Modular Feed Forward Networks to Predict Sugar Diffusivity from Date Pulp Part I. Model Validation

Abstract

In Tunisia, some 15,000 tons of fructose could be produced annually from second quality dates presently being left to rot. Extraction is the first step in producing sugar from these dates, and sugar diffusivity from the date paste governs the process. The objective of this project was therefore to measure in the laboratory, the sugar diffusivity of three date varieties (Manakher, Lemsi, and Alligue) under three different temperatures (50, 65 and 80°C), and from this data, develop an artificial neural network (ANN) model to predict sugar extraction. For each date variety, the laboratory procedure consisted of soaking a layer of date paste in water at one of the three temperatures and observing water sugar concentration at 20 mm from the date layer, every 15 min over a period of 240 min. This experimental data was then used to develop the ANN model, where several configurations were evaluated. Date sugar concentration with time was found to be significantly influenced by temperature and variety. The Lemsi variety allowed for the highest sugar extraction of 75% at 80°C. The optimal ANN model was found to be a network with two hidden layers and seven neurones in both the upper and lower levels of each hidden layers. This optimal model was capable of predicting sugar diffusivity from the different date varieties with a mean square error of 0.0037 and an 8.0% Error. The results show very good agreement between the predicted and the desired values of sugar diffusivity (R² = 0.98). The coefficient of determination was also very good (R² > 0.95), due to a small prediction error.

Keywords:

• Date pulp
• Sugar diffusivity
• Modelling
• ANN

INTRODUCTION

Date palm plantations occupy an important surface area in Southern Tunisia and represent thereafter a pillar for the regional and national economy. Tunisian palm plantations cover a surface of 22,500 ha with an annual production in clear evolution, increasing from 86,050 tons in 1993–1994 to 113,000 tons in 2005–2006.[1,2] Nevertheless, this tonnage represents only the best quality of dates consumed fresh, while another important quantity is left to rot because of its lower quality. Nevertheless, these non-consumed dates contain appreciable amounts of nutritive sugars such as fructose. The annual mass of non-consumed dates produced in Tunisia alone is estimated at 43,300 tons which contain 30,000 tons of sugar. The sugar contained in these lower quality dates could be exploited following the development of extraction and sugar separation technologies.

The first step in exploiting the sugar contained in dates is an extraction process, generally consisting of soaking the date pulp in water at an optimized temperature and for a specific period. The parameter governing this extraction process is the diffusivity of the date pulp sugar, which varies depending on the fibre content of the variety and the temperature of the water. Sugar diffusivity (D_S ) must therefore be maximised to optimise this initial extraction process.

Several models were proposed to predict moisture and sugar diffusivity from fruits under specific conditions. A finite element model was used to predict moisture diffusivity from a granular material.[3] The effective diffusion coefficient of hydrated starch material was calculated from data obtained during drying experiments, under an air velocity of 2 m/s, and at an air temperature of 60 to 100°C and a relative humidity of 5 to 10%. The effective moisture diffusivity was calculated by the slopes method, which is based on the solution of Fick's equation for non-steady-state conditions.[3] A finite element model was developed to determine moisture diffusion from the apple cultivars ‘Jonagold’ and ‘Elstar.’ For this purpose, geometrical models based on microscopic cuticle images were used.[4] Recently, an estimation procedure for the effective diffusivity of moisture from pear tissues used a numerical water diffusion model.[5] The moisture diffusivity of different tissues like that of the outer cortex, inner cortex, and cuticle was estimated as a function of picking date and temperature. However, modelling by means of finite element methods requires much computing time. To calculate the diffusion of sugar, simpler methods have therefore been investigated such as the artificial neural network (ANN) models, which are now known as powerful data-modeling tools.[6] The major benefits of such a technique include modelling without any assumptions about the nature of the phenomological mechanisms underlying the process; the ability to learn linear and nonlinear relationships between variables and directly from a set of examples; the capacity of modeling multiple outputs simultaneously and; a reasonable application of the model to unlearned data.[6]

The ANN technique was applied to food science and technology. In fact, some ANNs were developed to predict flavor intensity in blackcurrant concentrates. The ANN prediction was based on gas chromatographic data for 133 flavor component extracts from blackcurrant concentrates varying with seasons, geographical origin and processing technology.[7] Other artificial neural network (ANN) models were used to predict fruit juice viscosity (orange, peach, pear, malus floribunda, and blackcurrant) as a function of solids concentration and temperature.[8] In addition, an ANN model was used to predict the heat and mass transfer during the drying of cassava and mango. Such model provided on-line predictions of temperature and moisture kinetics during the drying process.[9] A neural network approach was used also to predict the specific heat, the thermal conductivity, and the density of milk[10] and predict the effect of milk constituents (protein, lactose, and fat) and temperature on the electrical conductivity of recombined milk.[11]

Recently, Ochoa-Martinez and Ayala-Aponte developed an ANN model to predict the water loss and solids gain for apples cut in cylinders and subjected to osmotic dehydration.[6] This investigation considered six processing variables, namely temperature and concentration of the osmotic solution, immersion time, surface area, solution to fruit mass ratio and agitation level. Very little work has been conducted on the measurement and prediction of sugar diffusivity from date paste. Because such prediction could optimize sugar extraction from dates, the objectives of the present research were to measure sugar diffusion from date paste in the laboratory and use this data to develop an ANN model to predict sugar diffusivity as a function of date variety, temperature, and diffusion period.

THEORY

Sugar diffusivity, D_S , is calculated based on Fisk's second law for unidirectional unsteady state diffusion:

(1)

where t is time (s); x is distance (cm); C (x, t) is the concentration of sugar at distance x and time t (g cm⁻³); and D_S is the diffusivity coefficient in (cm² s⁻¹). At time t = 0 s, where Q is the initial quantity of sugar per unit surface area (g cm⁻¹) in the date paste and the delta function, δ, is the Dirac distribution. Although the Dirac distribution assumes that the date paste layer has no thickness, the present analysis is assuming that this thickness is negligible with respect to the diffusion distance. Therefore, at any time, mass conservation must be respected and:

(2)

Eq. (2) can be solved as follow, when a sugar rich surface is exposed to water on both sides[16] (Fig. 1):

(3)

Figure 1 Experimental device where the volume of water measures 34 mm in height, 37 mm in width and 113 mm in length. The layer of date paste weights 20 g and was about 5 mm thick. Sugar concentration was measured at a distance of 20 mm from the date pulp layer surface and at a depth of 17 mm. (Figure provided in color online.)

To solve Eq. (3), experimental values of the concentrations are plotted against time, as the curve of as function of time :

(4)

In this project, the Table curve 2D Software[18] was used to determine the value of D_S from Eq. (4) using the Non-linear-Least-Square fitting method. The performance of the Table Curve 2D Software was measured by calculating the coefficient of determination R ².[19]

MATERIALS AND METHODS

The development of an ANN model involves: the generation of data required for the training/testing of the model; the actual training/testing of the ANN model; the evaluation of the ANN configuration leading to the selection of the optimal configuration; and the validation of the optimal ANN model with a data set other than that used for training.

Data Generation

The three varieties of dates used, Menakher, Lemsi, and Alligue, contained, respectively, 540, 580, and 800 g of sugar/kg of wet pulp. The initial sugar content of date pulp for each variety was determined by hydrolyzing the date pulp sucrose into glucose and fructose and using the Lane-Eynon method to measure these two sugars. Initially, date pulp contains reducing sugars (glucose and fructose) with some non-reducing sugars such as sucrose.[17] The Lane-Eynon method consisted of a colorimetric quantification using a copper sulfate solution and methylene blue indicator. A calibration curve was used before hand with a series of standard solutions of known sugar concentration.

The experimental apparatus used to measure D_S is described in Fig. 1. This apparatus consisted of a rectangular plastic box measuring 34 mm in height by 37 mm in width and 113 mm in length. The box temperature was kept constant by immerging in a water bath (Lauda 100, Germany). For each variety, 20 g samples of dates were crushed into a homogeneous paste using a food processor, rolled into a flat layer placed in the middle of the plastic box and gently immersed with water at the set experimental temperature. A net maintained the shape of layer of date pulp once inside the apparatus.

Sugar diffusion was assumed to occur symmetrically on both sides of the date pulp layer. Every 15 min, over a period of 240 min, a 0.1 ml sample of water was collected at a distance of 20 mm from the date pulp layer and at a water depth of 17 mm. The sugar concentration was assumed uniform over the face of the box, because of the cubic configuration of the setup. The sugar content of all samples was measured using a digital refractometer (model PA201, MISCO, Cleveland, USA) with a reading error of +/− 0.1 Brix.

Triplicate tests were repeated at random, using all three varieties and, for each variety, three water temperatures of 50, 65, and 80°C. The data sets were obtained from a full factorial experimental design for three factors. The effect of date variety and water solution temperature on sugar concentration at 20 mm from the date paste layer was analyzed using the method of General Linear Model Univariate Analysis. The Table curve 2D Software was used to determine the value of D_S from the sugar concentration data plotted against time as illustrated by Eq. (4) using the Non-linear-Least-Square fitting method. A non-linear fitting consists of an iterative procedure that begins with an initial set of estimates for the parameters. Although there are a variety of methods used to converge towards the minimum least squares solution, all procedures must compute a point-by-point sum of squared residuals for each iteration's set of coefficients. Table Curve 2D uses the Levenburg-Marquardt algorithm for fitting its non-linear equations and user-defined functions. Although this algorithm requires a matrix inversion and the computation of partial derivatives for individual iterations, its rate of convergence is among the best of available methods.[19] Sugar diffusivity, D_S , was the variable analyzed throughout the statistical procedure. A total of 161 Ds values were obtained and they ranged from 1.26 × 10⁻⁷ to 9.57 × 10⁻⁷ cm²/s.

Neural Network Selection

An artificial neural network (ANN) is a computational structure inspired by biological neural systems.[12] Among the many neural network models proposed, the NeuroSolutions commercial software was used to develop the ANN model and more specifically, the Modular Feed Forward networks (MFF) was selected because of its special classes of Multilayer Perceptrons (MLP) where layers are segmented into modules.[13] These networks process their inputs using several parallel MLP and then recombine the results. This operation creates a structure within the topology, which fosters specialization of functions in each sub-module. Modular Feed Forward (MFF) networks do not have full interconnectivity between the layers. Therefore, a smaller number of weights are required for the same size network or the same number of Processing Elements (PEs). This tends to speed the training and reduce the number of examples needed to train the network for the same degree of accuracy.[12]

To select the number of hidden layers and the number of processing elements (neurons) in the hidden layers, a trial and error procedure is conducted to reach the required behaviour. In the present study, the ranges of settings for the main configuration parameters are shown in Table 1. The optimal configuration was found using 1 and 2 hidden layers, with a range of 4 to 20 neurons in each hidden layers, and 1000–10,000 learning runs. In this study, the performance of the ANN model was tested using two different neural network topologies described in Fig. 2a, b.

Table 1 Main configuration parameters and their levels of neural networks used to predict D _S , the sugar diffusion coefficient

Factors	Levels
Learning rule	Momentum
Transfer	TanH  Sigmoid Bias
Number of hidden layer	1 and 2
Number of neurons in upper level	2 to 10
Number of neurons in lower level	2 to 10
Neural topology (Fig. 2)	I and II
Learning runs	1000 to 10,000

Figure 2 The neural network topologies tested. (a) Neural network topologies I, and (b) Neural network topologies II.

Training

Once the ANN architecture was defined, the training was initiated and repeated several times to get the best performance.[6] The training, cross validating and testing of the model used 81, 48, and 32 experimental data points, respectively, representing 50, 30, and 20% of the complete data set. Cross validation is highly recommended to stop network training because it monitors the error using an independent set of data and stops the training when this error begins to increase. This is considered the point of best generalization.

The model weights are frozen once the network is trained and the testing set is fed into the network to compare its output with the desired output.[12] The error minimization process is achieved using the momentum rule, which is an improvement as compare to the straight gradient descent since a memory term (the past increment to the weight) is used to speed up the process and stabilize convergence.

Selection of Optimal Configuration

For each specified problem, the following neural network parameters must be optimized: number of hidden layers, number of neurons in each hidden layer, and number of learning runs. The optimum configuration was decided based on minimizing the difference between the neural network and the desired outputs. The performance of the various ANN configurations were compared using: the mean squared error (MSE) and the % Error; the Akaike information criterion (AIC), which measures the trade-off between training performance and network size, and; the MDL criterion (minimum description length) which is similar to the AIC in that it tries to combine the model's error with the number of degrees of freedom to determine the level of generalization. The goal is to minimize respectively the ACI and MDL terms to produce a network with the best generalization. The coefficient of determination, R ², of the linear regression line between the values predicted by the neural network model and the desired output was also used as a measure of performance. The MSE, AIC, MDL and R ² equations used to compare the performance of various ANN configurations are:

(5)

(6)

(7)

(8)

where n is the number of exemplars of the training set, C_D and C_P are the desired and predicted values of sugar concentration, respectively and k is the number of network weights. The coefficients RSS and TSS represent the regression sum of squares and the total sum of squares, and are defined respectively as:

(9)

(10)

where and are the means of the observed data (Y_i ) and predicted values (f_i ), respectively.

RESULTS AND DISCUSSION

Sugar Diffusivity

The sugar diffusion data for all three varieties Menakher, Lemsi and Alligue is presented in Figs. 3a–c, respectively. All tests, using the three varieties exposed to the different temperatures, show three phases of sugar concentration changing with time: the latency phase with a sugar concentration of zero, which represents the time it takes for the sugar molecules to reach the point of measurement; the full diffusion phase with a sugar concentration increasing steadily with time which represents the transfer of sugar from the date paste to the water solution, and; the third final phase where the sugar concentration increases very slowly which represents a point close to the sugar concentration equilibrium between the date paste and the water solution. The beginning of this third phase indicates that the diffusion process has practically reached its optimal value.

Figure 3 Sugar concentrations obtained over time with (a) Menakher, (b) Lemsi, and (c) Alligue date varieties, at a distance of 20 mm from the date paste layer.

The change in sugar concentration was significantly affected by date variety with the effect of temperature varying with variety (p < 0.01). With the Menakher variety, temperature had little effect on sugar diffusion because all three concentration curves overlapped with time. The only temperature effect was the shorter latency phase of 30 min obtained at 80°C as compared to that of 45 min obtained at 50 and 65°C. With the Lemsi variety, temperatures of 65 and 80°C produced similar sugar concentrations with time, while at 50°C, sugar concentrations were lower indicating a lower sugar diffusivity. All three temperatures produced a 45 min latency phase. For the Alligue variety, temperatures of 65 and 80°C also produced similar sugar concentrations with time, while at 50°C, sugar concentrations were lower indicating a lower sugar diffusivity. Nevertheless, all three temperatures of 50, 65, and 80°C produced a different latency phase of 60, 45, and 30 min, respectively.

The resulting sugar concentrations differed significantly between date varieties (p < 0.01; Table 2). The Menakher variety produced a maximum sugar concentration of 40 g/L, from a date paste containing 540 g/kg (wet basis), after 180, 210, and 240 min for temperatures of 50, 65, and 80°C, respectively, where a relatively stable sugar concentration had been reached at 80 and 65°C but not at 50°C. The Lemsi variety produced a maximum sugar concentration of 30, 55, and 60 g/L, from a date paste containing 580 g/kg (wet basis), after 210, 225, and 225 min for temperatures of 50, 65, and 80°C, respectively, where all conditions had reached a relatively stable value. The Alligue varieties produced a maximum sugar concentration of 30, 55, and 60 g/L, from a date paste containing 800 g/kg (wet basis), after 225, 225, and 240 min for temperatures of 50, 65, and 80°C, respectively, where a relatively stable sugar concentration had been reached at 65 and 50°C but not at 80°C.

Table 2 Statistical analysis of the effect of temperature, variety and time on water sugar concentration

Source	Type III sum of squares	df	Mean square	F	Sig.
Corrected model	118,415.495(a)	143	828.080	239.085	.000
Intercept	256,303.255	1	256,303.255	74,000.338	.000
Variety	750.260	2	375.130	108.308	.000
Temperature	12,056.510	2	6028.255	1740.489	.000
Time	95,682.161	15	6378.811	1841.702	.000
Variety * Temperature	3431.771	4	857.943	247.707	.000
Variety * time	1058.073	30	35.269	10.183	.000
Temperature * Time	3626.823	30	120.894	34.905	.000
Variety * Temperature * Time	1809.896	60	30.165	8.709	.000
Error	997.500	288	3.464
Total	375,716.250	432
Corrected Total	119,412.995	431
Note: R^2 = 0.992; adjusted R^2 = 0.987. The effect was tested using the method of General Linear Model Univariate Analysis.

The statistical analysis presented in Table 2 indicates that temperature had the most significant effect on the sugar diffusivity followed by time and then variety. Temperature affects the viscosity of water, which impacts the drag on the moving solute molecules,[14] and increases the rate of sugar diffusion from the date paste, as molecules move more rapidly.[15] Time affected sugar diffusivity likely for two reasons: after 180 and 240 min, the solution sugar concentration had not reached a stable value yet, and; the structure of the date pulp tissues was changing with loss of sugar.

Considering the quantity of water (130 mL) and the mass of date pulp (20 g) used for the extraction, the efficiency of the extraction process was computed after 240 min using the sugar concentration values measured over time at a distance of 20 mm from the date paste layer. In calculating the total mass of sugar dissolved, it was assumed that the sugar concentration beyond 20 mm respected the trend observed up to this distance. For the Menakher variety, 55, 60, and 65% of the sugar was extracted at 50, 65 and 80°C, respectively. For the Lemsi variety, 40, 70, and 75% of the sugar was extracted at temperatures of 50, 65, and 80°C, respectively. For the Alligue variety, 30, 50, and 55% of the sugar was extracted at temperatures of 50, 65, and 80°C, respectively.

Although the highest concentration of sugar was obtained from the varieties Alligue and Lemsi (60 g/L after 225 to 240 min), as compared to that from Menakher (40 g/L after 180 to 240 min), the Lemsi variety lead to the highest exchange percentage of sugar. As compared to the Alligue variety with a paste sugar concentration of 800 g/kg, the Lemsi variety had a lower paste sugar concentration of 580 g/kg; furthermore, some lower quality date varieties such as Alligue and Menakher are more fibrous than others, and thus less suitable for sugar extraction. A higher sugar extraction percentage would be obtained with all varieties if the extraction time had been extended beyond 240 min.

The data set was used to compute the sugar diffusivity coefficient D_S obtained for all three varieties and temperatures (Table 3). Although sugar concentrations were influenced by variety, temperature and time, the diffusion coefficient D_S ranged from 1.26 × 10⁻⁷ to 1.05 × 10⁻⁶ cm²/s. The observed date paste diffusivity values were similar to the range of 1.12 × 10⁻⁷ to 4.33 × 10⁻⁸cm²/s observed for apple pulp.[4]

Table 3 Sugar diffusivity coefficient D _S by variety, diffusion time and temperature

Variety	Diffusion duration (min)	Temperature (°C)	Sugar diffusivity Ds (cm^2/s)
Menakher	180	50	3.32 × 10^−7
Menakher	180	65	3.51 × 10^−7
Menakher	180	80	5.42 × 10^−7
Menakher	240	50	4.47 × 10^−7
Menakher	240	65	4.99 × 10^−7
Menakher	240	80	6.31 × 10^−7
Lemsi	180	50	2.01 × 10^−7
Lemsi	180	65	6.75 × 10^−7
Lemsi	180	80	7.64 × 10^−7
Lemsi	240	50	2.82 × 10^−7
Lemsi	240	65	9.24 × 10^−7
Lemsi	240	80	10.5 × 10^−7
Alligue	180	50	1.26 × 10^−7
Alligue	180	65	6.73 × 10^−7
Alligue	180	80	7.22 × 10^−7
Alligue	240	50	2.17 × 10^−7
Alligue	240	65	9.33 × 10^−7
Alligue	240	80	9.57 × 10^−7

Artificial Neural Network Performance

The ANN model was tested and developed to predict sugar diffusivity as a function of variety, water temperature and extraction time. The performance of ANN configuration was evaluated several times using the data set and the various configurations (Tables 4 and 5). The ANN configuration that minimized the MSE value and the % Error, and that optimized R ², were considered to be optimal. The verification of the ANN model performance is illustrated in Figs. 4a and b. Although the networks with Topology I gave good performances (Table 4), two networks for Topology II were selected as the best. The best ANN configuration from the network topology II used two hidden layers, with seven neurones in both the upper and lower levels (Fig. 4a). The MSE and % Error for this optimal configuration were 0.0037 and 8.05%, respectively. The results showed very good agreement between the predicted and the desired values of sugar diffusivity (R ² = 0.98). The coefficient of determination was also very good (R ² > 0.95), as a result of the small prediction error.

Table 4 Performances of various ANN configurations once trained with the data set, for the neural network topology I

	Processing elements (PEs) in each Hl
Hidden layer (Hl) number	Upper PEs	Lower PEs	Transfer	MSE	% Error	AIC	MDL	R^2
1	2	2	TanH	0.238	47.64	282.83	109.51	0.006
1	3	3	TanH	0.015	15.609	269.33	215.637	0.968
1	4	4	TanH	0.342	39.56	396.63	156.38	0.037
1	5	5	TanH	0.013	15.02	333.422	270.378	0.921
1	6	6	TanH	1.08	31.68	370.38	149.26	0.229
1	7	7	TanH	0.086	35.81	323.71	270.019	0.501
1	8	8	TanH	0.284	41.26	171.716	65.33	0.047
1	9	9	TanH	0.082	29.850	311.983	203.971	0.504
1	10	10	TanH	0.583	25.37	369.305	146.983	0.504
2	2	2	TanH	0.276	34.21	307.56	119.908	0.582
2	3	3	TanH	0.006	11.34	1778.63	1519.2	0.96
2	4	4	TanH	0.257	24.45	671.22	266.02	0.693
2	5	5	TanH	0.007	30.21	1734.94	1492.39	0.505
2	6	6	TanH	0.251	21.36	587.089	232.091	0.722
2	7	7	TanH	0.013	14.37	2300.66	1573.42	0.940
2	8	8	TanH	0.257	26.12	529.21	208.87	0.672
2	9	9	TanH	0.005	9.62	1641.388	1399.635	0.972
2	10	10	TanH	0.286	31.54	325.74	127.331	0.452
1	2	2	Sigmoid	0.076	30.66	9.149	−3.998	0.045
1	3	3	Sigmoid	0.026	30.36	373.639	241.079	0.409
1	4	4	Sigmoid	0.072	33.08	42.899	9.431	0.064
1	5	5	Sigmoid	0.022	29.78	−69.83	−76.78	0.313
1	6	6	Sigmoid	0.074	32.8	9.03	−4.118	0.046
1	7	7	Sigmoid	0.022	45.69	−99.722	−102.66	0.320
1	8	8	Sigmoid	0.078	33.08	79.267	24.285	0.062
1	9	9	Sigmoid	0.021	23.08	139.41	104.416	0.303
1	10	10	Sigmoid	0.074	35.97	69.012	20.006	0.030
2	2	2	Sigmoid	0.075	33.26	1245.098	493.267	0.064
2	3	3	Sigmoid	0.022	32.77	181.933	141.32	0.483
2	4	4	Sigmoid	0.075	31.45	1841.069	733.034	0.096
2	5	5	Sigmoid	0.024	35.079	483.151	402.745	0.398
2	6	6	Sigmoid	0.22	32.899	327.962	267.962	0.462
2	7	7	Sigmoid	0.221	32.75	306.051	248.884	0.467
2	8	8	Sigmoid	0.022	32.908	469.92	390.853	0.467
2	9	9	Sigmoid	0.022	33.197	1534.016	1312.83	0.452
2	10	10	Sigmoid	0.0222	32.93	1106.176	942.157	0.449
1	2	2	Bias	0.101	35.319	−51.246	−54.185	0.356
1	3	3	Bias	0.155	51.261	−45.65	−47.519	0.106
1	4	4	Bias	0.101	35.319	−51.246	−54.185	0.356
1	5	5	Bias	0.101	35.32	−51.246	−54.185	0.356
1	6	6	Bias	0.101	35.319	−51.246	−54.185	0.356
1	7	7	Bias	0.101	35.319	−51.246	−54.185	0.356
1	8	8	Bias	0.101	35.318	−51.246	−54.185	0.356
1	9	9	Bias	0.155	35.223	−51.201	−53.605	0.270
1	10	10	Bias	0.155	51.261	−45.65	−47.519	0.106
2	2	2	Bias	0.101	35.349	−43.232	−47.239	0.356
2	3	3	Bias	0.101	35.476	−43.24	−47.25	0.356
2	4	4	Bias	0.101	35.335	−23.231	−29.909	0.356
2	5	5	Bias	0.101	35.476	−43.24	−47.25	0.356
2	6	6	Bias	0.114	35.570	−43.24	−46.714	0.269
2	7	7	Bias	0.006	10.005	820.184	689.289	0.956
2	8	8	Bias	0.006	8.678	1174.805	996.093	0.964
2	9	9	Bias	0.101	35.476	−43.24	−47.25	0.356
2	10	10	Bias	0.101	35.476	−43.244	−47.251	0.356

Table 5 Performances of various ANN configurations once trained with the data set, for the neural network topology II

	Processing elements (PEs) in each Hl
Hidden layer (Hl) number	Upper PEs	Lower PEs	Transfer	MSE	% Error	AIC	MDL	R^2
1	2	2	TanH	0.089	37.11	140.707	111.59	0.504
1	3	3	TanH	0.025	19.54	304.56	248.19	0.874
1	4	4	TanH	0.008	11.35	119.61	83.286	0.944
1	5	5	TanH	0.093	33.39	62.046	43.614	0.426
1	6	6	TanH	0.095	26.575	263	217.855	0.393
1	7	7	TanH	0.014	16.031	347.086	282.707	0.912
1	8	8	TanH	0.093	36.894	252.088	163.1	0.418
1	9	9	TanH	0.01	12.17	296.823	237.78	0.946
1	10	10	TanH	0.008	13.01	250.094	196.4	0.944
2	2	2	TanH	0.094	26.86	1104.518	946.91	0.399
2	3	3	TanH	0.007	10.75	2232.4	1913.17	0.974
2	4	4	TanH	0.009	3.85	1864.29	1585.41	0.994
2	5	5	TanH	0.007	11.09	1109.27	940.18	0.952
2	6	6	TanH	0.09	33.92	2249.003	1938.32	0.396
2	7	7	TanH	0.0037	8.05	702.979	585.174	0.98
2	8	8	TanH	0.01	13.04	726.935	610.66	0.931
2	9	9	TanH	0.025	20.5	1639.45	1118.42	0.846
2	10	10	TanH	0.004	8.092	758.8	634.05	0.974
1	2	2	Sigmoid	0.022	33.267	−99.786	−102.72	0.438
1	3	3	Sigmoid	0.022	24.038	−18.192	−28.755	0.271
1	4	4	Sigmoid	0.022	32.57	−99.801	−102.73	0.435
1	5	5	Sigmoid	0.022	33.14	−100.016	−102.95	0.443
1	6	6	Sigmoid	0.021	32.97	−50.15	−59.773	0.446
1	7	7	Sigmoid	0.022	33.267	−89.932	−94.206	0.444
1	8	8	Sigmoid	0.021	33.639	−80.319	−85.929	0.454
1	9	9	Sigmoid	0.022	33.34	−100.324	−103.263	0.457
1	10	10	Sigmoid	0.025	36.26	−98.8	−101.209	0.348
2	2	2	Sigmoid	0.021	32.92	337.612	276.172	0.451
2	3	3	Sigmoid	0.022	32.175	−9.76	−24.719	0.446
2	4	4	Sigmoid	0.021	33.251	335.119	273.94	0.458
2	5	5	Sigmoid	0.021	32.69	479.18	398.76	0.455
2	6	6	Sigmoid	0.015	30.905	696.425	345.923	0.197
2	7	7	Sigmoid	0.022	32.469	73.929	47.75	0.455
2	8	8	Sigmoid	0.022	32.782	115.785	83.996	0.446
2	9	9	Sigmoid	0.027	32	1006.29	679.801	0.4
2	10	10	Sigmoid	0.021	33.069	173.502	133.966	0.455
1	2	2	Bias	0.101	35.31	−51.246	−54.185	0.356
1	3	3	Bias	0.101	35.32	38.75	23.79	0.356
1	4	4	Bias	0.101	35.31	−51.246	−54.185	0.356
1	5	5	Bias	0.101	35.31	−51.246	−54.185	0.356
1	6	6	Bias	0.115	35.22	28.798	15.709	0.269
1	7	7	Bias	0.101	35.319	−1.246	−10.863	0.356
1	8	8	Bias	0.101	35.32	158.752	127.765	0.356
1	9	9	Bias	0.101	36.455	108.673	84.364	0.356
1	10	10	Bias	0.101	35.31	−51.246	−54.185	0.356
2	3	3	Bias	0.101	35.476	−43.24	−47.25	0.356
2	4	4	Bias	0.101	35.335	−23.231	−29.909	0.356
2	5	5	Bias	0.101	35.476	−43.24	−47.25	0.356
2	6	6	Bias	0.114	25. 750	−134.24	−146.714	0.612
2	7	7	Bias	0.006	10.005	820.184	689.289	0.956
2	8	8	Bias	0.006	8.678	1174.805	996.093	0.964
2	9	9	Bias	0.101	35.476	−43.24	−47.25	0.356
2	10	10	Bias	0.101	35.476	−43.244	−47.251	0.356

Figure 4 Correlation of desired versus neural network values of sugar diffusivity coefficient D_S after testing the data set (a) for the ANN with a network topology II, 2 hidden layers, and four neurons in both the upper and lower hidden layers, and (b) for the ANN with network topology II, 2 hidden layers, and seven neurons in both the upper and lower hidden layers.

The second best ANN model was obtained with the network topology II with two hidden layers, and four neurones in both the upper and lower level for each layer (Fig. 4b). This ANN model also demonstrated very good agreement between the predicted and the desired values of sugar diffusivity (R² = 0.99) but the MSE was larger as compared to the previously selected model (0.009 > 0.0037), and in addition the AIC and MDL were also larger.

CONCLUSION

The change in sugar concentration with time and in water next to the date pulp was found to vary with temperature and date variety. Time also influenced sugar diffusivity because a stable sugar concentration had not been reached after 180 and 240 min of testing, and also because the structure of the date paste was changing with sugar loss. A sugar extraction efficiency of 75% was obtained with the Lemsi variety after 240 min and at 80°C, while under the same conditions, only 50 and 65% of the sugar was extracted from both the Menakher and Alligue varieties, respectively. Nevertheless, sugar diffusivity values computed from the change in sugar concentration with time ranged from 4.5 to 10 × 10⁻⁷ cm²/s. A neural network based model was developed for the prediction of sugar diffusivity from the dates under a range of temperatures and diffusion periods. The optimal model, which consisted of two hidden layers with seven neurones in both the upper and lower levels in each layer, was able to predict sugar diffusivity values with a MSE of 0.0037 and 8% Error.

ACKNOWLEDGMENTS

The authors wish to thank the High Education and Research Ministry of Tunisia for the financial support to this project. The Natural Science and Engineering Research Council of Canada financed the participation of Dr. Suzelle Barrington.