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Designing experiments to differentiate between adsorption isotherms using T-optimal designs
Journal of Food Engineering 101 (2010) 386–393
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Designing experiments to differentiate between adsorption isotherms using T-optimal designs Stuart H. Munson-McGee *, Aravind Mannarswamy, Paul K. Andersen Chemical Engineering, New Mexico State University, Las Cruces, NM 88003, USA
a r t i c l e
i n f o
Article history: Received 24 March 2010 Received in revised form 13 July 2010 Accepted 23 July 2010 Available online 27 July 2010 Keywords: Adsorption Experimental design Optimal design T-optimal design
a b s t r a c t T-optimal experimental designs are developed to distinguish between the 3-parameter Guggenheim– Anderson–deBoer (GAB) and 2-parameter Brunauer–Emmett–Teller (BET) adsorption isotherms. The results show that the designs are not dependent on one of the GAB model parameters (the monolayer capacity, which is a linear scaling factor), but are dependent on the other two, which determine the curvature and shape of the isotherms. Further analysis of the designs show that for the range of model parameters considered, misspecification of the model parameters can reduce the efficiency of the experimental design to 30% or less: proper selection of the model parameters at which the design is developed within the model-parameter space can increase the maximum–minimum design efficiency to nearly 55%. The experimental replication inherent in T-optimal designs is then used in combination with a statistical lack-of-fit analysis to illustrate the role of experimental variability on the ability to distinguish between the two isotherm models. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Moisture adsorption is a common concern with respect to the storage, spoilage, and properties of foods. Reviewing a number of adsorption studies in the food literature (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003; Goula et al., 2008; Syamaladevi et al., 2009; Ikhu-Omoregbe, 2006; Viollaz and Rovedo, 1999; Rahman and Al-Belushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007) we found that most such studies consist of three components: 1. Measurement of the weight gain (adsorption) or loss (desorption) of a specific food as a function of the water activity at several temperatures, 2. Fitting of multiple sorption isotherms to the experimental data, and 3. Selection of the isotherm which best represents the data. The main interest in these studies is to identify the isotherm which best fits the experimental data. Typically in these studies, 10–15 levels of the water activity and 1–4 levels of temperature (the independent factors) are selected. One experiment is con* Corresponding author. Tel.: +1 646 6439. E-mail address: [email protected] (S.H. Munson-McGee). 0260-8774/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2010.07.024
ducted at each combination of water activity and temperature. Occasionally, repeated measurements are conducted and the average of these is used in the subsequent analysis. However, the use of an average value decreases the information that can be obtained during the regression curve-fitting analysis and should be discouraged. Several sorption isotherms are then fitted to the multiple isothermal data using a non-linear, sum-of-squares regression technique to determine the model parameters for each isotherm at each of the temperatures studied. The quality of the fit is indicated by the correlation coefficient, R2. This approach is severely limited in the experimental design aspect in that it fails to consider experimental variability and it also does not consider the use of modern optimal design techniques to select the levels of water activity at which the measurements should be made. These oversights thereby decrease the reliability of the conclusions and increase the experimental effort required to reach those conclusions. In this work, we propose and explore the use of T-optimal designs to overcome these limitations. Specifically, we consider using T-optimal designs to distinguish between two isotherm models. The attraction of T-optimal designs is that they not only identify the most advantageous levels of the independent factors at which the experiments should be conducted, they also incorporate experimental replication to allow experimental variability to be estimated. In this work, we develop T-optimal designs and analyze the results to identify which isotherm model parameters are important in determining the T-optimal design. We then develop the T-optimal designs for a wide range of these model parameters typical of the values seen in recent experimental studies. Next, we
S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393
examine the effects of model-parameter misspecification and model-parameter uncertainty on the efficiencies of the designs. Finally, we explore the role of experimental variability in terms of lack-of-fit to differentiate between isotherm models, looking in turn at the number of experiments required to distinguish between the models at a known level of variability, then at the level of variability permissible if the number of experiments is limited. The two isotherms that we consider here, for the purpose of explicit illustration of the techniques and results, are the 3-parameter Guggenheim–Anderson–de Boer (GAB) and the 2-parameter Brunauer–Emmett–Teller (BET) isotherms given, respectively, by
mmG cG ka ð1 kaÞ½1 þ ðcG 1Þka mmB cB a mcB ¼ ð1 aÞ½1 þ ðcB 1Þa
mcG ¼
ð1Þ ð2Þ
In these models, mc is the amount of absorbed material (grams water per gram dry solid), mm is the monolayer capacity (grams water per gram dry solid), c is related to the enthalpy of adsorption (dimensionless), k is a measure of the free energy of enthalpy of the sorbate molecules (dimensionless), and a is the water activity (dimensionless). The subscripts G and B refer to the GAB and BET models, respectively. In terms of the notation for T-optimal designs used later, the model-parameter vectors for the GAB and BET models are bG = (mmG, cG, k)T and bB = (mmB, cB)T, respectively, where the superscript T indicates the transpose of the vector. Note that the BET model can be obtained from the GAB model by setting k = 1. By assuming a priori that these models may be appropriate, the sorption isotherm for materials such as salted products that have a discontinuity should not be modeled using these expressions. However, once other appropriate sorption isotherms have been selected, the T-optimal design procedure discussed here would still be appropriate for distinguishing between them. 2. T-optimal designs Optimal design theory has been developed to provide experimentalists methods to minimize the amount of experimental work necessary to achieve a desired objective (Atkinson and Donev, 1992). By design, we explicitly mean the design measure, e, which includes the points (i.e. the levels of the independent factors x) at which the experiment is to be conducted and the fraction of experiments to be conducted at each point (i.e. the experimental weights w). Like all experimental designs, optimal designs require that the experimentalist explicitly decide on the objective of the experiment before selecting the design that is appropriate for that objective. Several optimal designs can be used when the objective involves estimating the parameters of an a priori chosen model. Examples of such optimal designs include the following: A-optimal designs minimize the trace of the inverse of the information matrix and are used to minimize the sum of the variances of the model parameters. C-optimal designs are used to minimize selected functions of the model parameters and frequently are used to estimate a few model parameters very accurately at the expense of estimation accuracy of other parameters. Dh-optimal designs minimize the determinant of the inverse of the information matrix and provide the smallest volume of the joint confidence ellipsoid of the parameters, i.e. they provide the best simultaneous estimate of the model parameters. Dh-optimal designs are a subset of D-optimal designs; they are useful when only a subset of the model parameters is of interest.
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E-optimal designs maximize the minimum eigenvalue of the information matrix and minimize the maximum variance of the estimated model parameters. Other optimal designs focus on the variance of the predicted values. Examples of these optimal designs include the following: G-optimal designs minimize the maximum value along the diagonal of the hat matrix and are used when the minimum variance in the predicted values is desired. I-optimal designs minimize the mean squared prediction error and are likewise useful when the minimum variance on the predicted values is desired. V-optimal designs minimize the average prediction variance. Herein we are interested in developing and exploring T-optimal designs for the GAB and BET adsorption isotherms. T-optimal designs are used to discriminate between two models that have been selected a priori. Theoretical development of T-optimal designs for discrimination between two models has been considered (Dette and Titoff, 2009; Ponce de Leon and Atkinson, 1991; Otsu, 2008; Atkinson and Fedorov, 1975; Wiens, 2009), as has the theoretical development of T-optimal designs for discrimination between several models (Atkinson and Fedorov, 1975), the use of T-optimal designs in compound designs with D-optimal designs (Atkinson, 2008; McGree et al., 2008), as well as a few practical applications of T-optimal designs (Atkinson et al., 1998; Rodriguez-Aragon and Lopez-Fidalgo, 2007; Lopez-Fidalgo et al., 2008). The basic concept of a T-optimal design is one in which the experiments are conducted at the points at which the two isotherms under consideration differ the most. Then, if the models are sufficiently different at these points compared to the measured experimental variability, one model can be selected as being more appropriate for the material. If there is not a statistically significant difference, then the models are equally appropriate and either can be used to represent the data. T-optimal design begins by first defining the two models of interest. One of these models is selected as the ‘‘true” model against which the other model can be tested for a statistically significant lack-of-fit. It is common practice to select the model having the greater number of parameters as the true model; hence, we select the GAB model and the model parameters associated with it as the true model, and develop the design to test the BET model for lack-of-fit. T-optimal designs are those for which the T-optimal statistic has been maximized. The T-optimal statistic is given as
T GB ¼ minbB
n X
^B;i y ^G;i Þ2 wi ðy
ð3Þ
i¼1
where wi is the fraction of experiments conducted at the ith exper^B;i and y ^G;i are the values predicted by the BET imental point and y and GAB models, respectively, at the ith experimental point. The T-optimal statistic is computed by selecting the BET model parameters bB = (mmB, cB)T, so the resulting BET model is the best-fit in a minimum sum-of-squares sense to a specific GAB model. We developed a different approach than previous researchers (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003; Goula et al., 2008; Syamaladevi et al., 2009; Ikhu-Omoregbe, 2006; Viollaz and Rovedo, 1999; Rahman and AlBelushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007) to find the maximum of the T-optimal statistic. Our approach, implemented in Matlab (MathWorks, 2007), followed the steps below:
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1. Specify the parameters in the GAB model bG = (mmG, cG, k)T. 2. Use an algorithm designed to optimize non-linear sum-ofsquares problems (LSQNONLIN in Matlab) to determine the initial estimate of the best-fit parameters for the BET model. 3. Use a multivariate optimization algorithm (FMINCON in Matlab) to determine the design measure that maximizes the Toptimal statistic. Within this function call, the BET model parameters are re-optimized within each iteration. Once a design was developed, the generalized equivalence theorem was applied to confirm that a T-optimal design had been obtained. According to the generalized equivalence (Atkinson and Donev, 1992) theorem, a T-optimal design must satisfy the following inequality at all points within the experimental region:
W ¼ ½y^B ðaÞ y^G ðaÞ2
n X
^B;i y ^G;i Þ2 6 0 wi ðy
ð4Þ
i¼1
^B ðaÞ and y ^G ðaÞ are the values predicted by the BET and GAB where y isotherms, respectively, at point a in the experimental space. For a T-optimal design, traditionally denoted as eT , the equality will be satisfied at the points at which the experiment is to be conducted; these points are known as the support points of the design. The designs and experimental regions considered here are one-dimensional and bounded only at the lower and upper limits. For such a design the number of support points equals the number of parameters in the model having the largest number of parameters.
Since we assumed that the GAB model is the true model and that we i is given are testing for lack-of-fit of the BET model, it follows that y ^B;i ) and that y ^i is given by the GAB preby the BET predicted value (y ^G;i ). The F statistic for lack-of-fit then becomes dicted value (y
F LOF ¼
m X
1 ðm pÞS
2
^B;i y ^G;i Þ2 ni ðy
ð11Þ
i¼1
From the T-optimal design results we get, among other information, the fraction of experiments done at each design point, wi. The number of experiments done at each point is
ni ¼ RNDðnwi Þ
ð12Þ
where RND(x) indicates that x has been rounded to the nearest integer. Our expression for the FLOF statistic for lack-of-fit finally becomes
F LOF ¼
m X
1 ðm pÞS
2
^B;i y ^G;i Þ2 RNDðnwi Þni ðy
ð13Þ
i¼1
The criteria for significant lack-of-fit at the (1 a/2) confidence level is given by
F LOF P F ð1a2Þ;ðmpÞ;ðNmÞ
ð14Þ
where
N¼
m X
RNDðnwi Þ
ð15Þ
i¼1
3. Lack-of-fit analysis The standard definition of the statistic for lack-of-fit analysis is given by Montgomery (2008); Bates and Watts (2007)
SSLOF =ðm pÞ ¼ SSPE =ðn mÞ
F LOF
ð5Þ
where m is the number of design points at which replicates are taken, n the total number of experimental points, and p the number of parameters in the model. The lack-of-fit sum-of-squares and pureerror sum-of-squares are given by
SSLOF
m X
i y^i Þ2 ni ðy
ð6Þ
Some rearranging gives the following criteria for determining the number of experiments that must be conducted to determine a significant lack-of-fit. m X
h i ^B;i y ^G;i Þ2 P ½ðm pÞS2 F ð1aÞ;ðmpÞ;ðNmÞ RNDðnwi Þðy 2
In the GAB isotherm, the model-parameter mmG is a scaling parameter. We can remove its influence in the lack-of-fit analysis by dividing both sides of the previous inequality by m2mG . The result is m X
RNDðnwi Þ
i¼1
2 h i ^G;i 2 ^B;i y S y P ðm pÞ F ð1a2Þ;ðmpÞ;ðNmÞ mmG mmG ð17Þ
i¼1
SSPE
m X
ni X
i¼1
j¼1
ðyij yi Þ2
2
ð7Þ
where ni is the number of replicates at the ith design point (which i is the will be proportional to the T-optimal design weights, wi), y ^i is the predicted average of the replicates at the ith design point, y value at the ith design point, and yij is the jth experimental value at the ith design point. We now introduce the sample variance at the ith design point, given by
S2i ¼
ni X i Þ2 ðyij y ni 1 j¼1
ð8Þ
Substituting this expression into the definition of the sum-ofsquares pure-error above yields
SSPE ¼
m X
S2i ðni 1Þ
ð9Þ
i¼1
If the sample variance is assumed to be constant over the experimental range (i.e. the experimental variation is the same at each point at which the experiment was conducted) and is given by S2, the above expression becomes
SSPE ¼ S2 ðn mÞ
ð16Þ
i¼1
ð10Þ
As we explain more fully below, the quantity [S/mmG] can be considered a measure of experimental variance which can be estimated from the experimental data provided replicate experiments have been made. 4. Results To determine the range of model parameters measured experimentally for foodstuffs, we reviewed GAB adsorption isotherms published over the last decade. This review included grains and seeds (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003), dried fruits and berries (Goula et al., 2008; Syamaladevi et al., 2009), tubers (IkhuOmoregbe, 2006; Viollaz and Rovedo, 1999), and other foodstuffs (Rahman and Al-Belushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007). Based on this review, we selected the following ranges of the model parameters as encompassing most of the reported values: 1 6 b1 6 20,1 6 b2 6 15,0. 60 6 b3 6 0.99. However, there were some materials for which the model parameters fell well outside these ranges, including rice (Togrul and Arslan, 2006), maize flour (Oyelade et al., 2008), walnut kernels (Togrul
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S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393 Table 1 Computed best-fit BET model parameters and design measures for the 23 factorial screening study. Model
GAB isotherm model parameters bG = (mm, c, k)T
BET isotherm model parameters bG = (mm, c)T
Design measure
1 5
bG1 = (1, 1, 0.60)T bG5 = (20, 1, 0.60)T
bB1 = (0.2036, 15.08)T bB5 = (4.072, 15.08)T
eT1 ¼ eT5 ¼
2 6
bG2 = (1, 15, 0.60)T bG6 = (20, 15, 0.60)T
bB2 = (0.4334, 3714)T bB6 = (8.668, 3714)T
eT2 ¼ eT6 ¼
3 7
bG3 = (1, 1, 0.99)T bG7 = (20, 1, 0.99)T
bB3 = (0.9349, 1.111)T bB7 = (18.70, 1.111)T
eT3 ¼ eT7 ¼
4 8
bG4 = (1, 15, 0.99)T bG8 = (20, 15, 0.99)T
bB4 = (0.9645, 17.88)T bB8 = (19.29, 17.88)T
eT4 ¼ eT8 ¼
e¼
a1 w1
a2 w2
a3 w3
0:1120 0:1278
0:6367 0:5792
0:8000 0:2931
0:0035 0:0050
0:4627 0:7258
0:8000 0:2692
0:3108 0:3680
0:7148 0:4140
0:8000 0:2180
0:0428 0:1230
0:6062 0:5936
0:8000 0:2834
and Arslan, 2007), freeze dried blueberries (Syamaladevi et al., 2009), freeze dried plantain (Johnson and Brennan, 2000), sorghum malt (Aviara et al., 2006), morel mushrooms (Mulet et al., 2002), cocoa beans (Sandoval and Barreiro, 2002) and yam chips (Sobukola et al., 2007). The range of model parameters selected above is slightly different from that recommended by Lewicki (1997), Blahovec (2004) based on theoretical considerations. Lewicki’s recommendations were 5.67 6 b2 6 1 and 0.24 6 b3 6 1 while those of Blahovec were 0 6 b2 6 2 for type II isotherms and 2 6 b2 6 1 for type III isotherms and 0 6 b3 6 1 for both isotherm types. However, the ranges used here were selected based on experimental results obtained for a wide range of foodstuffs and represent much of that data.
nificantly influenced the T-optimal design. To do this, a 23 factorial design (Montgomery, 2008) was executed using the eight possible combinations of the extreme values of the three parameters. The results of these calculations are given in Table 1. From the results in Table 1, the following observations can be made:
4.1. 23 Factorial design
The observation that the parameter b1 does not affect the design is not surprising since it is a scaling parameter that does not affect the curvature of the predicted response. The other two parameters, on the other hand, do affect the response curvature and therefore
Having selected a suitable range of model parameters to examine, the next task was to determine which of these parameters sig-
1. The designs for models 1 and 5, models 2 and 6, models 3 and 7, and models 4 and 8 are identical. From this we conclude that the parameter mmG (b1) does not affect the design but the other two parameters do. 2. The design measures for all models consist of one point for a1 < 0.35, one point 0.45 < a2 < 0.75, and one point at a3 = 0.8000, the maximum of the experimental space.
Fig. 1. Comparison of the best-fit BET isotherm to the GAB isotherm for models 1–4 (Figs. 1A–1D, respectively) given in Table 1 (note that in the bottom two figures the models are indistinguishable at this scale). The vertical lines are the levels of the independent factor which satisfy the T-optimal criteria.
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their importance in determining the T-optimal design is expected. Examining the results in Table 1, we notice that increasing b2 decreases a1, a2, w1, and w3 but increases the weight w2. Increasing b3 increases a1, a2, and w1 but decreases w2. The effect of the model parameters on the weight w3 is less straightforward: w3 decreases at low levels of b2 but increases at high levels of b2 indicating that there is a significant interaction between these two parameters in determining the value of w3. Shown in Fig. 1 are plots of the best-fit BET model for the first four GAB models used in the factorial design study along with the T-optimal design experimental locations. From this figure it is apparent that the locations of the T-optimal experiment are those at which there is the greatest difference between the GAB isotherm and the BET isotherm. The fact that these are T-optimal designs is shown in Fig. 2 where the scaled normalized variance has been plotted and is less than zero over the entire experimental space, thus satisfying the generalized equivalence theorem. 4.2. T-optimal designs for the model-parameter space Based on the results above, the model-parameter mmG (i.e. b1) was given no additional consideration; it was assigned the value b1 = 1. T-optimal designs were developed for the entire modelparameter space given by b1 = 1, 1 6 b2 6 15, 0.60 6 b3 6 0.99, and these results are given in Figs. 3 and 4. These results confirm the earlier observations about the effects of the model parameters on the design measure. Increasing b2 will decrease a1, a2, and w1 but will increase w2. Increasing b3 will increase a1, a2, and w1 but will decrease w2. The effect of changing the model parameters on the weight w3 is more complex in that a maximum occurs in the response surface. 4.3. Effect of parameter misspecification The third task in this investigation was to examine the effect of parameter misspecification on the design efficiency. By parameter misspecification, we mean using GAB model parameters that are not well known when developing the design. The reciprocal of the design efficiency is approximately the number of times the non-optimal design would have to be replicated to have the same ability to distinguish between the isotherms as does the optimal design. The design efficiency n is given by
n¼
T GB T GB
ð18Þ
where TGB is the T-optimal statistic for the non-optimal design and T GB is the T-optimal statistic for the optimal design. We examined the effect of parameter misspecification as follows:
Fig. 3. Contour plots of the components a1 (Fig. 3A) and a2 (Fig. 3B) of the T-optimal designs for the range of GAB model parameters b1 = 1, 1 6 b2 6 15,0.60 6 b3 6 0.99. For all levels of the model parameters a3 = 0.8000.
(1) Set (b1, b2, b3) = (1, 8, 0.8), approximately the center of the model-parameter space. (2) Determine the T-optimal design measure for this parameter combination. (3) Calculate the design efficiencies for the remainder of the model space using this design measure. The results of these calculations are presented in Fig. 5. By selecting the center of the model-parameter space, the design efficiency was 90% or more over approximately the half of the space corresponding to high values of b2 and low values of b3. However, the efficiency dropped quickly within the remainder of the space to slightly less than 30%. 4.4. Effect of parameter uncertainty When the precise values of the model parameters are unknown, the question of which design to use must still be addressed. In the current work, we consider this by finding the design that maximizes the minimum design efficiency within the space of possible model parameters. This was accomplished by the following procedure: (1) Select all possible pairwise combinations of parameters from b2 = {1, 2, . . . , 14, 15} and b3 = {0.60, 0.61, 0.62, . . . , 0.98, 0.99}. (2) Determine the T-optimal design for each pairwise combination. (3) Calculate the efficiency at the four corners of the model parameter space for each pairwise combination. (4) Select the smallest calculated efficiency as the minimum efficiency for this pairwise combination.
Fig. 2. General equivalence theorem for the models examined for the 23 factorial screening study.
The efficiencies for each combination of model parameters were calculated only at the corners of the space since the response
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surface was convex down and therefore the smallest efficiency had to occur at one of the corners. This significantly reduced the computational burden compared to either computing the efficiency at all points within the space or finding the smallest value numerically using a multifactor optimization approach. The results of these calculations are shown in Fig. 6. The maximum–minimum efficiency is at (b2, b3) (2.0, 0.82); we call this the T-optimal max–min design point. At the T-optimal max–min design point for the range of model parameters shown in the figure, the max–min efficiency has a value slightly greater than 55%. This means using the T-optimal design generated for this parameter combination that, regardless of the true values of the model parameters, the resulting design efficiency would be 55% or greater. At this point the design measure is given by
e¼
0:115 0:642 0:800 0:150 0:564 0:286
Of course, should the possible range of model parameters change, the T-optimal max–min design point may change as may the max–min efficiency. The smaller the range of model parameters, the larger the max–min efficiency should be. 4.5. Lack-of-fit analysis
Fig. 4. Contour plots of the design measure weightings w1, w2, and w3 (Figs. 4A–4C, respectively) for the T-optimal designs for the range of GAB model parametersb1 = 1,1 6 b2 6 15,0.60 6 b3 6 0.99.
Fig. 5. Design efficiency when values of the model parameters are incorrectly specified. The GAB model parameters were assumed to be bG(1, 8, 0.80)T and the plots show the affect of this choice on the T-optimal efficiency when the true model parameters are bG = (1, b2, b3)T.
The final objective of this project was to use the lack-of-fit analysis developed earlier to determine the number of experiments required to distinguish between the GAB and BET isotherms. This was done for the first four models given in Table 1. Before discussing the results from these calculations, a short discussion of the meaning of [S/mmG]2 is warranted. This term is a measure of the repeatability of the experimental measurements and a measure of the experimental variability. This variability may be due to a number of causes including variation in the materials, experimental apparatus variability, and variability of the experimentalist. In this sense, it should not be considered as an ‘‘error”, i.e. a mistake that can be corrected. Some of this variability might be reducible by using better experimental techniques or more accurate instrumentation. However, a portion of this variability will always exist and cannot be eliminated regardless of the care or expertise of the experimentalist. Since S is the sample standard deviation based on repeated measurements, the term [S/mmG] approximates a scaled standard deviation. It should be noted, however, that this scaling does not include the effects of the other model parameters. As illustrated in Fig. 1 where all the models had the same value of mmG the range of the response varied by a factor of almost 5; this may affect the magnitude of the standard deviation of the experiment. The results of the lack-of-fit calculations at the 95% confidence level (i.e. a = 0.05) are given in Fig. 7 where the left side of Eq. (17) was plotted for models 1–4 from Table 1 while the right side of Eq.
Fig. 6. Minimum design efficiencies of models 1–4 in Table 1 when the T-optimal design is computed using b1 = 1 and the values of b2 and b3 as given in the figure.
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come more similar, the acceptable level of experimental variability decreases. For model 4 and 3, the variability drops to 0.57 and 0.015, respectively, for designs consisting of only 20 points. Similar trends are seen in the maximum variability possible if the isotherms are to be distinguished using an infinite number of experiments. This knowledge can be used to establish, once the experimental variability of a particular experimental set-up and experimental protocol has been established, the whether or not two isotherms could be distinguished and whether or not it is worth the effort to attempt such an undertaking or if improvements needed to be made in the experiment to reduce the variability prior to conducting the study.
Fig. 7. Lack-of-fit criteria for 4 of the models used for the 23 factorial study and given in Table 1. The solid lines are the left-hand side of the lack-of-fit criteria for the indicated models and the dashed lines are the right-hand side of the lack-of-fit criteria for the indicated levels of [S/mmG].
(17) was plotted for five levels of the parameter [S/mmG]. Since reports on the sample variation for repeated measurements are rarely given in the literature, a wide range of values were examined. For model 2 given in Table 1, which shows the greatest difference between the two isotherm models, even for a highly variably experiment (i.e. for [S/mmG] = 0.50), the two isotherms can be differentiated using 10–20 experiments. At this level of variability it would be impossible to differentiate between the isotherms for models 1, 3, and 4 even with an unlimited number of experiments. To differentiate between the isotherms for model 1 with a decreased variability of [S/mmG] = 0.20 would require between 30 and 40 experiments or between 10 and 20 experiments if the variability was decreased to [S/mmG] = 0.10. For model 4, the isotherms could be differentiated with an experimental variability of [S/mmG] = 0.10 using between 50 and 60 experiments or with an experimental variability of [S/mmG] = 0.05 and 10–20 experiments. Similar results are obtained for model 3 if the experimental variability can be reduced to [S/mmG] = 0.02 (30–40 experiments) or [S/mmG] = 0.01 (10–20 experiments). The second portion of the lack-of-fit analysis was to examine the required level of experimental variability necessary to distinguish between the isotherms for the various models using either approximately 20 or 100 points. Twenty points was selected as a reasonable number of points and 100 points was selected to approximate an infinite number of points, i.e. the maximum possible experimental variability that would still allow differentiation between the isotherms. These results are only for approximately the specified number of points since the rounding operation specified by Eq. (15) may change that value by plus or minus one. The results of these calculations are given in Table 2 for each of the first four models given in Table 1. As expected for the models for which the isotherms were very different, the levels of experimental variability that can be tolerated and still distinguish between the isotherms is relatively high. For models 2 and 1, the minimum variability for distinguishing between the two isotherms using only 20 points is 0.68 and 0.15, respectively. As the isotherms beTable 2 Maximum permissible experimental variability measured by (S/mmG) to differentiate between the GAB and BET isotherms at a = 0.05 using approximately 20 or 100 T-optimal designed experiments.
5. Conclusions In this work we have shown that T-optimal designs can be developed that will efficiently distinguish between the GAB and BET isotherm models for adsorption. The process is general enough that it could be applied to any a priori selected pair of isotherm models. The specific examples here used a range of model parameters reflective of moisture adsorption in foods. It was shown that the monolayer coverage did not play a role in determining the experimental design but that the other two parameters in the GAB model did. The role of these parameters was fully explored and the ranges of the design measures were determined. The designs placed the experimental points at the locations where the GAB and BET models differed the most; one of the three support points was always for a water activity less than about 0.35, one in the middle of the space between 0.45 and 0.75, and the third was always at 0.80, the upper limit of the independent factor space. The design weightings were such that most of the experiments were always in the center of the independent factor space where the 45% and 75% of the experiments would be conducted. The fraction of experiments conducted at the lowest level of water activity was 35% or less, while the fraction conducted at the highest level was between 20% and 30%. Only when the GAB and BET models were very similar were an approximately equal number of experiments conducted at each of the three support points. The effect of model-parameter misspecification was studied and it was shown that by assuming incorrect values of the model parameters, the design efficiency could be reduced to 30% or less. However, by judiciously selecting the model parameters for which the design was developed and using a maximum–minimum efficiency criteria at the corners of the model-parameter space, the maximum–minimum efficiency could be increased to 55%. Finally, it was shown how experimental variability affected the number of experiments necessary to distinguish between the isotherms. In addition to illustrating how the number of experiments required was a function of the experimental variability, it was also shown what level of variability was permissible to distinguish between the isotherms for a specified number of experiments. This work has addressed observed shortcomings in many adsorption studies: the lack of experimental replication and the lack of efficient designs to minimize experimental effort while maximizing the information content of the experimental results. T-optimal designs, when developed appropriately, can be useful to alleviate these shortcomings and should lead to more reliable and statistically justifiable conclusions about isotherm model selection.
Model
GAB isotherm model parameters bG = (mm, c, k)T
[S/mmG]n 20
[S/mmG]n 100
References
1 2 3 4
bG1 = (1, 1, 0.60)T bG2 = (1, 15, 0.60)T bG3 = (1, 1, 0.99)T bG4 = (1, 15, 0.99)T
0.15 0.68 0.015 0.057
0.34 1.6 0.037 0.14
Arslan, N., Togrul, H., 2006. The fitting of various models to water sorption isotherms of tea stored in a chamber under controlled temperature and humidity. Journal of Stored Products Research 42, 112–135. Atkinson, A.C., 2008. DT-optimum designs for model discrimination and parameter estimation. Journal of Statistical Planning and Inference 138, 56–64.
S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393 Atkinson, A.C., Donev, A.N., 1992. Optimal Experimental Designs. Oxford Science Publications, Oxford University Press, Inc., New York, NY. Atkinson, A.C., Fedorov, V.V., 1975. The design of experiments for discrimination between two rival models. Biometrika 62 (1), 57–70. Atkinson, A.C., Fedorov, V.V., 1975. Optimal design: experiments for discrimination between several models. Biometrika 62 (2), 289–303. Atkinson, A.C., Bogacka, B., Bogacki, M.B., 1998. D- and T-optimum designs for the kinetics of a reversible chemical reaction. Chemometrics and Intelligent Laboratory Systems 43, 185–198. Aviara, N.A., Ajibola, O.O., Aregbesola, O.A., et al., 2006. Moisture sorption isotherms of sorghum malt at 40 and 50 °C. Journal of Stored Products Research 42, 290– 301. Bates, D.M., Watts, D.G., 2007. Nonlinear Regression Analysis and Its Applications. John Wiley and Sons, New York, NY. Blahovec, J., 2004. Sorption isotherms in materials of biological origin mathematical and physical approach. Journal of Food Engineering 65, 489–495. Chen, C.C., 2003. Moisture sorption isotherms of pea seeds. Journal of Food Engineering 58 (1), 45–51. Dette, H., Titoff, S., 2009. Optimal discrimination designs. Annals of Statistics 37 (4), 2056–2082. Goula, A.M., Karapantsios, T.D., Achilias, D.S., Adamopoulos, K.G., 2008. Water sorption isotherms and glass transition temperature of spray dried tomato pulp. Journal of Food Engineering 85 (1), 73–83. Ikhu-Omoregbe, D.I.O., 2006. Comparison of the sorption isotherm characteristics of two cassava products. International Journal of Food Properties 9, 167–177. Johnson, P.N.T., Brennan, J.G., 2000. Moisture sorption isotherm characteristics of plantain (Musa, AAB). Journal of Food Engineering 44 (2), 79–84. Kaymak-Ertekin, F., Sultanoglu, M., 2001. Moisture sorption isotherm characteristics of peppers. Journal of Food Engineering 47 (3), 225–231. Lewicki, P.P., 1997. The applicability of the GAB model to food water sorption isotherms. International Journal of Food Science and Technology 32, 553–557. Lopez-Fidalgo, J., Tommasi, C., Trandafir, P.C., 2008. Optimal designs for discriminating between some extensions of the Michaelis–Menten model. Journal of Statistical Planning and Inference 138, 3797–3804. The MathWorks, Inc., 2007. Matlab, version 7, 3 Apple Hill Drive, Natick, MA 017602098, USA. McGree, J.M., Eccleston, J.A., Duffull, S.B., 2008. Compound optimal design criteria for nonlinear models. Journal of Biopharaceutical Statistics 18, 646–661. Mohamad Saad, M., Gaiani, C., Scher, J., Cuq, B., Ehrhardt, J.J., Desobry, S., 2009. Impact of re-grinding on hydration properties and surface composition of wheat flour. Journal of Cereal Science 49, 134–140. Montgomery, D.C., 2008. Design and Analysis of Experiments, seventh ed. John Wiley and Sons, New York, NY.
393
Mulet, A., Garcia-Pascual, P., Sanjuan, N., Garcia-Reverter, J., 2002. Equilibrium isotherms and isosteric heats of morel (Morchella esculenta). Journal of Food Engineering 53 (1), 75–81. Otsu, T., 2008. Optimal experimental design criterion for discriminating semiparametric models. Journal of Statistical Planning and Inference 138, 4141–4150. Oyelade, O.J., Tunde-Akintunde, T.Y., Igbeka, J.C., et al., 2008. Modelling moisture sorption isotherms for maize flour. Journal of Stored Products Research 44, 179– 185. Pagano, A.M., Mascheroni, R.H., 2003. Water sorption of Amaranthus cruentus L. seeds modelled by GAB equation. International Journal of Food Properties 6 (3), 369–391. Peng, G.L., Chen, X.G., Wu, W.F., Jiang, X.J., 2007. Modeling of water sorption isotherm for corn starch. Journal of Food Engineering 80 (2), 562–567. Ponce de Leon, A.C., Atkinson, A.C., 1991. Optimum experimental design for discrimination between two rival models in the presence of prior information. Biometrika 78 (3), 601–608. Rahman, M.S., Al-Belushi, R.H., 2006. Dynamic isopiestic method (DIM): measuring moisture sorption isotherm of freeze-dried garlic powder and other potential uses of DIM. International Journal of Food Properties 9, 421–437. Resio, A.C., Aguerre, R.J., Suarez, C., 1999. Analysis of the sorptional characteristics of amaranth starch. Journal of Food Engineering 42 (1), 51–57. Rodriguez-Aragon, L.J., Lopez-Fidalgo, J., 2007. T-, D-, and C-optimum designs for BET and GAB isotherms. Chemometrics and Intelligent Laboratory Systems 89 (1), 36–44. Sandoval, A.J., Barreiro, J.A., 2002. Water sorption isotherms of non-fermented cocoa beans (Theobroma cacao). Journal of Food Engineering 51 (2), 119–123. Sobukola, O.P., Dairo, O.U., Afe, T.T., Coker, O.J., 2007. Water sorption isotherms and crispness of fried yam chips in the temperature range from 293 K to 313 K. International Journal of Food Properties 10, 561–575. Syamaladevi, R.M., Sablani, S.S., Tang, J.M., Powers, J., Swanson, B.G., 2009. State diagram and water adsorption isotherm of raspberry (Rubus idaeus). Journal of Food Engineering 91 (3), 460–467. Togrul, H., Arslan, N., 2006. Moisture sorption behaviour and thermodynamic characteristics of rice stored in a chamber under controlled humidity. Biosystems Engineering 95, 181–195. Togrul, H., Arslan, N., 2007. Moisture sorption isotherms and thermodynamic properties of walnut kernels. Journal of Stored Products Research 43, 252–264. Viollaz, P.E., Rovedo, C.O., 1999. Equilibrium sorption isotherms and thermodynamic properties of starch and gluten. Journal of Food Engineering 40 (4), 287–292. Wiens, D.P., 2009. Robust discrimination designs. Journal of the Royal Statistical Society, B 71 (4), 805–829.
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Designing experiments to differentiate between adsorption isotherms using T-optimal designs Stuart H. Munson-McGee *, Aravind Mannarswamy, Paul K. Andersen Chemical Engineering, New Mexico State University, Las Cruces, NM 88003, USA
a r t i c l e
i n f o
Article history: Received 24 March 2010 Received in revised form 13 July 2010 Accepted 23 July 2010 Available online 27 July 2010 Keywords: Adsorption Experimental design Optimal design T-optimal design
a b s t r a c t T-optimal experimental designs are developed to distinguish between the 3-parameter Guggenheim– Anderson–deBoer (GAB) and 2-parameter Brunauer–Emmett–Teller (BET) adsorption isotherms. The results show that the designs are not dependent on one of the GAB model parameters (the monolayer capacity, which is a linear scaling factor), but are dependent on the other two, which determine the curvature and shape of the isotherms. Further analysis of the designs show that for the range of model parameters considered, misspecification of the model parameters can reduce the efficiency of the experimental design to 30% or less: proper selection of the model parameters at which the design is developed within the model-parameter space can increase the maximum–minimum design efficiency to nearly 55%. The experimental replication inherent in T-optimal designs is then used in combination with a statistical lack-of-fit analysis to illustrate the role of experimental variability on the ability to distinguish between the two isotherm models. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Moisture adsorption is a common concern with respect to the storage, spoilage, and properties of foods. Reviewing a number of adsorption studies in the food literature (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003; Goula et al., 2008; Syamaladevi et al., 2009; Ikhu-Omoregbe, 2006; Viollaz and Rovedo, 1999; Rahman and Al-Belushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007) we found that most such studies consist of three components: 1. Measurement of the weight gain (adsorption) or loss (desorption) of a specific food as a function of the water activity at several temperatures, 2. Fitting of multiple sorption isotherms to the experimental data, and 3. Selection of the isotherm which best represents the data. The main interest in these studies is to identify the isotherm which best fits the experimental data. Typically in these studies, 10–15 levels of the water activity and 1–4 levels of temperature (the independent factors) are selected. One experiment is con* Corresponding author. Tel.: +1 646 6439. E-mail address: [email protected] (S.H. Munson-McGee). 0260-8774/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2010.07.024
ducted at each combination of water activity and temperature. Occasionally, repeated measurements are conducted and the average of these is used in the subsequent analysis. However, the use of an average value decreases the information that can be obtained during the regression curve-fitting analysis and should be discouraged. Several sorption isotherms are then fitted to the multiple isothermal data using a non-linear, sum-of-squares regression technique to determine the model parameters for each isotherm at each of the temperatures studied. The quality of the fit is indicated by the correlation coefficient, R2. This approach is severely limited in the experimental design aspect in that it fails to consider experimental variability and it also does not consider the use of modern optimal design techniques to select the levels of water activity at which the measurements should be made. These oversights thereby decrease the reliability of the conclusions and increase the experimental effort required to reach those conclusions. In this work, we propose and explore the use of T-optimal designs to overcome these limitations. Specifically, we consider using T-optimal designs to distinguish between two isotherm models. The attraction of T-optimal designs is that they not only identify the most advantageous levels of the independent factors at which the experiments should be conducted, they also incorporate experimental replication to allow experimental variability to be estimated. In this work, we develop T-optimal designs and analyze the results to identify which isotherm model parameters are important in determining the T-optimal design. We then develop the T-optimal designs for a wide range of these model parameters typical of the values seen in recent experimental studies. Next, we
S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393
examine the effects of model-parameter misspecification and model-parameter uncertainty on the efficiencies of the designs. Finally, we explore the role of experimental variability in terms of lack-of-fit to differentiate between isotherm models, looking in turn at the number of experiments required to distinguish between the models at a known level of variability, then at the level of variability permissible if the number of experiments is limited. The two isotherms that we consider here, for the purpose of explicit illustration of the techniques and results, are the 3-parameter Guggenheim–Anderson–de Boer (GAB) and the 2-parameter Brunauer–Emmett–Teller (BET) isotherms given, respectively, by
mmG cG ka ð1 kaÞ½1 þ ðcG 1Þka mmB cB a mcB ¼ ð1 aÞ½1 þ ðcB 1Þa
mcG ¼
ð1Þ ð2Þ
In these models, mc is the amount of absorbed material (grams water per gram dry solid), mm is the monolayer capacity (grams water per gram dry solid), c is related to the enthalpy of adsorption (dimensionless), k is a measure of the free energy of enthalpy of the sorbate molecules (dimensionless), and a is the water activity (dimensionless). The subscripts G and B refer to the GAB and BET models, respectively. In terms of the notation for T-optimal designs used later, the model-parameter vectors for the GAB and BET models are bG = (mmG, cG, k)T and bB = (mmB, cB)T, respectively, where the superscript T indicates the transpose of the vector. Note that the BET model can be obtained from the GAB model by setting k = 1. By assuming a priori that these models may be appropriate, the sorption isotherm for materials such as salted products that have a discontinuity should not be modeled using these expressions. However, once other appropriate sorption isotherms have been selected, the T-optimal design procedure discussed here would still be appropriate for distinguishing between them. 2. T-optimal designs Optimal design theory has been developed to provide experimentalists methods to minimize the amount of experimental work necessary to achieve a desired objective (Atkinson and Donev, 1992). By design, we explicitly mean the design measure, e, which includes the points (i.e. the levels of the independent factors x) at which the experiment is to be conducted and the fraction of experiments to be conducted at each point (i.e. the experimental weights w). Like all experimental designs, optimal designs require that the experimentalist explicitly decide on the objective of the experiment before selecting the design that is appropriate for that objective. Several optimal designs can be used when the objective involves estimating the parameters of an a priori chosen model. Examples of such optimal designs include the following: A-optimal designs minimize the trace of the inverse of the information matrix and are used to minimize the sum of the variances of the model parameters. C-optimal designs are used to minimize selected functions of the model parameters and frequently are used to estimate a few model parameters very accurately at the expense of estimation accuracy of other parameters. Dh-optimal designs minimize the determinant of the inverse of the information matrix and provide the smallest volume of the joint confidence ellipsoid of the parameters, i.e. they provide the best simultaneous estimate of the model parameters. Dh-optimal designs are a subset of D-optimal designs; they are useful when only a subset of the model parameters is of interest.
387
E-optimal designs maximize the minimum eigenvalue of the information matrix and minimize the maximum variance of the estimated model parameters. Other optimal designs focus on the variance of the predicted values. Examples of these optimal designs include the following: G-optimal designs minimize the maximum value along the diagonal of the hat matrix and are used when the minimum variance in the predicted values is desired. I-optimal designs minimize the mean squared prediction error and are likewise useful when the minimum variance on the predicted values is desired. V-optimal designs minimize the average prediction variance. Herein we are interested in developing and exploring T-optimal designs for the GAB and BET adsorption isotherms. T-optimal designs are used to discriminate between two models that have been selected a priori. Theoretical development of T-optimal designs for discrimination between two models has been considered (Dette and Titoff, 2009; Ponce de Leon and Atkinson, 1991; Otsu, 2008; Atkinson and Fedorov, 1975; Wiens, 2009), as has the theoretical development of T-optimal designs for discrimination between several models (Atkinson and Fedorov, 1975), the use of T-optimal designs in compound designs with D-optimal designs (Atkinson, 2008; McGree et al., 2008), as well as a few practical applications of T-optimal designs (Atkinson et al., 1998; Rodriguez-Aragon and Lopez-Fidalgo, 2007; Lopez-Fidalgo et al., 2008). The basic concept of a T-optimal design is one in which the experiments are conducted at the points at which the two isotherms under consideration differ the most. Then, if the models are sufficiently different at these points compared to the measured experimental variability, one model can be selected as being more appropriate for the material. If there is not a statistically significant difference, then the models are equally appropriate and either can be used to represent the data. T-optimal design begins by first defining the two models of interest. One of these models is selected as the ‘‘true” model against which the other model can be tested for a statistically significant lack-of-fit. It is common practice to select the model having the greater number of parameters as the true model; hence, we select the GAB model and the model parameters associated with it as the true model, and develop the design to test the BET model for lack-of-fit. T-optimal designs are those for which the T-optimal statistic has been maximized. The T-optimal statistic is given as
T GB ¼ minbB
n X
^B;i y ^G;i Þ2 wi ðy
ð3Þ
i¼1
where wi is the fraction of experiments conducted at the ith exper^B;i and y ^G;i are the values predicted by the BET imental point and y and GAB models, respectively, at the ith experimental point. The T-optimal statistic is computed by selecting the BET model parameters bB = (mmB, cB)T, so the resulting BET model is the best-fit in a minimum sum-of-squares sense to a specific GAB model. We developed a different approach than previous researchers (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003; Goula et al., 2008; Syamaladevi et al., 2009; Ikhu-Omoregbe, 2006; Viollaz and Rovedo, 1999; Rahman and AlBelushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007) to find the maximum of the T-optimal statistic. Our approach, implemented in Matlab (MathWorks, 2007), followed the steps below:
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1. Specify the parameters in the GAB model bG = (mmG, cG, k)T. 2. Use an algorithm designed to optimize non-linear sum-ofsquares problems (LSQNONLIN in Matlab) to determine the initial estimate of the best-fit parameters for the BET model. 3. Use a multivariate optimization algorithm (FMINCON in Matlab) to determine the design measure that maximizes the Toptimal statistic. Within this function call, the BET model parameters are re-optimized within each iteration. Once a design was developed, the generalized equivalence theorem was applied to confirm that a T-optimal design had been obtained. According to the generalized equivalence (Atkinson and Donev, 1992) theorem, a T-optimal design must satisfy the following inequality at all points within the experimental region:
W ¼ ½y^B ðaÞ y^G ðaÞ2
n X
^B;i y ^G;i Þ2 6 0 wi ðy
ð4Þ
i¼1
^B ðaÞ and y ^G ðaÞ are the values predicted by the BET and GAB where y isotherms, respectively, at point a in the experimental space. For a T-optimal design, traditionally denoted as eT , the equality will be satisfied at the points at which the experiment is to be conducted; these points are known as the support points of the design. The designs and experimental regions considered here are one-dimensional and bounded only at the lower and upper limits. For such a design the number of support points equals the number of parameters in the model having the largest number of parameters.
Since we assumed that the GAB model is the true model and that we i is given are testing for lack-of-fit of the BET model, it follows that y ^B;i ) and that y ^i is given by the GAB preby the BET predicted value (y ^G;i ). The F statistic for lack-of-fit then becomes dicted value (y
F LOF ¼
m X
1 ðm pÞS
2
^B;i y ^G;i Þ2 ni ðy
ð11Þ
i¼1
From the T-optimal design results we get, among other information, the fraction of experiments done at each design point, wi. The number of experiments done at each point is
ni ¼ RNDðnwi Þ
ð12Þ
where RND(x) indicates that x has been rounded to the nearest integer. Our expression for the FLOF statistic for lack-of-fit finally becomes
F LOF ¼
m X
1 ðm pÞS
2
^B;i y ^G;i Þ2 RNDðnwi Þni ðy
ð13Þ
i¼1
The criteria for significant lack-of-fit at the (1 a/2) confidence level is given by
F LOF P F ð1a2Þ;ðmpÞ;ðNmÞ
ð14Þ
where
N¼
m X
RNDðnwi Þ
ð15Þ
i¼1
3. Lack-of-fit analysis The standard definition of the statistic for lack-of-fit analysis is given by Montgomery (2008); Bates and Watts (2007)
SSLOF =ðm pÞ ¼ SSPE =ðn mÞ
F LOF
ð5Þ
where m is the number of design points at which replicates are taken, n the total number of experimental points, and p the number of parameters in the model. The lack-of-fit sum-of-squares and pureerror sum-of-squares are given by
SSLOF
m X
i y^i Þ2 ni ðy
ð6Þ
Some rearranging gives the following criteria for determining the number of experiments that must be conducted to determine a significant lack-of-fit. m X
h i ^B;i y ^G;i Þ2 P ½ðm pÞS2 F ð1aÞ;ðmpÞ;ðNmÞ RNDðnwi Þðy 2
In the GAB isotherm, the model-parameter mmG is a scaling parameter. We can remove its influence in the lack-of-fit analysis by dividing both sides of the previous inequality by m2mG . The result is m X
RNDðnwi Þ
i¼1
2 h i ^G;i 2 ^B;i y S y P ðm pÞ F ð1a2Þ;ðmpÞ;ðNmÞ mmG mmG ð17Þ
i¼1
SSPE
m X
ni X
i¼1
j¼1
ðyij yi Þ2
2
ð7Þ
where ni is the number of replicates at the ith design point (which i is the will be proportional to the T-optimal design weights, wi), y ^i is the predicted average of the replicates at the ith design point, y value at the ith design point, and yij is the jth experimental value at the ith design point. We now introduce the sample variance at the ith design point, given by
S2i ¼
ni X i Þ2 ðyij y ni 1 j¼1
ð8Þ
Substituting this expression into the definition of the sum-ofsquares pure-error above yields
SSPE ¼
m X
S2i ðni 1Þ
ð9Þ
i¼1
If the sample variance is assumed to be constant over the experimental range (i.e. the experimental variation is the same at each point at which the experiment was conducted) and is given by S2, the above expression becomes
SSPE ¼ S2 ðn mÞ
ð16Þ
i¼1
ð10Þ
As we explain more fully below, the quantity [S/mmG] can be considered a measure of experimental variance which can be estimated from the experimental data provided replicate experiments have been made. 4. Results To determine the range of model parameters measured experimentally for foodstuffs, we reviewed GAB adsorption isotherms published over the last decade. This review included grains and seeds (Togrul and Arslan, 2006, 2007; Oyelade et al., 2008; Mohamad Saad et al., 2009; Resio et al., 1999; Peng et al., 2007; Chen, 2003; Pagano and Mascheroni, 2003), dried fruits and berries (Goula et al., 2008; Syamaladevi et al., 2009), tubers (IkhuOmoregbe, 2006; Viollaz and Rovedo, 1999), and other foodstuffs (Rahman and Al-Belushi, 2006; Kaymak-Ertekin and Sultanoglu, 2001; Arslan and Togrul, 2006; Johnson and Brennan, 2000; Aviara et al., 2006; Mulet et al., 2002; Sandoval and Barreiro, 2002; Sobukola et al., 2007). Based on this review, we selected the following ranges of the model parameters as encompassing most of the reported values: 1 6 b1 6 20,1 6 b2 6 15,0. 60 6 b3 6 0.99. However, there were some materials for which the model parameters fell well outside these ranges, including rice (Togrul and Arslan, 2006), maize flour (Oyelade et al., 2008), walnut kernels (Togrul
389
S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393 Table 1 Computed best-fit BET model parameters and design measures for the 23 factorial screening study. Model
GAB isotherm model parameters bG = (mm, c, k)T
BET isotherm model parameters bG = (mm, c)T
Design measure
1 5
bG1 = (1, 1, 0.60)T bG5 = (20, 1, 0.60)T
bB1 = (0.2036, 15.08)T bB5 = (4.072, 15.08)T
eT1 ¼ eT5 ¼
2 6
bG2 = (1, 15, 0.60)T bG6 = (20, 15, 0.60)T
bB2 = (0.4334, 3714)T bB6 = (8.668, 3714)T
eT2 ¼ eT6 ¼
3 7
bG3 = (1, 1, 0.99)T bG7 = (20, 1, 0.99)T
bB3 = (0.9349, 1.111)T bB7 = (18.70, 1.111)T
eT3 ¼ eT7 ¼
4 8
bG4 = (1, 15, 0.99)T bG8 = (20, 15, 0.99)T
bB4 = (0.9645, 17.88)T bB8 = (19.29, 17.88)T
eT4 ¼ eT8 ¼
e¼
a1 w1
a2 w2
a3 w3
0:1120 0:1278
0:6367 0:5792
0:8000 0:2931
0:0035 0:0050
0:4627 0:7258
0:8000 0:2692
0:3108 0:3680
0:7148 0:4140
0:8000 0:2180
0:0428 0:1230
0:6062 0:5936
0:8000 0:2834
and Arslan, 2007), freeze dried blueberries (Syamaladevi et al., 2009), freeze dried plantain (Johnson and Brennan, 2000), sorghum malt (Aviara et al., 2006), morel mushrooms (Mulet et al., 2002), cocoa beans (Sandoval and Barreiro, 2002) and yam chips (Sobukola et al., 2007). The range of model parameters selected above is slightly different from that recommended by Lewicki (1997), Blahovec (2004) based on theoretical considerations. Lewicki’s recommendations were 5.67 6 b2 6 1 and 0.24 6 b3 6 1 while those of Blahovec were 0 6 b2 6 2 for type II isotherms and 2 6 b2 6 1 for type III isotherms and 0 6 b3 6 1 for both isotherm types. However, the ranges used here were selected based on experimental results obtained for a wide range of foodstuffs and represent much of that data.
nificantly influenced the T-optimal design. To do this, a 23 factorial design (Montgomery, 2008) was executed using the eight possible combinations of the extreme values of the three parameters. The results of these calculations are given in Table 1. From the results in Table 1, the following observations can be made:
4.1. 23 Factorial design
The observation that the parameter b1 does not affect the design is not surprising since it is a scaling parameter that does not affect the curvature of the predicted response. The other two parameters, on the other hand, do affect the response curvature and therefore
Having selected a suitable range of model parameters to examine, the next task was to determine which of these parameters sig-
1. The designs for models 1 and 5, models 2 and 6, models 3 and 7, and models 4 and 8 are identical. From this we conclude that the parameter mmG (b1) does not affect the design but the other two parameters do. 2. The design measures for all models consist of one point for a1 < 0.35, one point 0.45 < a2 < 0.75, and one point at a3 = 0.8000, the maximum of the experimental space.
Fig. 1. Comparison of the best-fit BET isotherm to the GAB isotherm for models 1–4 (Figs. 1A–1D, respectively) given in Table 1 (note that in the bottom two figures the models are indistinguishable at this scale). The vertical lines are the levels of the independent factor which satisfy the T-optimal criteria.
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their importance in determining the T-optimal design is expected. Examining the results in Table 1, we notice that increasing b2 decreases a1, a2, w1, and w3 but increases the weight w2. Increasing b3 increases a1, a2, and w1 but decreases w2. The effect of the model parameters on the weight w3 is less straightforward: w3 decreases at low levels of b2 but increases at high levels of b2 indicating that there is a significant interaction between these two parameters in determining the value of w3. Shown in Fig. 1 are plots of the best-fit BET model for the first four GAB models used in the factorial design study along with the T-optimal design experimental locations. From this figure it is apparent that the locations of the T-optimal experiment are those at which there is the greatest difference between the GAB isotherm and the BET isotherm. The fact that these are T-optimal designs is shown in Fig. 2 where the scaled normalized variance has been plotted and is less than zero over the entire experimental space, thus satisfying the generalized equivalence theorem. 4.2. T-optimal designs for the model-parameter space Based on the results above, the model-parameter mmG (i.e. b1) was given no additional consideration; it was assigned the value b1 = 1. T-optimal designs were developed for the entire modelparameter space given by b1 = 1, 1 6 b2 6 15, 0.60 6 b3 6 0.99, and these results are given in Figs. 3 and 4. These results confirm the earlier observations about the effects of the model parameters on the design measure. Increasing b2 will decrease a1, a2, and w1 but will increase w2. Increasing b3 will increase a1, a2, and w1 but will decrease w2. The effect of changing the model parameters on the weight w3 is more complex in that a maximum occurs in the response surface. 4.3. Effect of parameter misspecification The third task in this investigation was to examine the effect of parameter misspecification on the design efficiency. By parameter misspecification, we mean using GAB model parameters that are not well known when developing the design. The reciprocal of the design efficiency is approximately the number of times the non-optimal design would have to be replicated to have the same ability to distinguish between the isotherms as does the optimal design. The design efficiency n is given by
n¼
T GB T GB
ð18Þ
where TGB is the T-optimal statistic for the non-optimal design and T GB is the T-optimal statistic for the optimal design. We examined the effect of parameter misspecification as follows:
Fig. 3. Contour plots of the components a1 (Fig. 3A) and a2 (Fig. 3B) of the T-optimal designs for the range of GAB model parameters b1 = 1, 1 6 b2 6 15,0.60 6 b3 6 0.99. For all levels of the model parameters a3 = 0.8000.
(1) Set (b1, b2, b3) = (1, 8, 0.8), approximately the center of the model-parameter space. (2) Determine the T-optimal design measure for this parameter combination. (3) Calculate the design efficiencies for the remainder of the model space using this design measure. The results of these calculations are presented in Fig. 5. By selecting the center of the model-parameter space, the design efficiency was 90% or more over approximately the half of the space corresponding to high values of b2 and low values of b3. However, the efficiency dropped quickly within the remainder of the space to slightly less than 30%. 4.4. Effect of parameter uncertainty When the precise values of the model parameters are unknown, the question of which design to use must still be addressed. In the current work, we consider this by finding the design that maximizes the minimum design efficiency within the space of possible model parameters. This was accomplished by the following procedure: (1) Select all possible pairwise combinations of parameters from b2 = {1, 2, . . . , 14, 15} and b3 = {0.60, 0.61, 0.62, . . . , 0.98, 0.99}. (2) Determine the T-optimal design for each pairwise combination. (3) Calculate the efficiency at the four corners of the model parameter space for each pairwise combination. (4) Select the smallest calculated efficiency as the minimum efficiency for this pairwise combination.
Fig. 2. General equivalence theorem for the models examined for the 23 factorial screening study.
The efficiencies for each combination of model parameters were calculated only at the corners of the space since the response
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surface was convex down and therefore the smallest efficiency had to occur at one of the corners. This significantly reduced the computational burden compared to either computing the efficiency at all points within the space or finding the smallest value numerically using a multifactor optimization approach. The results of these calculations are shown in Fig. 6. The maximum–minimum efficiency is at (b2, b3) (2.0, 0.82); we call this the T-optimal max–min design point. At the T-optimal max–min design point for the range of model parameters shown in the figure, the max–min efficiency has a value slightly greater than 55%. This means using the T-optimal design generated for this parameter combination that, regardless of the true values of the model parameters, the resulting design efficiency would be 55% or greater. At this point the design measure is given by
e¼
0:115 0:642 0:800 0:150 0:564 0:286
Of course, should the possible range of model parameters change, the T-optimal max–min design point may change as may the max–min efficiency. The smaller the range of model parameters, the larger the max–min efficiency should be. 4.5. Lack-of-fit analysis
Fig. 4. Contour plots of the design measure weightings w1, w2, and w3 (Figs. 4A–4C, respectively) for the T-optimal designs for the range of GAB model parametersb1 = 1,1 6 b2 6 15,0.60 6 b3 6 0.99.
Fig. 5. Design efficiency when values of the model parameters are incorrectly specified. The GAB model parameters were assumed to be bG(1, 8, 0.80)T and the plots show the affect of this choice on the T-optimal efficiency when the true model parameters are bG = (1, b2, b3)T.
The final objective of this project was to use the lack-of-fit analysis developed earlier to determine the number of experiments required to distinguish between the GAB and BET isotherms. This was done for the first four models given in Table 1. Before discussing the results from these calculations, a short discussion of the meaning of [S/mmG]2 is warranted. This term is a measure of the repeatability of the experimental measurements and a measure of the experimental variability. This variability may be due to a number of causes including variation in the materials, experimental apparatus variability, and variability of the experimentalist. In this sense, it should not be considered as an ‘‘error”, i.e. a mistake that can be corrected. Some of this variability might be reducible by using better experimental techniques or more accurate instrumentation. However, a portion of this variability will always exist and cannot be eliminated regardless of the care or expertise of the experimentalist. Since S is the sample standard deviation based on repeated measurements, the term [S/mmG] approximates a scaled standard deviation. It should be noted, however, that this scaling does not include the effects of the other model parameters. As illustrated in Fig. 1 where all the models had the same value of mmG the range of the response varied by a factor of almost 5; this may affect the magnitude of the standard deviation of the experiment. The results of the lack-of-fit calculations at the 95% confidence level (i.e. a = 0.05) are given in Fig. 7 where the left side of Eq. (17) was plotted for models 1–4 from Table 1 while the right side of Eq.
Fig. 6. Minimum design efficiencies of models 1–4 in Table 1 when the T-optimal design is computed using b1 = 1 and the values of b2 and b3 as given in the figure.
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come more similar, the acceptable level of experimental variability decreases. For model 4 and 3, the variability drops to 0.57 and 0.015, respectively, for designs consisting of only 20 points. Similar trends are seen in the maximum variability possible if the isotherms are to be distinguished using an infinite number of experiments. This knowledge can be used to establish, once the experimental variability of a particular experimental set-up and experimental protocol has been established, the whether or not two isotherms could be distinguished and whether or not it is worth the effort to attempt such an undertaking or if improvements needed to be made in the experiment to reduce the variability prior to conducting the study.
Fig. 7. Lack-of-fit criteria for 4 of the models used for the 23 factorial study and given in Table 1. The solid lines are the left-hand side of the lack-of-fit criteria for the indicated models and the dashed lines are the right-hand side of the lack-of-fit criteria for the indicated levels of [S/mmG].
(17) was plotted for five levels of the parameter [S/mmG]. Since reports on the sample variation for repeated measurements are rarely given in the literature, a wide range of values were examined. For model 2 given in Table 1, which shows the greatest difference between the two isotherm models, even for a highly variably experiment (i.e. for [S/mmG] = 0.50), the two isotherms can be differentiated using 10–20 experiments. At this level of variability it would be impossible to differentiate between the isotherms for models 1, 3, and 4 even with an unlimited number of experiments. To differentiate between the isotherms for model 1 with a decreased variability of [S/mmG] = 0.20 would require between 30 and 40 experiments or between 10 and 20 experiments if the variability was decreased to [S/mmG] = 0.10. For model 4, the isotherms could be differentiated with an experimental variability of [S/mmG] = 0.10 using between 50 and 60 experiments or with an experimental variability of [S/mmG] = 0.05 and 10–20 experiments. Similar results are obtained for model 3 if the experimental variability can be reduced to [S/mmG] = 0.02 (30–40 experiments) or [S/mmG] = 0.01 (10–20 experiments). The second portion of the lack-of-fit analysis was to examine the required level of experimental variability necessary to distinguish between the isotherms for the various models using either approximately 20 or 100 points. Twenty points was selected as a reasonable number of points and 100 points was selected to approximate an infinite number of points, i.e. the maximum possible experimental variability that would still allow differentiation between the isotherms. These results are only for approximately the specified number of points since the rounding operation specified by Eq. (15) may change that value by plus or minus one. The results of these calculations are given in Table 2 for each of the first four models given in Table 1. As expected for the models for which the isotherms were very different, the levels of experimental variability that can be tolerated and still distinguish between the isotherms is relatively high. For models 2 and 1, the minimum variability for distinguishing between the two isotherms using only 20 points is 0.68 and 0.15, respectively. As the isotherms beTable 2 Maximum permissible experimental variability measured by (S/mmG) to differentiate between the GAB and BET isotherms at a = 0.05 using approximately 20 or 100 T-optimal designed experiments.
5. Conclusions In this work we have shown that T-optimal designs can be developed that will efficiently distinguish between the GAB and BET isotherm models for adsorption. The process is general enough that it could be applied to any a priori selected pair of isotherm models. The specific examples here used a range of model parameters reflective of moisture adsorption in foods. It was shown that the monolayer coverage did not play a role in determining the experimental design but that the other two parameters in the GAB model did. The role of these parameters was fully explored and the ranges of the design measures were determined. The designs placed the experimental points at the locations where the GAB and BET models differed the most; one of the three support points was always for a water activity less than about 0.35, one in the middle of the space between 0.45 and 0.75, and the third was always at 0.80, the upper limit of the independent factor space. The design weightings were such that most of the experiments were always in the center of the independent factor space where the 45% and 75% of the experiments would be conducted. The fraction of experiments conducted at the lowest level of water activity was 35% or less, while the fraction conducted at the highest level was between 20% and 30%. Only when the GAB and BET models were very similar were an approximately equal number of experiments conducted at each of the three support points. The effect of model-parameter misspecification was studied and it was shown that by assuming incorrect values of the model parameters, the design efficiency could be reduced to 30% or less. However, by judiciously selecting the model parameters for which the design was developed and using a maximum–minimum efficiency criteria at the corners of the model-parameter space, the maximum–minimum efficiency could be increased to 55%. Finally, it was shown how experimental variability affected the number of experiments necessary to distinguish between the isotherms. In addition to illustrating how the number of experiments required was a function of the experimental variability, it was also shown what level of variability was permissible to distinguish between the isotherms for a specified number of experiments. This work has addressed observed shortcomings in many adsorption studies: the lack of experimental replication and the lack of efficient designs to minimize experimental effort while maximizing the information content of the experimental results. T-optimal designs, when developed appropriately, can be useful to alleviate these shortcomings and should lead to more reliable and statistically justifiable conclusions about isotherm model selection.
Model
GAB isotherm model parameters bG = (mm, c, k)T
[S/mmG]n 20
[S/mmG]n 100
References
1 2 3 4
bG1 = (1, 1, 0.60)T bG2 = (1, 15, 0.60)T bG3 = (1, 1, 0.99)T bG4 = (1, 15, 0.99)T
0.15 0.68 0.015 0.057
0.34 1.6 0.037 0.14
Arslan, N., Togrul, H., 2006. The fitting of various models to water sorption isotherms of tea stored in a chamber under controlled temperature and humidity. Journal of Stored Products Research 42, 112–135. Atkinson, A.C., 2008. DT-optimum designs for model discrimination and parameter estimation. Journal of Statistical Planning and Inference 138, 56–64.
S.H. Munson-McGee et al. / Journal of Food Engineering 101 (2010) 386–393 Atkinson, A.C., Donev, A.N., 1992. Optimal Experimental Designs. Oxford Science Publications, Oxford University Press, Inc., New York, NY. Atkinson, A.C., Fedorov, V.V., 1975. The design of experiments for discrimination between two rival models. Biometrika 62 (1), 57–70. Atkinson, A.C., Fedorov, V.V., 1975. Optimal design: experiments for discrimination between several models. Biometrika 62 (2), 289–303. Atkinson, A.C., Bogacka, B., Bogacki, M.B., 1998. D- and T-optimum designs for the kinetics of a reversible chemical reaction. Chemometrics and Intelligent Laboratory Systems 43, 185–198. Aviara, N.A., Ajibola, O.O., Aregbesola, O.A., et al., 2006. Moisture sorption isotherms of sorghum malt at 40 and 50 °C. Journal of Stored Products Research 42, 290– 301. Bates, D.M., Watts, D.G., 2007. Nonlinear Regression Analysis and Its Applications. John Wiley and Sons, New York, NY. Blahovec, J., 2004. Sorption isotherms in materials of biological origin mathematical and physical approach. Journal of Food Engineering 65, 489–495. Chen, C.C., 2003. Moisture sorption isotherms of pea seeds. Journal of Food Engineering 58 (1), 45–51. Dette, H., Titoff, S., 2009. Optimal discrimination designs. Annals of Statistics 37 (4), 2056–2082. Goula, A.M., Karapantsios, T.D., Achilias, D.S., Adamopoulos, K.G., 2008. Water sorption isotherms and glass transition temperature of spray dried tomato pulp. Journal of Food Engineering 85 (1), 73–83. Ikhu-Omoregbe, D.I.O., 2006. Comparison of the sorption isotherm characteristics of two cassava products. International Journal of Food Properties 9, 167–177. Johnson, P.N.T., Brennan, J.G., 2000. Moisture sorption isotherm characteristics of plantain (Musa, AAB). Journal of Food Engineering 44 (2), 79–84. Kaymak-Ertekin, F., Sultanoglu, M., 2001. Moisture sorption isotherm characteristics of peppers. Journal of Food Engineering 47 (3), 225–231. Lewicki, P.P., 1997. The applicability of the GAB model to food water sorption isotherms. International Journal of Food Science and Technology 32, 553–557. Lopez-Fidalgo, J., Tommasi, C., Trandafir, P.C., 2008. Optimal designs for discriminating between some extensions of the Michaelis–Menten model. Journal of Statistical Planning and Inference 138, 3797–3804. The MathWorks, Inc., 2007. Matlab, version 7, 3 Apple Hill Drive, Natick, MA 017602098, USA. McGree, J.M., Eccleston, J.A., Duffull, S.B., 2008. Compound optimal design criteria for nonlinear models. Journal of Biopharaceutical Statistics 18, 646–661. Mohamad Saad, M., Gaiani, C., Scher, J., Cuq, B., Ehrhardt, J.J., Desobry, S., 2009. Impact of re-grinding on hydration properties and surface composition of wheat flour. Journal of Cereal Science 49, 134–140. Montgomery, D.C., 2008. Design and Analysis of Experiments, seventh ed. John Wiley and Sons, New York, NY.
393
Mulet, A., Garcia-Pascual, P., Sanjuan, N., Garcia-Reverter, J., 2002. Equilibrium isotherms and isosteric heats of morel (Morchella esculenta). Journal of Food Engineering 53 (1), 75–81. Otsu, T., 2008. Optimal experimental design criterion for discriminating semiparametric models. Journal of Statistical Planning and Inference 138, 4141–4150. Oyelade, O.J., Tunde-Akintunde, T.Y., Igbeka, J.C., et al., 2008. Modelling moisture sorption isotherms for maize flour. Journal of Stored Products Research 44, 179– 185. Pagano, A.M., Mascheroni, R.H., 2003. Water sorption of Amaranthus cruentus L. seeds modelled by GAB equation. International Journal of Food Properties 6 (3), 369–391. Peng, G.L., Chen, X.G., Wu, W.F., Jiang, X.J., 2007. Modeling of water sorption isotherm for corn starch. Journal of Food Engineering 80 (2), 562–567. Ponce de Leon, A.C., Atkinson, A.C., 1991. Optimum experimental design for discrimination between two rival models in the presence of prior information. Biometrika 78 (3), 601–608. Rahman, M.S., Al-Belushi, R.H., 2006. Dynamic isopiestic method (DIM): measuring moisture sorption isotherm of freeze-dried garlic powder and other potential uses of DIM. International Journal of Food Properties 9, 421–437. Resio, A.C., Aguerre, R.J., Suarez, C., 1999. Analysis of the sorptional characteristics of amaranth starch. Journal of Food Engineering 42 (1), 51–57. Rodriguez-Aragon, L.J., Lopez-Fidalgo, J., 2007. T-, D-, and C-optimum designs for BET and GAB isotherms. Chemometrics and Intelligent Laboratory Systems 89 (1), 36–44. Sandoval, A.J., Barreiro, J.A., 2002. Water sorption isotherms of non-fermented cocoa beans (Theobroma cacao). Journal of Food Engineering 51 (2), 119–123. Sobukola, O.P., Dairo, O.U., Afe, T.T., Coker, O.J., 2007. Water sorption isotherms and crispness of fried yam chips in the temperature range from 293 K to 313 K. International Journal of Food Properties 10, 561–575. Syamaladevi, R.M., Sablani, S.S., Tang, J.M., Powers, J., Swanson, B.G., 2009. State diagram and water adsorption isotherm of raspberry (Rubus idaeus). Journal of Food Engineering 91 (3), 460–467. Togrul, H., Arslan, N., 2006. Moisture sorption behaviour and thermodynamic characteristics of rice stored in a chamber under controlled humidity. Biosystems Engineering 95, 181–195. Togrul, H., Arslan, N., 2007. Moisture sorption isotherms and thermodynamic properties of walnut kernels. Journal of Stored Products Research 43, 252–264. Viollaz, P.E., Rovedo, C.O., 1999. Equilibrium sorption isotherms and thermodynamic properties of starch and gluten. Journal of Food Engineering 40 (4), 287–292. Wiens, D.P., 2009. Robust discrimination designs. Journal of the Royal Statistical Society, B 71 (4), 805–829.