experimental condition, defined as brp ¼
@yr @yp
ð2Þ
that is, the derivative of the response r with respect to the parameter p (for a specific model m and experimental condition i). In order to improve the quality of the model parameters, Box and Lucas (1959) originally proposed the selection of the experimental condition that minimizes the determinant of the posterior covariance matrix of the parameter estimates, since this determinant is proportional to the volume of the hyper-ellipsoid that defines the confidence region of the parameter estimates under assumptions of normal distribution. Alternatively, Hosten (1974) proposed the minimization of the largest characteristic value of the posterior covariance matrix of parameter estimates, which is equivalent to the minimization of the maximum axis of the ellipsoidal confidence region of parameter estimates and to the minimization of the maximum parameter uncertainty. However, experiments can also be designed to reduce the correlations among the parameter estimates, as this can significantly prejudice parameter identification. For this reason, Pritchard and Bacon (1978) proposed the minimization of the average squared correlation among the parameter estimates as the optimum experimental design criterion. Bernaerts et al. (2000) used a modified criterion, where the ratio between the larger and the smaller axis of the ellipsoidal confidence region is minimized. Recently, Franceschini and Macchietto (2008) proposed a more involving criterion in order to reduce the correlations among the parameter estimates, taking into account the information content of the new experimental conditions. It is important to observe that correlations among parameter estimates can be minimized efficiently through the appropriate reparameterization of the model equations, as described by Schwaab and Pinto (2007, 2008) and Schwaab et al. (2008b). The uncertainties of the parameter estimates can also be reduced when the trace of the posterior covariance matrix of parameter estimates is minimized (Pinto et al., 1990; Asprey and Macchietto, 2002). In order to normalize the magnitudes of the distinct model parameters, Pinto et al. (1991) proposed using the relative posterior covariance matrix of parameter estimates for sequential design of experiments. Vanrolleghem et al. (1995) and Versyck et al. (1997) applied the experimental design criteria for precise parameter estimation in order to allow identifiability and identification in biological processes. Donckels et al. (2009a, 2010) analyzed the design of experiments that allowed the simultaneous improvement of parameter estimates of several rival models, in order to improve the efficiency of model discrimination procedure, since accurate parameter estimates lead to accurate model predictions facilitating model discrimination. In order to perform the discrimination of rival models, experimental conditions are normally designed for maximization
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of some measure of the difference of the responses obtained with the distinct plausible models. Hunter and Reiner (1965) originally proposed a very simple criterion, based on the maximization of the squared deference between the responses of two rival models: D1,2 ¼ ðy^ 1 y^ 2 Þ2
ð3Þ
where D1,2 is the discriminant value, y^ 1 and y^ 2 are the predictions of Models 1 and 2 at experimental condition x and with model parameters h1 and h2 estimated with the available N experiments (x, h1 and h2 are omitted for simplicity). It must be observed that Eq. (3) implicitly considers that experimental and model prediction variances are constant throughout the experimental region. In order to overcome this drawback, Box and Hill (1967) proposed a design criterion that takes into account the prediction variances and model probabilities as follows: " M 1 M X X ^ 2m s ^ 2n Þ2 ðs D¼ þ ðy^ m Pm Pn 2 2 ^ m Þðs2 þ s ^ 2n Þ ðs þ s n ¼ 1 n ¼ mþ1 !# 1 1 ð4Þ þ y^ n Þ2 s2 þ s^ 2m s2 þ s^ 2n ^ 2m represent where Pm is the probability of model m and s2 and s the experimental variance and the prediction variance of model m. The definition of model probabilities allows for faster model discrimination, as the discrimination procedure is influenced more strongly by the models that present the best performances. However, as discussed by Buzzi-Ferraris and Forzatti (1983), when model probabilities are updated as suggested by Box and Hill (1967), the obtained probabilities are sensitive to ordering of available experiments (which is a major drawback, as this cannot be supported by physical reasoning). Besides, maximization of Eq. (4) can lead to selection of experiments that lead to maximization of model prediction variances, which cannot be supported by sound physical reasoning either. Atkinson and Fedorov (1975a, b) developed a T-optimum design criterion for model discrimination, assuming that one of the rival models was the true model. This method can be particularly interesting for experimental design when there is no available experimental information. Ponce de Leon and Atkinson (1991) also made use of prior model probabilities and assumed that any of the rival models could be regarded as the true one, leading to more involving representation of the design equation. Buzzi-Ferraris and Forzatti (1983) proposed a design criterion for sequential model discrimination as follows: D¼
M 1 X
M X
ðy^ m y^ n Þ2 PM 2 ^2 n ¼ 1 n ¼ m þ 1 ðM1ÞðM s þ m ¼ 1 sm Þ
ð5Þ
which is the ratio between the variance of model deviations and the mean values of model prediction variances. In Eq. (5) all models are considered simultaneously during calculation of the discriminant D. The discrimination between pairs of models can be more interesting many times, as it allows for fast elimination of bad candidates. In this case, Eq. (5) can be written as Dm,n ¼
ðy^ m y^ n Þ2 ^ 2m þ s ^ 2n 2s2 þ s
ð6Þ
The criterion defined in Eq. (6) can be extended for models with multiple responses (Buzzi-Ferraris et al., 1984) as follows: T ^ ^ Dm,n ¼ ðy^ m y^ n Þ V1 m,n ðym yn Þ
ð7Þ
where ym is a vector of responses for model m and Vm,n is the covariance matrix of differences between model predictions,
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defined as Vm,n ¼ 2V þVm þ Vn
ð8Þ
where V is the covariance matrix of experimental deviations and Vm is the covariance matrix of model prediction for model m, calculated as Vm ¼ Bm Vh,m BTm
ð9Þ
where Bm is the matrix of sensitivities of model m defined in Eq. (2). An interesting feature of the criteria defined in Eqs. (6) and (7) is that they suggest that the model discrimination procedure should be halted when the value of the discriminant between models m and n is lower than a certain critical value (calculated with the help of the classical w2-distribution, for instance), which indicates that model discrimination is not possible. In order to avoid the execution of experiments for discrimination of bad candidate models, Schwaab et al. (2006) extended the criteria defined in Eq. (7) as T
^ ^ Dm,n ¼ ðPm Pn ÞZ ðy^ m y^ n Þ V1 m,n ðym y n Þ
ð10Þ
where the parameter Z is used to modulate the importance of the model probabilities. As the value of Z approaches 0, the effect of the model probabilities is decreased; as the value of Z increases, the effect of the model probabilities increases and the discrimination criterion concentrates the effort to discriminate among the more plausible models. It must be emphasized that the model probabilities used in Eq. (9) can be calculated with the help of standard statistical tools (such as the classical w2-distribution) and are not sensitive to the ordering of available experiments, as shown by Schwaab et al. (2006). It must be observed that the covariance matrix of parameter estimates used to calculate the covariance matrix of model predictions in Eq. (9) does not take into account the effect of the new experimental conditions on the parameter uncertainties. Schwaab et al. (2008c) were the first to observe this important numerical aspect and for this reason proposed the use of the posterior covariance matrix of parameter estimates defined in Eq. (1) for the computation of the posterior covariance matrix of model predictions, as follows: ^ m ¼ Bm V ^ h, mBT V m
ð11Þ
This procedure was developed independently by Donckels et al. (2009b), where emphasis was given to the beneficial effects of this ‘anticipatory approach’ on model discrimination. Eq. (9) can then be used for calculation of the posterior covariance matrix of differences between model predictions, as follows: ^ m, n ¼ 2V þ V ^ m þV ^n V
ð12Þ
Finally, Eq. (12) can be inserted into the design criterion to provide the following new discriminant equation: ^ m, n ¼ ðPm Pn ÞZ ðy^ y^ ÞT V ^ m, n1 ðy^ y^ Þ D m n m n
ð13Þ
As observed by Schwaab et al. (2008c), the use of the posterior covariance matrix of parameter estimates in Eq. (13) increases the model discrimination power of the design technique and increases the reliability of the experimental design procedure, since the effect of the new experimental condition on the model performance is not ignored. It is interesting to observe that the use of Eq. (13) also leads to experimental designs that reduce the uncertainties of the parameter estimates, as reduction of the variances of the model parameters cause the reduction of the variances of the model responses and improvement of the discriminant index. One must note that this certainly depends on the experimental results and on the quality of the posterior covariance matrix of parameter estimates, which is based on the current parameter values. If the parameter estimates change
significantly after re-estimation of model parameters, it cannot be assured that parameter uncertainties will in fact decrease. Anyway, this can be regarded as a very interesting result, as previous attempts to design experiments for the simultaneous discrimination of rival models and estimation of precise model parameters were not successful (Hill et al., 1968; Borth, 1975). Some previous works present propositions for achievement of the simultaneous model discrimination and precise parameters estimation. Generally, the conciliation of these distinct objectives is pursued by compound design criteria, maximizing a function compounded by one term that represents the model discrimination function and the other term that represents the precise parameter estimation function (Hill et al., 1968; O’Brien and Rawlings, 1996; Atkinson, 2008; Tommasi, 2009). It is generally believed that experiments that are good for model discrimination are seldom useful for precise parameter estimation. It must be observed that experimental designs for precise parameter estimation are model specific and that, due to the different model structures, optimal experimental conditions may vary from one model to another. Consequently, experimental designs for model discrimination and precise parameter estimation are expected to be conflicting to some extent. However, as discussed by Schwaab et al. (2008c) and shown in Eq. (13), these two pursued objectives are not necessarily conflicting. As a matter of fact, the two objectives can be conciliated when significant model divergence and significant reduction of model prediction uncertainties occur simultaneously at similar experimental regions. Although it cannot be formally guaranteed that these conditions will always be fulfilled, previous experience indicates that these conditions occur simultaneously quite often, as significant reduction of model prediction uncertainties make model divergence more evident. Therefore, the use of Eq. (13) often leads to the selection of experimental regions where good discrimination potential and good precise parameters estimation potential can be obtained simultaneously. It is important to emphasize that most of the previously reported works can be formally described in terms of the Fisher Information Matrix (Atkinson et al., 2007). For this reason, the Fisher Information Matrix approach has also been used for formulation of some of the previously reported multiobjective experimental design problems (O’Brien and Rawlings, 1996; Donckels et al., 2010). Based on the previous discussion, a multiobjective optimization method based on the particle swarm optimization procedure is used in this work to build the Pareto fronts in experimental design problems where distinct design criteria used for discrimination of rival models and/or estimation of precise model parameters are considered simultaneously. The Pareto fronts can provide the proper characterization of how conflicting the experimental design criteria for model discrimination and precise parameter estimation are.
2. Numerical approach The multiobjective optimization problem consists in the simultaneous optimization of N objective functions (F1(x), F2(x), y, FN(x), N Z2), where x is a vector containing the search variables. Generally, it is not possible to find one vector x that maximizes simultaneously all objective functions. For this reason, the Pareto optimality criterion must be defined. A candidate point xA in the search region is said to dominate another candidate point xB when all objective functions evaluated at xA are higher (or lower, if the objective functions are being minimized) than the respective objective functions evaluated at point xB. If at least one of the N objective functions evaluated at xB is higher (lower) than
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the respective objective function evaluated at xA, the points xA and xB are said to be non-dominated. The set of non-dominated solutions is often referred to as the Pareto front (Miettinen, 1999). Multiobjective optimizations are usually performed with the help of non-deterministic methods, such as the evolutionary, genetic and particle swarm algorithms (Srinivas and Deb, 1994; Zitzler et al., 2000; Deb et al., 2002; Parsopoulos and Vrahatis, 2002; Coello et al., 2004). In the present work, the algorithm based on particle swarm optimization is used, as originally proposed by Coello et al. (2004). (The interested reader should refer to the original reference for detailed presentation of the optimization technique.) The multiobjective optimization procedure is defined here as the simultaneous maximization of design criteria used for estimation of precise model parameters and discrimination of rival models, assuming that a single experiment must be designed each time. Eqs. (10) and (13) are used for discrimination of rival models for the reasons presented in the previous section. In order to improve the quality of the presentation and normalize the values of the objective functions in the range [0,1], Eqs. (10) and (13) were rewritten as relative objective functions, as shown below FD1 ¼ Dm,n =Dmax r,s max
^ m,n =D ^ FD2 ¼ D r,s
ð14aÞ ð14bÞ
where the subscripts r and s are related with the pair of models that leads to the maximum value of the discrimination criterion. The experimental design criterion used for precise parameter estimation was the minimization of the determinant of the posterior covariance matrix of parameter estimates (or maximization of the determinant of its inverse), as defined in Eq. (1). It must be observed that M distinct objectives are defined, since one can search for the optimal experimental condition for precise estimation of model parameters of any of the M rival models. As shown in the previous case, the objective function was also normalized to lie within the interval [0,1], as follows: min min FEm ¼ Detm =Detm ¼ detðV1 =detðV1 h,m Þ h,m Þ
ð15Þ
min represents the minimum value observed in the whereDetm search region. As a consequence, Eqs. (14) and (15) must be maximized in the search region. When two rival models are considered, the Pareto front can be observed in a 3D plot, since there are 3 objective functions. However, even in this simple case, it is almost always preferable to observe the Pareto front for pairs of objectives, particularly by confronting the discrimination and the precise parameter estimation criteria for one of the models. There are unavoidable difficulties to treat the problem as the number of objective functions increases in a multiobjective optimization, as the interpretation of projections in two dimensions can be very difficult if the multidimensional Pareto front is not regularly shaped. Besides, some objectives may be conciliated but other objectives may not, leading to a complex scenario for the choice of one specific solution inside the Pareto front. The choice of one specific solution is usually done adopting some specific criterion, such as maximin, combined criteria, ideal point, among others (Atkinson, 2008; Donckels et al., 2010), or using alternative approaches, such as the ones based on lp distance functions (Donckels et al., 2010). Nevertheless, the construction of the Pareto fronts, even in pairs of objective functions, helps to investigate how conflicting these objectives are. When several rival models are considered simultaneously, the user can concentrate the analysis of the Pareto fronts on the most promising models (presenting the highest model probabilities), avoiding the unnecessary analysis of poor models that are likely
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to be discarded. In this case, the multiobjective approach and construction of the Pareto fronts still constitute an important tool in order to evaluate whether the remaining and most important objective functions can be optimized simultaneously.
3. Examples Two examples are presented here to illustrate the application of the proposed procedure. In both examples the estimation of model parameters was performed with a hybrid estimation method, as described by Schwaab et al. (2008a). According to this method, the particle swarm technique (Kennedy and Eberhart, 1995, 2001) is used to initiate the optimization procedure. Afterwards, the obtained solution is used as the initial guess in a second optimization round with the help of a Gauss– Newton optimization procedure. During the experimental design procedure, the optimization of the individual design criteria was performed initially with the particle swarm optimization (Kennedy and Eberhart, 1995, 2001). Afterwards, the multiobjective particle swarm optimization (MOPSO) technique described by Coello et al. (2004) was used in a second optimization round, using the individual optimum conditions as the initial Pareto front. The examples investigated below are simple and used to illustrate the application of the proposed optimal design procedure. Nevertheless, it is important to emphasize that more complex models, such as dynamic systems, may introduce additional numerical difficulties during the parameter estimation and experimental design stages. As the complexity of models increases, the difficulty to implement consistent and robust numerical procedures also increases. This particular issue should be investigated in future works. However, it must be clear that the proposed techniques should not only be useful for the analysis of complex models, but also mainly for analysis of real systems, regardless of their numerical complexity. 3.1. Example 1: Adsorption models This first example considers the discrimination between two adsorption models used to describe the adsorption of a gaseous component onto a solid matrix. The models describe the equilibrium adsorbate concentration as a function of the gas pressure. Model 1 represents the Langmuir adsorption isotherm, while Model 2 represents the Freundlich adsorption isotherm, as described as follows:
y1,2 x 1 þ y1,2 x
Model 1 :
y1 ¼ y1,1
Model 2 :
y2 ¼ y2,1 xy2,2
ð16aÞ ð16bÞ
where ym,p is the parameter p of model m, x is the gas pressure in bar and y is the concentration of adsorbate in the solid material in mol kg 1. This problem was studied previously by Schwaab et al. (2008c) and constitutes a simple benchmark for comparative analysis. It was observed that the inclusion of the posterior covariance matrix of parameter estimates in the discrimination procedure, as defined in Eq. (13), leads to simultaneous improvement of parameter estimates and model discrimination. For this reason, this problem is analyzed here with the help of the multiobjective approach. Initialization of the experimental design procedure is performed as described by Schwaaab et al. (2008c). It is assumed that Model 1 is the correct model and model outputs are corrupted by a random normal deviation with zero mean and variance of 0.01. Parameters used to generate the ‘‘experimental data’’ were y1,1
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equal to 3 mol kg 1 and y1,2 equal to 2 bar 1. Initially, three experiments were proposed (Table 1) and used for parameter estimation, as shown in Table 2. Fm and Pm (the minimum value of
Table 1 Initial experimental data set for Example 1. Run
x (bar)
y (mol kg 1)
1 2 3
0.50 1.00 2.00
1.40 1.99 2.41
Table 2 Initial parameter estimates in Example 1. Model
Fm
Pm
ym,1 7 Dym,1
ym,2 7 Dym,2
1 2
0.135 1.478
76.1 23.9
3.175 7 0.283 1.892 7 0.060
1.613 7 0.402 0.372 7 0.055
Fig. 1. Relative design criteria as functions of the design variable (red line: FD1; blue line: FD2; black line: FE1; green line: FE2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Pareto fronts for two cases:
J:
the objective function and the relative probability of model m) are also shown in Table 2. Dym,p represents the standard deviation of parameter ym,p. Model 1 obviously leads to best fit of the available experimental data, although the performances of both models can be regarded as statistically equivalent. As a consequence, new experiments must be designed to discriminate the best model. Since only two models are considered in this example, model probabilities and the parameter Z do not affect the experimental design. The relative design criteria defined in Eqs. ((14)–(15)) are plotted as functions of the independent variable x in Fig. 1. It can be seen that the experimental design for model discrimination (with the inclusion of the posterior covariance matrix of parameter estimates, function FD2) and the experimental designs for precise parameter estimation of both models (functions FE1 and FE2) present a maximum value at x equal to 5 bar. Therefore, all objectives can be maximized simultaneously. In this case, the Pareto front is formed by a single point, where all design criteria reach the maximum value of 1. However, when one compares the experimental design for model discrimination without the inclusion of the posterior covariance matrix of parameter estimates (function FD1) with the experimental designs for precise parameter estimation of any of the two rival models, the simultaneous maximization of the distinct objective functions cannot be attained. Eq. (14a) presents a maximum value at very low values of x, while both relative precise estimation criteria present a maximum value at x equal to 5 bar. Consequently, a Pareto front can be drawn for each pair of objective functions, that is, for objective functions FD1 and FE1 (denoted by circles in Fig. 2) and objective functions FD1 and FE2 (denoted by crosses in Fig. 2). It can be observed in Fig. 2 that the Pareto fronts are not continuous and comprise two different regions, confirming the conflicting experimental design scenario. It must also be observed that the Pareto front comprises the range of xA(0.047, 0.133) bar and x equal to 5 bar, when the precise parameter estimation of Model 1 is considered, and the range of xA(0.045, 0.095) bar and x equal to 5 bar, when the precise parameter estimation of Model 2 is considered. One must note that the optimum design conditions constitute a very narrow range in the independent variable space. The discrimination power of the designed experimental condition can be calculated as suggested by Buzzi-Ferraris and Forzatti (1983) and Buzzi-Ferraris et al. (1984): if the discriminant value is higher than the number of model responses, the discrimination is possible; otherwise, the discrimination is not
functions FD1 and FE1; + functions FD1 and FE2.
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possible and the experimental design procedure must be halted and revised. The maximum discriminant value is equal to 4.4 at x equal to 0.045 bar (the normalized discriminant value is equal to 1) and, consequently, the discrimination is possible. (In this case, it is important to note that the discriminant value is higher than the number of model responses for all points in the Pareto front.) In fact, the lower discriminant value is equal to 2.9 at x equal to 5 bar, where the design criterion for parameter estimation attains its maximum. One could suggest this experimental condition to perform the new experiment, since the estimation criterion is maximum and discrimination is still possible. It must be noticed that when the posterior covariance matrix of parameter estimates is used, this conflicting scenario does not appear and both model discrimination and precise parameter estimation design criteria lead to the same experimental condition, as described by Schwaab et al. (2008c). Additional experiments were designed, performed and analyzed by Schwaab et al. (2008c) and it was shown that model discrimination could be attained after execution of a single additional experiment, regardless of the proposed discrimination design criteria. However, good parameter estimates could be obtained only when the design variable x was close to 5 bar, showing that experiments performed for low x values would be adequate for model discrimination, but not for precise parameter estimation, reflecting the conflicting Pareto set presented in Fig. 2. Therefore, the multiobjective analysis of this very simple problem indicates that the use of the posterior covariance matrix of parameter estimates during the discrimination of rival models can indeed reduce the conflicting nature of the experimental design problem, when model discrimination and estimation of precise model parameters are considered simultaneously.
3.2. Example 2: Discrimination of multi-response kinetic models This example considers the experimental design for discrimination among three multi-response rival models. This example was originally proposed by Buzzi-Ferraris et al. (1984) and was also analyzed by Schwaab et al. (2006). The original example contains four rival models; however, original Model 3 is not considered here because it provides very poor fits of available data and can be removed from the analysis after execution of a single additional experiment. Model candidates are presented in Eq. (17). Model 1 is assumed here to be the correct one and is used Table 3 Initial experimental data set for Example 2. Run
x1
x2
y1
y2
1 2 3 4 5
20.0 30.0 20.0 30.0 25.0
20.0 20.0 30.0 30.0 25.0
13.443 13.817 17.809 21.139 16.039
1.299 1.433 1.885 2.118 1.635
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to generate pseudo-experimental data with parameters K1,1 ¼0.1, K2,1 ¼0.01, K3,1 ¼ 0.1 and K4,1 ¼ 0.01. Random normal experimental deviations with zero mean and variances equal to 0.35 for y1 and 2.3 10 3 for y2 are added to model responses. Five initial experiments are performed in accordance with a standard 22 full factorial design, as shown in Table 3, as reported by Schwaab et al. (2006): Model 1 :
y1 ¼
K1,1 x1 x2 , 1þ K3,1 x1 þ K4,1 x2
y2 ¼
K2,1 x1 x2 1 þ K3,1 x1 þ K4,1 x2 ð17aÞ
Model 2 :
y1 ¼
Model 3 :
y1 ¼
K1,2 x1 x2 ð1 þ K3,2 x1 þ K4,2 x2 Þ2 K1,3 x1 x2 , 1þ K3,3 x1 þ K4,3 x2
,
y2 ¼
y2 ¼
K2,2 x1 x2 ð1þ K3,2 x1 Þ2
K2,3 x1 x2 1 þ K3,3 x1
ð17bÞ
ð17cÞ
In this example, seven design criteria can be considered simultaneously: four distinct discrimination criteria (maximization of the discriminant value with and without taking into account the posterior covariance matrix of parameter estimates, with Z values equal to 0 and 1) and three precise parameter estimation criteria (minimization of the determinant of the posterior covariance matrix of parameter estimates of each model). The experimental region is limited to the range 0rxi r55 for both design variables. The optimum experimental conditions obtained after optimization of each individual design criteria and the respective function values are shown in Table 4, where the underlined values indicate the maximum values for the analyzed criteria. One must observe that the experimental condition that allows for minimization of the determinant of the posterior covariance matrix of parameter estimates of Model 1 also allows for maximization of the discriminant value (for both Z equals 0 and 1), when the posterior covariance matrix of parameter estimates is taken into account, indicating once more that these objectives can be reached simultaneously. This occurs because the reduction of the parameter uncertainties also leads to reduction of model prediction uncertainties, as discussed previously. As Model 1 is the most probable model and fits the experimental data better, discriminating Model 1 from the other competitors becomes easier when parameter estimates become more accurate. From this point of view, analyses of the remaining models are biased, as the performances and qualities of parameter estimates of the remaining models are much poorer than the performance and quality of parameter estimates of Model 1. Fig. 3 presents the Pareto fronts when two objectives are considered simultaneously. It must be observed that the best value reached by any criterion is normalized and, for this reason, is equal to 1. The best scenario will be found when both objectives give a single point where both objective functions are equal to 1. In Fig. 3A the relative criteria for precise parameter estimation for Model 1 are confronted with the two relative discriminant criteria, with and without the posterior covariance matrix of
Table 4 Optimum values of the design criteria after five initial experiments. Optimum experiment
Criteria values
x1
x2
Dm,n (Z ¼0)
^ m,n (Z¼ 0) D
Dm,n (Z¼ 1)
^ m,n (Z¼ 1) D
Det1 1021
Det2 1026
Det3 1022
55.0 55.0 18.2 51.1 6.9
43.2 55.0 55.0 55.0 55.0
2.93 2.50 2.87 1.82 2.40
19.02 31.98 16.52 21.77 23.74
0.29 0.25 0.37 0.18 0.31
1.90 3.20 2.12 2.18 3.05
17.62 6.71 7.07 7.08 8.20
24.41 4.82 0.77 0.47 0.71
6.75 3.31 1.60 3.33 0.27
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Fig. 3. Pareto fronts for the relative estimation function FEi (i¼ 1, 2 and 3) and the relative discrimination functions FD1 (red) and FD2 (blue) with parameter Z equal to 0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Pareto fronts for the relative estimation function FEi (i¼ 1, 2 and 3) and the relative discrimination functions FD1 (red) and FD2 (blue) with parameter Z equal to 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Table 5 Sequentially designed experiments obtained with different criteria. Criteria
Run
x1
x2
y1
y2
P1
P2
P3
Initial Dm,n (Z ¼0)
1–5 6 7 8
– 55.0 4.8 28.3
– 43.2 55.0 55.0
– 34.212 11.933 34.925
– 3.424 1.267 3.519
40.0 52.7 59.1 74.1
31.2 12.7 24.3 0.0
38.9 58.5 0.0 0.0
^ m,n (Z¼0) D
6
55.0
55.0
43.037
4.293
50.2
15.6
47.4
7 8
5.3 27.4
55.0 55.0
13.623 35.064
1.363 3.556
61.9 70.3
27.8 0.0
0.0 0.0
Dm,n (Z ¼1)
6 7a 8a
18.2 55.0 55.0
55.0 55.0 55.0
29.194 43.037 41.124
3.004 4.293 4.401
61.0 59.6 23.1
0.4 0.8 0.2
0.2 0.0 0.0
^ m,n (Z¼1) D
6
55.0
55.0
43.037
4.293
50.2
15.6
47.4
7 8
7.9 28.4
55.0 55.0
18.110 34.872
1.876 3.573
60.5 72.5
24.5 0.0
0.0 0.0
Det1
6 7 8
55.0 9.8 55.0
55.0 55.0 55.0
43.037 21.835 41.124
4.293 2.160 4.401
50.2 64.9 18.4
15.6 23.7 4.8
47.4 0.0 0.0
Det2
6 7 8
51.1 14.8 55.0
55.0 55.0 55.0
41.322 27.490 43.037
4.223 2.708 4.293
59.9 63.6 67.5
29.2 2.8 2.4
59.1 0.0 0.0
Det3
6 7 8
6.9 55.0 55.0
55.0 55.0 55.0
17.506 43.037 41.124
1.746 4.293 4.401
59.5 67.3 21.9
28.4 25.9 6.2
9.3 0.0 0.0
MO
6 7a 8a
13.0 55.0 55.0
55.0 55.0 55.0
25.310 43.037 41.124
2.458 4.293 4.401
42.2 52.3 10.0
0.9 1.5 0.2
0.3 0.0 0.0
a Experiments designed with the precise estimation criteria for Model 1, as discrimination was already achieved.
parameter estimates. When the posterior covariance matrix of parameter estimates is not considered (FD1), the Pareto front is formed by 3 disconnected regions. The region that includes the maximum value for the function FE1 presents the lowest values of function FD1, while the region that includes the maximum value for the function FD1 presents the lowest values of function FE1. The central region is formed by values of function FE1 ranging from 0.99 to 0.96 and by values of function FD1 ranging from 0.94 to 0.98. In this central region, the objective function values for both precise parameter estimation and model discrimination are high. Consequently, an experimental condition from this central region can be used to guarantee that both objectives can be achieved simultaneously. When the posterior covariance matrix of parameter estimates is considered in the discrimination criteria (FD2), the optimal experimental condition is the same as that obtained for the precise parameter estimation of Model 1 (FE1), as shown in Table 4. Consequently, the Pareto front is formed by a single point, as shown in Fig. 3A. In Fig. 3B the relative criterion for precise parameter estimation of Model 2 (FE2) is confronted with the two relative discriminant criteria (FD1 and FD2). Again three disconnected regions are obtained when the discrimination criterion does not take into account the posterior covariance matrix of parameter estimates. The upper side of the central region seems to constitute the best choice for the experimental design, since it presents a relative estimation criterion value FE2 equal to 0.999 and a relative discriminant criterion value FD1 equal to 0.913. The other two regions present high values for one of the analyzed relative criteria and low values for the other one, indicating that it is not possible to optimize both criteria simultaneously. When the relative discriminant FD2 is considered, the Pareto front is formed by a region where all points present a relative precise parameter estimation criterion that is very close to 1. Therefore, one can
Fig. 5. Pareto fronts for the relative estimation function FEi (i¼ 1, 2 and 3) and the relative discrimination functions FD1 (red) and FD2 (blue) with parameter Z equal to 0 after inclusion of the sixth experiment (x1 equal to 55.0 and x2 equal to 43.2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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select the next experimental condition as the point (55.0, 55.0), since the precise parameter estimation criterion is high and the discriminant criterion attains its maximum value at this condition. In Fig. 3C, the scenario is more conflicting. Regardless of the relative discriminant criterion FD1 or FD2 adopted, it seems very difficult to select one point that allows for the simultaneous discrimination and precise parameter estimation for Model 3. When the parameter Z is set to 1, the qualitative behavior of the Pareto sets changes a bit, as shown in Fig. 4. When the relative discrimination criterion FD1 is considered, the Pareto fronts shown in Fig. 4A and B present two disconnected regions (and not three, as in Fig. 3). Despite that, it is still possible to select solutions that can be regarded as good for both precise parameter estimation and model discrimination, as discussed previously. From Table 4 and Figs. 3 and 4 it can be observed that the experimental conditions with values of x1 around 13.0 and x2 equal to 55.0 lead simultaneously to high values for the model discrimination and precise parameter estimation criteria for Models 1 and 2, when the posterior covariance matrix of parameter estimates is not considered in the discrimination criterion (function FD1). When the posterior covariance matrix of parameter estimates is considered in the discrimination criterion (function FD2), the optimum experimental condition is close to the point where x1 and x2 are equal to 55.0, leading to higher values for both model discrimination and precise parameter estimation criteria for Models 1 and 2. For Model 3, a conflicting scenario appears in almost all cases and it is difficult to define one experimental condition that provides high values for both precise parameter estimation and model discrimination criteria. Besides, the optimum experimental value in this case is different from the optimum experimental values obtained for Models 1 and 2. Despite that, it can be concluded once more that the multiobjective analysis indicates that the use of the posterior covariance matrix of parameter estimates during the discrimination of rival models can indeed reduce the conflicting nature of the experimental design problem, when model discrimination and estimation of precise model parameters are considered simultaneously. In order to proceed with the sequential experimental design, four different experimental conditions were selected and analyzed. The first experimental conditions is x1 equal to 55.0 and x2 equal to 43.2 and is the one that maximizes the discriminant function FD1, with Z equal to 0 and without taking into account the posterior covariance of parameter estimates. (One experiment was simulated at this condition with Model 1. The simulated values for y1 and y2 were equal to 34.212 and 3.424, after addition of the random normal deviation. Similar procedure was adopted in the following cases.) After inclusion of this new observation in the experimental data set, all three models can still provide good fits, as shown in Table 5. For this reason, the multiobjective optimization procedure was applied for design of the next experiment and the Pareto fronts obtained are shown in Fig. 5, while the optimum conditions for each individual criterion are
presented in Table 6. For the sake of consistency, the Z value was kept constant and equal to 0. It can be observed in Table 6 that the optimum experimental conditions for both discrimination criteria are essentially the same. Besides, the Pareto fronts presented in Fig. 5 for both functions FD1 and FD2 are very close to each other. These results indicate that the use of the posterior covariance matrix of parameter estimates does not lead to different results in this case, although the Pareto fronts in Fig. 5A and C indicate that the use of the posterior covariance matrix of parameter estimates (function FD2) may lead to selection of one experimental condition that presents high values (higher than 0.90) for both precise parameter estimation and model discrimination criteria (functions FE1 and FD2 in Fig. 5A and functions FE3 and FD2 in Fig. 5C). The second experimental condition selected for the sixth experiment is x1 and x2 equal to 55.0, since this condition is optimal for both model discrimination (when the criterion takes into account the posterior covariance matrix of parameter estimates, with Z equal to 0 and 1) and precise parameter estimation criteria for Model 1, as shown in Table 4. Although not optimal, this condition is also very good for the precise parameter estimation criterion for Model 2. After including this new observation in the experimental data set, new parameter estimates and model performances can be obtained, as presented in Table 5. Sequentially, the multiobjective optimization procedure is used once more and the obtained Pareto fronts can be obtained, as presented in Fig. 6 for Z equal to 0 and presented in Fig. 7 for Z equal to 1. Table 7 presents the optimal experimental conditions for each individual analyzed criterion. It can be seen in Table 7 that the discriminant values calculated with Z equal to 0 and 1 lead to similar optimal experimental conditions when the discriminant does not consider the posterior covariance matrix of parameter estimates. It can also be seen that the optimal experimental condition when the discriminant takes into account the posterior covariance matrix of parameter estimates with Z equal to 1 is practically the same one obtained for the precise parameter estimation of Model 3. In Fig. 6, where Z is equal to 0, the Pareto fronts are very similar, no matter whether the posterior covariance matrix of parameter estimates is considered or not. However, when Z is equal to 1 and the posterior covariance matrix of parameter estimates is considered (Fig. 7), the Pareto front indicates that the model discrimination criterion (FD2) is much closer to the precise parameter estimation criteria (FEi, with i equal to 1, 2 and 3), when compared with the results obtained with the model discrimination criterion (FD1) that does not take the posterior covariance matrix of parameter estimates into consideration. The third experimental condition selected as the sixth observation is the one that maximizes the discriminant criterion without considering the posterior covariance matrix of parameter estimates with Z equal to 1. After including the new experiment in the data set, the new parameter estimation results indicate that Models 2 and 3 are not adequate, as shown in Table 5. Consequently, model discrimination is achieved. This result had
Table 6 Optimum values of the design criteria after six experiments (sixth experiment defined as x1 equal to 55.0 and x2 equal to 43.2). Optimum experiment
Criteria values
x1
x2
Dm,n (Z¼ 0)
^ m,n (Z ¼ 0) D
Det1 1024
Det2 1028
Det3 1024
4.8 5.6 11.6 17.1 8.6
55.0 55.0 55.0 55.0 55.0
14.63 14.48 8.54 3.04 12.09
28.58 29.08 18.32 14.25 24.67
3.74 3.31 2.45 2.72 2.59
1.81 1.42 0.55 0.46 0.76
2.82 2.41 2.20 3.54 1.97
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Fig. 6. Pareto fronts for the relative estimation function FEi (i ¼1, 2 and 3) and the relative discrimination functions FD1 (red) and FD2 (blue) with parameter Z equal to 0 after inclusion of the sixth experiment (x1 equal to 55.0 and x2 equal to 55.0). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 7. Pareto fronts for the relative estimation function FEi (i¼ 1, 2 and 3) and the relative discrimination functions FD1 (red) and FD2 (blue) with parameter Z equal to 1 after inclusion of the sixth experiment (x1 equal to 55.0 and x2 equal to 55.0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Table 7 Optimum values of the design criteria after six experiments (sixth experiment defined as x1 equal to 55.0 and x2 equal to 55.0). Optimum experiment
Criteria values
x1
x2
Dm,n (Z¼ 0)
^ m,n (Z ¼0) D
Dm,n (Z¼ 1)
^ m,n (Z ¼1) D
Det1 1024
Det2 1028
Det3 1024
4.7 5.3 7.9 9.8 15.8 8.0
55.0 55.0 55.0 55.0 55.0 55.0
16.54 16.42 14.05 11.39 3.81 13.96
29.37 29.69 25.41 19.80 9.65 25.20
0.96 0.95 0.81 0.66 0.36 0.81
2.17 2.30 2.52 2.44 1.79 2.52
1.25 1.15 0.94 0.91 1.09 0.94
0.82 0.70 0.43 0.34 0.27 0.42
1.25 1.12 0.93 0.98 1.66 0.93
Fig. 8. Determinant of the covariance matrix of parameter estimates as a function of the designed experiment. (D0 and D1 indicate the discrimination criterion with Z equal to 0 or 1; SP and CP represent without or with the posterior covariance matrix of parameter estimates.)
already been observed by Schwaab et al. (2006). The fourth experimental condition selected as the sixth observation is the experimental point where x1 is equal to 13.0 and x2 is equal to
55.0, since this condition is promising for both model discrimination and precise parameter estimation, as shown in Figs. 3 and 4 and discussed previously. After including the new observation in the experimental data set, the new parameter estimation results indicate that model discrimination can be achieved, with elimination of Models 2 and 3, as shown in Table 5. In order to observe how the sequential design procedure evolves after the inclusion of the new experimental points, Table 5 presents the obtained sequential designs for all cases (MO corresponds to the best multiobjective performance, when both criteria are close to 1). In general, model discrimination is achieved after inclusion of three additional experiments. Criteria E1 and E3 could not discriminate among Models 1 and 2, confirming that precise parameter estimation does not necessarily lead to model discrimination. It is interesting to observe that some replicated experiments were designed when a precise parameter estimation criterion was used, indicating hat the experimental condition with x1 and x2 equal to 55.0 is indeed a good condition for precise parameter estimation. This condition was also selected by the model discrimination criterion that takes into account the posterior covariance matrix of parameter estimates, indicating that this discrimination criterion is very sensitive to modification of the parameter uncertainties when the precision of the parameter estimates can be improved. In order to observe how the precisions of the parameter estimates change along the experimental designs and how they are influenced by the design criteria, Fig. 8 shows how the determinant of the covariance matrix of the parameter estimates changes along the sequence of designed experiments for each model. It can be observed for Model 1 that, after inclusion of Run 6, the experiments designed with criteria Dm,n (with Z equal to 1, D1_SP), Det3 and MO lead to the poorest parameter estimates. It must be observed that model discrimination is achieved with a single additional experiment with two of these criteria, confirming the conflicting scenario. Besides, the best parameter estimates for Model 1 were obtained with the criteria that indicated x1 and x2 equal to 55.0 as optimal experimental conditions. After inclusion of Runs 7 and 8, the precision of the parameter estimates remained essentially the same for all criteria, with the exception of the discrimination criteria Dm,n with Z equal to 0, which led to determinant values approximately 5 times higher than in the other cases. The determinant of Model 2 decreased along the experiments for all criteria; however, as observed for Model 1, the determinant values obtained with criteria Dm,n and Z equal to 0 were significantly higher than in the other cases. For Model 3, an opposite behavior was observed: the designs that led to the worst parameter estimates for Model 1 led to the best parameter estimates for Model 3. After inclusion of Runs 7 and 8, all determinant values presented similar values. It can be observed in all cases that, after execution of Run 6, the precision of the parameter estimates become very different,
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which is particularly true for Models 1 and 3. When Run 7 is added, the parameter precisions become similar in all cases. This can possibly explain why the Pareto fronts presented in Figs. 5 and 6 and in Tables 5 and 6 suggest the execution of similar experimental conditions. Finally, one must observe that the Pareto fronts can be used as a guide for shifting the experimental design problem from model discrimination to precise parameter estimation, as they clearly show when both objectives are conflicting and when they are not. Furthermore, as already discussed, when several rival models are considered simultaneously, the user can concentrate the analysis of the Pareto fronts on the most promising models (presenting the highest probabilities), avoiding the unnecessary analysis of poor models that are likely to be discarded.
4. Conclusion In this work a multiobjective optimization approach was used for analysis of sequential experimental designs when model discrimination and precise parameter estimation are sought simultaneously. The results were presented as Pareto fronts where both objectives (model discrimination and estimation of precise parameters) can be confronted with each other in distinct cases. In the first example, regarding the analysis of two rival adsorption models, it was shown that the use of the posterior covariance matrix of parameter estimates during model discrimination can lead to the design of optimum experiments for both model discrimination and precise parameter estimation simultaneously. In the second example, regarding the analysis of three rival kinetic models, it was shown that both objectives are conflicting many times. However, it was also shown that the use of the posterior covariance matrix of estimated model parameters for model discrimination makes the design of experiments for the simultaneous optimum model discrimination and estimation of model parameters possible in many experimental design problems, especially when the models presenting the best performances are considered. Based on the obtained results, it can be said that the use of multiobjective optimization procedures for sequential design of experiments can constitute a very useful tool for anyone who is interested in the simultaneous discrimination of rival models and estimation of precise parameters, as the computed Pareto sets provide a measure of how conflicting the pursued objectives really are. As observed in the analyzed examples, some experimental regions can provide adequate model discrimination and parameter estimation power, even when these regions do not lead to optimal responses when the design criteria are analyzed separately.
Acknowledgment The authors thank CNPq (Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico) for providing scholarships and for supporting this work. References Asprey, S.P., Macchietto, S., 2002. Designing robust optimal dynamic experiments. Journal of Process Control 12, 545–556. Atkinson, A.C., Dovev, A.N., Tobias, R.D., 2007. Optimal Experimental Design, With SAS. Oxford University Press, New York. Atkinson, A.C., 2008. DT-optimum designs for model discrimination and parameter estimation. Journal of Statistical Planning and Inference 138, 56–64. Atkinson, A.C., Fedorov, V.V., 1975a. The design of experiments for discriminating between two rival models. Biometrika 62, 57–70.
5493
Atkinson, A.C., Fedorov, V.V., 1975b. Optimal design: experiments for discriminating between several models. Biometrika 62, 289–303. Bard, Y., 1974. Nonlinear Parameter Estimation. Academic Press, New York. Bernaerts, K., Versyck, K.J., Van Impe, J.F., 2000. On the design of optimal dynamic experiments for parameter estimation of a ratkowskytype growth kinetics at suboptimal temperatures. International Journal of Food Microbiology 54, 27–38. Borth, D.M., 1975. A total entropy criterion for the dual problem of model discrimination and parameter estimation. Journal of the Royal Statistical Society B 37, 77–87. Box, G.E.P., Hill, W.J., 1967. Discrimination among mechanistic models. Technometrics 9, 57–71. Box, G.E.P., Lucas, H.L., 1959. Design of experiments in non-linear situations. Biometrika 46, 77–90. Buzzi-Ferraris, G., Forzatti, P., 1983. A new sequential experimental design procedure for discriminating among rival models. Chemical Engineering Science 38, 225–232. Buzzi-Ferraris, G., Forzatti, P., Emig, G., Hofmann, H., 1984. Sequential experimental design for model discriminating in the case of multiresponse models. Chemical Engineering Science 39, 81–85. Coello, C.A.C., Pulido, G.T., Lechuga, M.S., 2004. Handling multiple objectives with particle swarm optimization. IEEE Transactions on Evolutionary Computation 8, 256–279. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6, 182–197. Donckels, B.M.R., De Pauw, D.J.W., Vanrolleghem, P.A., De Baets, B., 2009a. A kernel-based method to determine optimal sampling times for the simultaneous estimation of the parameters of rival mathematical models. Journal of Computational Chemistry 30, 2064–2077. Donckels, B.M.R., De Pauw, D.J.W., De Baets, B., Maertens, J., Vanrolleghem, P.A., 2009b. An anticipatory approach to optimal experimental design for model discrimination. Chemometrics and Intelligent Laboratory Systems 95, 53–63. Donckels, B.M.R., De Pauw, D.J.W., Vanrolleghem, P.A., De Baets, B., 2010. An ideal point method for the design of compromise experiments to simultaneously estimate the parameters of rival mathematical models. Chemical Engineering Science 65, 1705–1719. Franceschini, G., Macchietto, S., 2008. Novel anticorrelation criteria for model-based experiment design: theory and formulations. AIChE Journal 54, 1009–1024. Hill, W.J., Hunter, W.G., Wichern, D.W., 1968. A joint design criterion for the dual problem of model discrimination and parameter estimation. Technometrics 19, 145–160. Hosten, L.H., 1974. A sequential experimental design procedure for precise parameter estimation based upon the shape of the joint confidence region. Chemical Engineering Science 29, 2247–2252. Hunter, W.G., Reiner, A.M., 1965. Designs for discriminating between two rival models. Technometrics 7, 307–323. Kennedy, J., Eberhart, R.C., 1995. Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. IV, Perth, Australia, pp. 1942–1948. Kennedy, J., Eberhart, R.C., 2001. Swarm Intelligence. Morgan Kaufmann Publishers, San Diego. Miettinen, K.M., 1999. Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Norwell. O’Brien, T.E., Rawlings, J.O., 1996. A nonsequential design procedure for parameter estimation and model discrimination in nonlinear regression models. Journal of Statistical and Planning Inference 55, 77–93. Parsopoulos, K.E., Vrahatis, M.N., 2002. Particle swarm optimization method in multiobjective problems. In: Proceedings of the 2002 ACM Symposium on Applied Computing (SAC’2002), Madrid, Spain, pp. 603–607. Pinto, J.C., Loba~ o, M.W., Monteiro, J.L., 1990. Sequential experimental design for parameter estimation: a different approach. Chemical Engineering Science 45, 883–892. Pinto, J.C., Loba~ o, M.W., Monteiro, J.L., 1991. Sequential experimental design for parameter estimation: analysis of relative deviations. Chemical Engineering Science 46, 3129–3138. Ponce de Leon, A.C., Atkinson, A.C., 1991. Optimum experimental design for discriminating between two rival models in the presence of prior information. Biometrika 78, 601–608. Pritchard, D.J., Bacon, D.W., 1978. Prospects for reducing correlations among parameter estimates in kinetic models. Chemical Engineering Science 33, 1539–1543. Schwaab, M., Biscaia Jr., E.C., Monteiro, J.L, Pinto, J.C., 2008a. Nonlinear parameter estimation through particle swarm optimization. Chemical Engineering Science 63, 1542–1552. Schwaab, M., Lemos, L.P., Pinto, J.C., 2008b. Optimum reference temperature for reparameterization of the Arrhenius equation. Part 2: Problems involving multiple reparameterizations. Chemical Engineering Science 63, 2895–2906. Schwaab, M., Monteiro, J.L, Pinto, J.C., 2008c. Sequential experimental design for model discrimination. Taking into account the posterior covariance matrix of differences between model predictions. Chemical Engineering Science 63, 2408–2419. Schwaab, M., Pinto, J.C., 2007. Optimum reference temperature for reparameterization of the Arrhenius equation. Part 1: Problems involving one kinetic constant. Chemical Engineering Science 62, 2750–2764. Schwaab, M., Pinto, J.C., 2008. Optimum reparameterization of power function models. Chemical Engineering Science 63, 4631–4635.
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A.L. Alberton et al. / Chemical Engineering Science 65 (2010) 5482–5494
Schwaab, M., Queipo, C.Q., Silva, F.M., Barreto Jr., A.G., Nele, M., Pinto, J.C., 2006. A new approach for sequential experimental design for model discrimination. Chemical Engineering Science 61, 5791–5806. Srinivas, N., Deb, K., 1994. Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation 2, 221–248. Tommasi, C., 2009. Optimal designs for both model discrimination and parameter estimation. Journal of Statistical Planning and Inference 139, 4123–4132.
Vanrolleghem, P.A., Van Daele, M., Dochain, D., 1995. Practical identifiability of a biokinetic model of activated sludge respiration. Water Research 29, 2561–2570. Versyck, K.J., Claes, J.E., Van Impe, J.F., 1997. Practical identification of unstructured growth kinetics by application of optimal experimental design. Biotechnology Progress 13, 524–531. Zitzler, E., Deb, K., Thiele, L., 2000. Comparison of multiobjective evolutionary algorithms: empirical results. Evolutionary Computation 8, 173–195.
