Adaptive Neuro-fuzzy Inference System and Artificial Neural Network

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model exhibited better performance with high accuracy for the prediction of unmeasured values of apparent ... Key words: Fuzzy inference system, artificial neural networks, apparent viscosity, ice-cream mix, ... ferent conditions in the absence of accurate ... work systems. ..... the application of the ANFIS modeling using the.

Adaptive Neuro-fuzzy Inference System and Artificial Neural Network Estimation of Apparent Viscosity of Ice-cream Mixes Stabilized with Different Concentrations of Xanthan Gum Ömer Said Toker1, Mustafa Tahsin Yilmaz2*, Safa Karaman3, Mahmut Dog˘an3, Ahmed Kayacier3 1 Igdur

University, Engineering Faculty, Food Engineering Department, 76000 Igdur, Turkey 2 Yildiz Technical University, Chemical and Metallurgical Engineering Faculty, Food Engineering Department, 34210 Istanbul, Turkey 3 Erciyes University, Engineering Faculty, Food Engineering Department, 38039 Kayseri, Turkey Corresponding author: [email protected] Fax: x90.212.3834571 Received: 9.7.2012, Final version: 31.10.2012

Abstract: An adaptive neuro-fuzzy inference system (ANFIS) was used to accurately model the effect of gum concentration (GC) and shear rate (SR) on the apparent viscosity (h) of the ice-cream mixes stabilized with different concentrations of xanthan gum. ANFIS with different types of input membership functions (MFs) was developed. Membership function “the gauss2” generally gave the most desired results with respect to MAE, RMSE and R2 statistical performance testing tools. The ANFIS model was compared with artificial neural network (ANN) and multiple linear regression (MLR) models. The estimation by ANFIS was superior to those obtained by ANN and MLR models. The ANFIS and ANN model resulted in a good fit with the observed data, indicating that the apparent viscosity values of the ice-cream can be estimated using the ANFIS and ANN models. Comparison of the constructed models indicated that the ANFIS model exhibited better performance with high accuracy for the prediction of unmeasured values of apparent viscosity h parameter as compared to ANN although the performance of ANFIS and ANN were similar to each other. Comparison of the constructed models indicated that the ANFIS model exhibited better performance with high accuracy for the prediction of unmeasured values of apparent viscosity h parameter as compared to ANN although the performance of ANFIS and ANN were similar to each other. Zusammenfassung: Ein adaptives, sogenanntes Neuro-Fuzzy-Inferenzsystem (ANFIS) wurde angewandt, um den Einfluss der Gummiharz-Konzentration (GC) und der Schergeschwindigkeit (SR) auf die scheinbare Viskosität (h) von EiscremeMischungen zu untersuchen, die mit unterschiedlichen Konzentrationen von Xanthan stabilisiert wurden. ANFIS mit unterschiedlichen Typen von Eingabefunktionen (MFs) wurden entwickelt. Die Eingabefunktion „the gauss2“ führte generell zu den besten Resultaten hinsichtlich der statistischen Auswertetools MAE, RMSE und R2. Das ANFIS-Modell wurde mit künstlichen neuronalen Netzwerken (ANN) und multiplen linearen Regressions (MLR)-Modellen verglichen. Die Abschätzung durch das ANFIS-Modell war besser als die durch die ANN und MLR-Modelle erhaltenen Abschätzungen. Das ANFIS und das ANN-Modell resultierten in einen guten Fit der Messdaten. Dies zeigt, dass die scheinbare Viskosität der Eiscreme durch das ANFIS und das ANN-Modell abgeschätzt werden können. Der Vergleich der entwickelten Modelle zeigte, dass das ANFIS-Modell eine bessere Darstellung mit höherer Genauigkeit für die Vorhersange nicht gemessener Werte der scheinbaren Viskosität h im Vergleich zum ANN-Modell aufwies, obgleich das ANFIS- und das ANN-Modell eine ähnliche Darstellung aufwiesen. Résumé: Un système adaptif “neuro-fuzzy inference” (ANFIS) a été utilisé pour modéliser précisément l’effet de la concentration en gomme (GC) et de la vitesse de cisaillement (SR) sur la viscosité apparente (h) de mélanges de crèmes glacées stabilisées avec des concentrations différentes de gomme de xanthan. Des ANSIS avec des types différents de fonctions de données membres (MFs) ont été développés. La fonction « le gauss2 » a donné les resultats les plus adéquat relativement aux outils de tests de performance statistique MAE, RMSE et R2. le modèle ANFIS a été comparé avec le réseau neural artificiel (ANN) et les modèles de régression linéaire multiples (MLR). L’estimation par ANFIS est supérieure à celles obtenues avec ANN et MLR. Les modèles ANFIS et ANN ont produit de bons ajustements avec les données observées, ce qui indique que les valeurs de la vis-

© Appl. Rheol. 22 (2012) 63918

DOI: 10.3933/ApplRheol-22-63918

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cosité apparente de la crème glacée peuvent être estimées avec ces modèles. La comparaison entre les modèles construits indique que le modèle ANFIS présente une performance meilleure avec une grande précision pour la prédiction des valeurs non mesurées de la viscosité apparente, bien que les performances de ANFIS et ANN soient similaires. Key words: Fuzzy inference system, artificial neural networks, apparent viscosity, ice-cream mix, xanthan gum

1

INTRODUCTION

Xanthan gum is a hydrocolloid, being a heteropolysaccharide with a primary structure, consisting of repeated pentasaccharide units that are formed by two glucose units, and one glucuronic acid unit [1]. Being produced 30.000 tons per annum, it adresses a market of $ 408 million, which indicates that it is one of the most commmercially industrial gum; thus, that it is an important industrial biopolymer [2, 3]. Xanthan gum has a unique textural and rheological properties; therefore, it is commonly used as a thickening or stabilizing agent in the food industry. Even at low shear rate values, it can exhibit high viscosity and pseudoplastic behavior as well as stability within a wide range of temperature and pH values. Furthermore, it can show high resistance against degradation caused by shear force in aqueous solutions [3, 4]. In the ice-cream production, among the rheological properties, apparent viscosity is known to be one of the most important quality parameters to get high quality of product [5, 6]. To achieve desired quality attributes, it is very important to take many factors into consideration simultaneously, such as kind, quality and concentration of ingredients, processing and handling of the mix and composition; mainly fat and stabilizer. Rheological properties are one of the properties influencing the texture of the ice-cream [6 – 8]. Among these properties are smooth texture and cooling sensation that are the most frequently desired icecream characteristics requested by consumers. The mentioned properties can be achieved by optimization of the rheological properties of the icecream mix [9]. Ice-cream mix is a complex colloidal system and it includes some substances present in true solution, in which some of them are colloidally suspended and some of them are in dispersion [10, 11]. Stabilizers and/or their combinations with other constituents are among these substances that

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have been used in the ice-cream industry [12]. In order to increase ice-cream quality, some parameters of the products like the apparent viscosity (AV) of mix can be manipulated [13]. Increasing of the AV is important because, with the increased AV values, a resistance occurs against melting and smoothness of the body, which gives rise to a decrease in the rate of whipping [5]. Additionally, increase in resistance to melting of final product and enhanced smoothness can be achieved by the increased AV values. Therefore, desirable apparent viscosity of the mix can be obtained by controlling composition of the mix [5, 6]. It should be also noted here that, sensory quality, and consequently consumer preference are considerably influenced by the rheological properties of fluid foods. Knowledge of the flow behavior is beneficial not only in quality control of the product, but also for energy input calculations, process design and equipment selection, particularly for heat exchangers and pumps [14]. Ice-cream mix is a product based on water in oil emulsion. Due to ill-defined and complicated viscosity (apparent viscosity) nature of the emulsions; it is very difficult to perform mathematical modeling of the possible viscoelastic structure of the emulsion systems using differential equations at the present [15, 16]. Therefore, selection of an accurate model defining the viscoelastic properties of the such emulsions is of great importance [17]. In this respect, in food science and technology area, there are useful models like adaptive neuro-fuzzy inference system (ANFIS) and artificial neural networks (ANN) that have been succesfully and commonly used so far in modeling studies. Identifying and modeling some parameters are possible by using these effective and versatile approaches, in non-linear systems. This allows researchers to efficiently solve their problems and predict the effect of processing factors on these parameters under different conditions in the absence of accurate mathematical models [18, 19].

Artificial Neural Network (ANN) is a mathematical algorithm method which relates input and output parameters. ANN modeling technique has the advantage that prior information is not required about relations of the process parameters and computational models inspired by structure or functional aspects of biological neural network systems. Furthermore, this modeling technique can also be used efficiently in the modeling of non-linear systems having complex relationships between inputs and outputs thanks to being a non-linear statistical data modeling tool [20, 21]. ANN model was succesfully used to predict the antioxidant activity of essential oils [22] and creep and recovery properties of the meat emulsion systems [17]. On the other hand, ANFIS was known to be better non-linear modeling technique compared to ANN and multiple linear regression (MLR). There have been a great number of studies conducted on the modeling and identification of food characteristics using the fuzzy inference system in the food engineering field. Abu Ghoush et al. [18] modeled the emulsion stability and viscosity of a gum-protein emulsifier in a model mayonnaise system using the adaptive neuro fuzzy inference system (ANFIS). The fuzzy model was also used to evaluate viscosity of an orangeflavored carboxymethylcellulose-whey protein isolate beverage [23]. Similarly, Yilmaz [17] compared the effectiveness of ANFIS and ANN for estimation of linear creep and recovery properties of model meat emulsions and found that the ANFIS modeling performed better than ANN and MLR models. In addition, a great number of studies have been carried out based on different applications [19, 24 – 26]. In this study, the primary goal was to construct predictive models to estimate the effect of gum concentration (GC) and shear rate (SR) on the apparent viscosity using three modeling techniques such as ANFIS, ANN and MLR and to compare the performances of these different models.

2

MATERIALS AND METHODS

2.1 INGREDIENTS AND PREPARATION OF ICECREAM MIXES The ice-cream mixes were prepared according to the traditional ice-cream manufacturing method [6, 27]. In this respect, the following ingredients

were used in the ice-cream mix production (in g/100 g): 10 % sucrose, 7.0 % cream (including 70 % fat), 0.5 % emulsifier (saturated mono- and di glycerides) and stabilizer system. The system was consisted of 0.5 % salep and different concentrations (0.0, 0.4, or 0.8 %) of xanthan gum (Sigma, G1253 USA). After homogenization of mix in a single state homogenizer (20.5 MPa at 60º C), the ingredients were pasteurized at 85º C for 1 min with constant stirring. After the pasteurization, the ice cream mixes were rapidly cooled down to 4º C. For ageing, the ice cream mixes were then stored at 4º C for 24 h until the analysis. 2.2 STEADY SHEAR ANALYSIS Apparent viscosity values in the study of Dogan et al. [27] were used as data set for ANFIS, ANN and MLR modeling procedures in this study. A controlled stress rheometer (RheoStress 1, HAAKE, Karlsruhe, Germany) equipped with a temperature control unit (Water bath K15, Thermo-Haake Karlsruhe, Germany) was used to conduct rheological analysis of the ice-cream mixes. The mixes were sheared using a cone-plate configuration (cone diameter 35 mm, angle 4°, gap size 0.14 mm). Measurements were carried out in the shear rate range of 1 – 100 s-1 at 5º C. 0.85 ml sample was placed between cone and plate and the measurement was started immediately. During the shearing, a total of 25 data points were recorded at 10 s intervals. Each measurement was replicated with two repetitions. The apparent viscosity was determined as a function of shear rate. Obtained data were fitted to the Oswald de Waele model using RheoWin Data Manager (RheoWin Pro V. 4.0, HAAKE, Karlsruhe, Germany) and consistency coefficient and flow behavior index values were calculated according to the following model used to describe shear-induced behavior of the icecream mix samples: (1) where h is apparent viscosity (Pa s), K is consistency coefficient (Pa sn), g· is shear rate (s-1) and n is flow behavior index (dimensionless). The steady shear data were processed with respect to the effect of xanthan gum concentration on apparent viscosity on the ice-cream mixes. The variation of apparent viscosity with concentration at the specified shear rate of 50 s-1 can be

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Figure 1 (left): Two-input first-order sugeno fuzzy model with two rules. Figure 2: ANFIS and ANN architectures: (a) the first-order Takagi-Sugeno inference system and (b) the multilayer artificial neural network architecture design. ANFIS, adaptive neuro-fuzzy inference system; ANN, artificial neural network.

described by several models [28, 29]. These are generally power-law and exponential type models as following: (5)

(2)

(3) where h50 is apparent viscosity at 50 s-1 (Pa·s), C is concentration of xanthan gum (%), h1 and h2 are constant for concentration effect (Pa·s), a1 is constant (%-1) of Power law model and a2 is constant (dimensionless) of exponential type model. 2.3 ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM (ANFIS) MODELING Mamdani’s system, Tsukamoto’s system and Sugeno’s system are the fuzzy inference systems that are classified depending on the types of inference operations upon ‘‘if-then rules”. Mamdani’s system is known to be the most widely used system; however, the more compact and computationally efficient one is also reported to be Sugeno’s system because the output is crisp and without the time consuming and mathematically intractable defuzzification operation as well as based on sample-data. That is why the Sugeno’s system is regarded to be the most popular tool for fuzzy modeling and appropriate for the use of adaptive techniques [30, 31]. The Sugeno type fuzzy system was used in this study because it has neural learning ability. In order to briefly describe the Sugeno type fuzzy system used, it is necessary to give a knowledge on its basic structure based on the rules with several inputs x, y, … and one output f. An example can be given as an assumption that a fuzzy inference system has two inputs x and y and one output z. Then, it is possible define a typical rule set of the first-order Sugeno type fuzzy system that is described as IF/THEN rules that can be expressed as: (4)

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The resulting Sugeno fuzzy reasoning system is shown in Figure 1. In this figure, x and y are the inputs, Ai and Bi are the fuzzy sets; fi are the outputs within the fuzzy region specified by the fuzzy rule; p1, q1 and r1 are the design parameters determined during training phase. Figure 2a represents the ANFIS architecture implementing these two rules, showing the circles which indicates a fixed node, while the square which indicates an adaptive node. Five layers constitute the ANFIS structure. These layers are fuzzification layer (1), rule layer (2), normalization layer (3), defuzzification layer (4) and output layer (5). The output of the ith node in layer l is shown as Ql,i. In the fuzzification layer, all the nodes are adaptive (square) nodes with a node function defined as: (6) or

(7) where x (or) y is the input to the ith node and Ai (or Bi-2) is a linguistic label (e.g. “low” or “high”) associated with this node and mAi and mBi-z can adopt fuzzy membership function. In other words, Ql,i is the membership grade of a fuzzy set A(= A1, A2, B1 or B2) and it specifies the degree to which the given input x (or y) satisfies the quantifier A. The following equation is chosen if the bell shaped membership function is used; then, mAi is calculated by:

(8)

Figure 3: ANFIS model structures: multiple input single output models consisting of (a) two inputs (gum concentration, GC and shear rate, SR) with one output (apparent viscosity, h) with three MFs and (b) two inputs (GC and SR) with one output (AV) with six MFs. ANFIS, adaptive neuro-fuzzy inference system; MF, membership function.

where ai, bi and ci are the parameters of the membership function, operating the bell shaped functions. In the rule layer, every node is the fixed (circle) node labeled with M, which indicates that they perform as a simple multiplier. The output of this layer is represented as following: (9) In the normalization layer, the nodes are also fixed nodes, but labeled with N, indicating their playing a normalization role to the firing strengths from the previous layer. The outputs of this layer are called normalized firing strengths, as represented following:

(10) In the defuzzification layer, every node in this layer is adaptive node whose output is simply the product of the normalized firing strength and a first order polynomial Sugeno model. The outputs are presented as: (11) In the output layer, there is only one single fixed node labeled with E, which computes the overall output as the summation of all incoming signals, as represented following:

(12) In this study, modeling was performed using ANFIS toolbox in MATLAB 7.0.1 and Sugeno-type fuzzy inference systems were used to model the effect of xanthan gum concentration and shear rate on the apparent viscosity of ice-cream mixes. The grid partition method [32] was utilized to classify the input data make the rules. Figure 3a shows the ANFIS structure of gum concentration (GC) and shear rate (SR) values as the two input parameters and linguistic values (apparent viscosity values) as the output parameters with three membership functions. On the other hand,

Figure 3b shows the ANFIS structure of GC and SR values as the two input parameters and linguistic values (apparent viscosity values) as the output parameters with six membership functions. In order to find the best function describing the effects of input parameters, eight different types of membership functions (MFs); namely, ‘trimf’, ‘trapmf’, ‘gbellmf’, ‘gaussmf’, ‘gauss2mf’, ‘pimf’, ‘dsigmf’ and ‘psigmf’ were used for ANFIS modeling. To perform the ANFIS modeling, the data sets were divided into three groups of sets for training, testing and checking (validating) randomly. In order to estimate the apparent viscosity values of ice-cream mixes stabilized with xanthan gum using ANFIS modeling, 132 data were used. They were divided randomly into three subsets of 66 data for training, and the remaining 33 data for testing and 33 data for checking (validation) periods. Table 1 shows the number of nodes (NON), number of linear parameters (NLP), number of non-linear parameters (NNLP), number of training data pairs (NTDP), number of checking data pairs (NCDP) and number of fuzzy rules (NFR) for ANFIS modeling of AV values. The only variable changing by membership functions used is the number of non-linear parameters (NNLP) among these parameters. Similar results were determined in the study of Yilmaz [17] who reported that only variable changing by MFs used

Table 1: Details of ANFIS Model (NMFs: Number of membership function, MFTI: Membership function type of input, MFTO: Membership function type of output, NON: Number of nodes, NLP:Number of linear parameters, NNLP: Number of non-linear parameters, NTDP: Number of training data pairs, NCDP: Number of checking data pairs, NFR: Number of fuzzy rules).

Details of ANFIS model NMFs

MFTI

MFTO

NON NLP

NNLP NTDP NCDP NFR

66

trimf trapmf gbellmf gaussmf gauss2mf pimf dsigmf psigmf

Linear Linear Linear Linear Linear Linear Linear Linear

101 101 101 101 101 101 101 101

36 48 36 24 48 48 48 48

108 108 108 108 108 108 108 108

66 66 66 66 66 66 66 66

0 0 0 0 0 0 0 0

36 36 36 36 36 36 36 36

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two methods in the training phase, called a) back propagation and b) a hybrid learning method. Back propagation (steepest descent method) was used for all parameters. The hybrid learning method was comprised of back propagation for parameters that are related to input membership and least squares estimation for parameters associated with output MF [32, 33]. In the the steepest descent method based on the basic learning rule of the adaptive network, the gradient is derived by repeated application of the chain rule. To calculate the gradient in a network structure, the ordered derivative indicated as ¶+ as opposed to the ordinary partial derivative ¶ is used. This technique is the back propagation rule [33 – 35]. Below is the update formula for the simple steepest descent for the generic parameter α:

(13) where h is the learning rate. The back propagation learning rule is slow to converge; therefore, the hybrid learning algorithm [33] can be used to rapidly train and adapt the equivalent fuzzy inference system. This algorithm combines back propagation and the least squares method. The overall output can be given as a linear combination of the consequent parameters as can be seen in the Figure 2. The output f can be given as:

(14)

Figure 4: Membership functions (eight different types) for two inputs after developing the ANFIS by grid partition: (a) trimf, (b) trapmf, (c) gbellmf, (d) gaussmf, (e) gauss2mf, (f) pimf, (g) dsigmf and (h) psigmf. The output result of ANFIS was for apparent viscosity (h) values. ANFIS, adaptive neuro-fuzzy inference system.

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was the NNLP in the ANFIS modeling of linear creep and recovery properties of model sytem meat emulsions. An ANN learning algorithm was used to construct a set of fuzzy in ANFIS modeling of the apparent viscosity values since appropriate mem bership functions (MFs) were ruled from specified input–output pairs. In this study, ANFIS was perfomed in three phases; namely, training, testing and checking phases. The MFs were updated using

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which is linear in the consequent parameters p1, q1, r1, p2, q2, and r2. Then, S = set of total parameters S1 = set of premise (non-linear) parameters S2 = set of consequent (linear) parameters. If values of S1 are taken into consideration, it can be seen that P training data are integrated into the Equation 21 and the following matrix equation is obtained: (15) where θ is an unknown vector whose elements are parameters in S2, the set of consequent (lin-

ear parameters) [34]. Then, the set S2 of consequent parameters is determined using the standard least-squares estimator (LSE) as shown below:

(16) where AT is the transpose of A and (ATA)-1AT is the pseudo inverse of A if ATA is nonsingular. θ* can also be calculated by the recursive least-square estimator (RLS) [33]. Normalization of the training input and output data was the first step in the application of the ANFIS modeling using the following equation:

were used to calculate these statistical testing tools:

(17) where xmin and xmax are the minimum and maximum of the training data set, respectively. In this study, a and b values were taken as 0.6 and 0.2, respectively according to the Altun et al. [36]. It was reported that the different values can be assigned for the scaling factors a and b because there is no fixed rules as to which standardization approach should be used in particular circumstances. [36]. The data were scaled between 0.2 and 0.8. Then, the grid partition was used to generate the initial ANFIS model, and for the fuzzification of input data, the eight different types of MFs were used in this study [17]. In order to minimize error, the hybrid learning was used to update the parameters of adaptive neurons in each epoch [37]. In Figure 4, the MFs are shown after development of the ANFIS model by grid partition procedure. In order to construct a neuro-fuzzy inference system, data obtained should be partitioned at the first step [38]. Training process of the ANFIS model is followed by testing of the estimation performance. In the Figure 5, the performance of ANFIS for h values with different number of membership functions 3 3 gaussmf and 6 6 gaussmf in training, testing and checking phases is illustrated. The RMSE (root means square error), MAE (mean absolute error) and R2 (coefficient of determination) statistical tools were used to test the accuracy of the ANFIS model in estimating the apparent viscosity parameter and their values are presented in Table 2. The following equations

(18)

(19)

(20) where N is the number of the data set, Yi is the apparent viscosity parameter h. It is known that as R2 values increase and RMSE and MAE values decrease, the fitting performance becomes better [39]. Accordingly, the best estimation was achieved under this ANFIS setup: number of membership functions (NMFs) was 6 6 and membership function type of output (MFTO) was linear (Table 2). However, the success of the membership functions of input (MFTI) were different regarding the MAE, RMSE and R2 values. The lowest MAE, RMSE and the highest R2 values were

Figure 5: The performance of ANFIS for estimation of apparent viscosity h values with different number of membership functions (gauss2mf ) in training, testing and checking phases: (a) 3 3 gauss2mf (b) 6 6 gauss2mf. Actual degree (+) and estimated degree (*) ANFIS, adaptive neuro-fuzzy inference system. Table 2: Comparision of the performances of membership function (MF) types in checking period of ANFIS model (NMFs: Number of membership function, MFTI: Membership function type of input, MFTO: Membership function type of output, MAE: Mean absolute error, RMSE: Root mean square error, R2: Coefficient of determination).

Comparison tools NMFs

MFTI

MFTO

R2

RMSE

MAE

66

trimf trapmf gbellmf gaussmf gauss2mf pimf dsigmf psigmf

Linear Linear Linear Linear Linear Linear Linear Linear

0.9934 0.9919 0.9986 0.9918 0.9923 0.9891 0.9912 0.9912

0.1833 0.203 0.2713 0.2075 0.1684 0.1988 0.1795 0.1795

8.31 11.8717 9.6284 9.0191 18.4663 11.8412 11.0453 11.0453

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a three-layered ANN architecture consisted of three layers i, j and k with the connection weights (Wij and Wjk) between the input, hidden and output layers. During the learning phase, initial assigned weights are connected. In order to obtain the appropriate weight adjustments necessary to minimize errors in this process, estimated outputs are compared with known inputs, and errors are back propagated (from right to left in Figure 2b). Levenberg-Marquardt technique is used as it is more powerful and faster than the conventional descent techniques [40, 41]; therefore, ANN was trained using this technique in this study. 2.5 MULTIPLE LINEAR REGRESSION (MLR) ANALYSIS

Figure 6: Viewer of rules for apparent viscosity h values of ice cream samples.

achived with the MFs: “trimf”, “gauss2mf” and “gbellmf”, respectively. However, the gauss2mf generally gave the the most desired results with respect to MAE, RMSE and R2 statistical performance testing tools. Accordingly, the achievement of the only gauss2mf is shown in the Figure 5 where the performance of ANFIS for h values with different number gauss2mf is displayed in training, testing and and checking phases. On the other hand, all the reamining MFs exhibited the similar performances. But, increased number of epochs after 10 iteration resulted in a model with similar performances. 2.4 ARTIFICIAL NEURAL NETWORK (ANN) MODELING ANN has one or more hidden layers and their computation nodes are called hidden neurons. A neural network with one hidden layer consists of a weight A, a weight matrix α and a sigmoid function g:

(21) where xk(t) are the m input values at time t. Equation 21 describes a neural network with m input neurons and l neurons in the hidden layer. F(t) indicates the output value. Figure 2b represents

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In order to determine a relationship between independent variables and dependent variable [42], MLR analysis is also used to test its prediction performance. Below is the linear form of multiple regression that is given by:

(22) where y is the output; xi the input and bi the constants of factors.

3

RESULTS AND DISCUSSION

3.1 ESTIMATION BY ANFIS, ANN AND MLR MODELING Viewer of rules from the fuzzy model developed for ice-cream mixes stabilized with different concentrations of xanthan gum was illustrated in Figure 6 indicating that there are 30 rules in the structure and each row represents one rule. The first two columns are xanthan gum concentration, and shear rate selected as the inputs and the last column is apparent viscosity selected as the output. The rule viewer shows an example for predicting of the h values of the ice-cream mixes. Figure 6 shows the selected concentration and shear rate values as 0.4 % and 100 1/s, respectively and h was predicted to be 0.296 Pa·s using the developed fuzzy model. From Table 2, it can be inferred that the computed apparent viscosity value is very close to observed values given in checking data matrix (Table 2).

Figure 7: Estimated versus experimental apparent viscosity (h) values of ice cream samples shown in scattering graphs plotted using (a) ANFIS, (b) ANN and (c) MLR models in checking (validation) period. Experimental (•) and estimated (—) by ANFIS, ANN or MLR models. ANFIS, adaptive neuro-fuzzy inference system; ANN, artificial neural network; MLR, multiple linear regression.

As to the estimation of the apparent viscosity values of ice-cream mixes using ANN, the ANFIS and ANN approaches followed the same steps for training, testing and validating using the MATLAB software, as described above. Regarding details for ANN modeling, the final neural network modeling for apparent viscoity parameter consisted of one hidden layer with 2 nodes, which was obtained by trial and error approach and gave the best prediction results. In MLR modeling, training, testing and validating steps are not applied. It is a mathematical tool quantifying the relationship between a dependent variable and one or more independent variables. Evaluation by MLR modeling was also conducted using MATLAB software.

Table 3: Comparison of the testing performances of ANFIS, ANN and MLR models for estimation of apparent viscosity parameter of ice-cream mixes in training, testing and checking (validation) periods (R2: Coefficient of determination, RMSE: Root mean square error, MAE: Mean absolute error. *For the best testing performance of ANFIS model, NMFs, MFTI and MFTO were 6 6, gauss2mf and linear, respectively. ** For the best testing performance of ANN model, NON was 2 10 1 (NMFs, number of membership function; MFTI, membership function type of input; MFTO, membership function type of output; NON, number of nodes in layer)).

3.2 COMPARISON OF ANFIS, ANN AND MLR MODELS The R2, RMSE and MAE statistical tools were used to compare the accuracy of the ANFIS, ANN and MLR models in estimating the apparent viscosity parameter. Comparison of the testing performances of ANFIS, ANN and MLR models for estimation of h parameter of ice-cream mixes in training, testing and checking (validation) periods are presented in Table 3. ANFIS showed the best estimation performance; namely, the highest R2 and the lowest RMSE values were obtained when the data were modeled using ANFIS (Table 3). When the calculated MAE values were taken into consideration; however, ANN was observed to exhibit the best prediction performance because the lowest MAE values were obtained (Table 3). On the other hand, the MLR had the lowest accuracy regarding the R2, RMSE and MAE statistical accuracy testing tools. It should be also noted here that the data used in the training phase resulted in the best R2, RMSE and MAE values, indicating that the data in the training phase were more suitable than those in testing and checking phases in ANFIS and and MLR modeling. Figure 7 indicates estimated versus experimental apparent viscosity values of ice cream samples shown in scattering graphs plotted using ANFIS, ANN and MLR models in checking (validation) period. It can be clearly seen from the figure that the performance of ANFIS model in estimating the h values was quite similar to that of ANN model; however, the MLR had bad accu-

racy. It can be obviously seen that the ANFIS and ANN model estimates were closer to the corresponding experimental values compared to MLR model. The MLR estimates were far from the corresponding experimental values; in other words, the estimates of MLR model are much more scattered than those of the ANFIS and ANN models. 3D graphs presented in Figure 8 clearly reflects this fact, indicating that the performances of ANFIS and ANN models were quite similar, but that of MLR model was very bad. These results meant the non-linearity of the studied phenomenon. In other words, the behaviors of the inputs (xanthan gum concentration and shear rate) and output (h values) were non-linear. Based on these results; therefore, the ANFIS and ANN models can be suggested to be sufficient for estimation of h values. R2

RMSE

MAE

Parameter

Data set

ANFIS*

ANN**

MLR

ANFIS

ANN

MLR

ANFIS

ANN

MLR

Apparent viscosity

Training Testing Checking

0.9988 0.9981 0.9906

0.9987 0.9966 0.9872

0.7385 0.7388 0.7092

0.0279 0.0850 0.1846

0.0668 0.0875 0.2193

0.9468 0.9513 1.1490

1.0627 2.9474 8.4225

0.0267 0.0077 0.0837

557.2 566.7 508.9

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[3]

[4]

[5] [6]

[7]

Figure 8: Comparison of three-dimensional plots generated using the apparent viscosity h values of ice cream samples estimated by (a) ANFIS, (b) ANN and (c) MLR models in checking (validation) period. ANFIS, adaptive neuro-fuzzy inference system; ANN, artificial neural network; MLR, multiple linear regression.

4

CONCLUSIONS

Comparison of the constructed models indicated that the ANFIS model exhibited better performance with high accuracy for the prediction of unmeasured values of apparent viscosity parameter as compared to ANN although the performance of ANFIS and ANN was similar to each other. However, MLR model was found to be inadequate for estimating the h values. These results might be useful for ice-cream industry aiming to control the rheological properties of their products added with different concentrations of xanthan gum because it may enable the ice-cream industry to previously estimate how the product viscosity would be before a large scale of production. Early prediction would also pave the way for the industry to save time and cost if it aims to produce a product with acceptable rheological properties. As a conclusion, ANFIS could be poposed to be the best model in order to estimate unmeasured or untested interval values of rheological properties of the icecream mixes added with different levels of xanthan gum.

[8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

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