HYBRID NEURAL MODELING OF THE ELECTRICAL CONDUCTIVITY PROPERTY OF RECOMBINED MILK

Abstract

This paper focuses on modeling the electrical conductivity of recombined milk by a hybrid neural modeling technique. It aims to establish a model that accounts for the effect of milk constituents (protein, lactose, and fat) and temperature on the electrical conductivity of recombined milk. Such a model should provide physical insight to the underlying relationship, in addition to its high precision.

A hybrid neural model was established by combining a mechanistic model to explain the major interrelation and an ANN model to deal with the difference and noise. Two mechanistic models and two 4-layer ANN models were developed. The best mechanistic model in terms of the smallest sum square error (SSE) combined a linear equation describing the effect of milk component concentration and a non-linear equation describing the effect of temperature. For this model, the correlation coefficient between the actual electrical conductivity and the modelled electrical conductivity was 0.9878 and SSE was 1.3376. Combining it with the 4-layer ANN model that produced the smallest SSE, the resulted hybrid neural model provided the best performance, with a correlation coefficient of 0.9982 between the actual electrical conductivity and the modelled electrical conductivity and a SSE of 0.1410.

Abbreviations: ANN, artificial neural network; EC, electrical conductivity (mS/cm); c, concentration (mol.m⁻³); D, diffusion coefficient (m².s⁻¹); F, Faraday constant (A.s.mol⁻¹); ΔG, Gibbs free energy of activation for the reaction (kJ.mol⁻¹); K, specific conductivity (Ω⁻¹.m⁻¹); k, dissociation constant (mol.m⁻³); R, ideal gas constant (J.K⁻¹.mol⁻¹); T, temperature (K); Λ₀, molar conductivity at infinite dilution (Ω⁻¹.m².mol⁻¹); Λ_c, molar conductivity at finite dilution (Ω⁻¹.m².mol⁻¹); z _i , charge number

INTRODUCTION

Electrical conductivity is a property of a solution involving the movement of ions and the electron transportation to complete the current path.[1] It has been used widely in dairy industry. The electrical conductivity is one of the testing methods used to determine quantities such as soluble salts[2], protein content in whey powder[3], and casein content during renneting[4]. It can also be used as a diagnostic tool for intramammary infection[5]. Electrical conductivity measurement has long been used during cleaning-in-place (CIP) as a quality control indicator.

Typically the electrical conductivity of milk at 25°C is in the range from 4.0 to 5.5 mS/cm.[6] It increases with temperature and ion concentration in milk[1], so protein has a positive influence on it.[7] On the other hand, it decreases as the concentrations of fat and lactose increase because their structures cannot conduct current.[8]

Although there were many studies on the effect of a single factor (such as temperature, protein, lactose) on electrical conductivity,8-10, [7]> there was generally a lack of study on the comprehensive effect of temperature and milk constituents on electrical conductivity, particularly their interactions. Consequently, Hutabarat[11] studied the electrical conductivity property of recombined milk at various temperatures and milk concentrations by applying multiple regression technique. It was found that the electrical conductivity change was caused by the variation in temperature and milk constituents, which include protein, fat, and lactose. Furthennore, the interaction between the milk constituents and temperature was non-linear.

Due to its superb capability of learning non-linear relationships from experimental data, feed-forward artificial neural network (ANN) with backpropagation training was used in our previous study to establish a non-linear model for the electrical conductivity of recombined milk.[12] Various ANN models of 4 layers (1 input layer, 2 hidden layers and 1 output layer) and 3 layers (1 input layer, 1 hidden layer and 1 output layer) were developed. It was found that the performance of the 4-layer models was better than that of the 3-layer ones. The sum square error (SSE) of the best ANN model was only 0.4864 and the correlation coefficient between the actual electrical conductivity and the modelled electrical conductivity was as high as 0.9937. However, the model provided no physical insight to the relationship, because ANN is a black box approach expressing a relationship in the form of numeric weights and bias. This disadvantage can be alleviated by hybrid neural modelling, which integrates an ANN model with a mechanistic model.[13] The integration reduces the disadvantages of both ANN modelling and mechanistic modelling while combining their advantages. ANN is able to deal with fuzzy or noisy data that are too complex to be explained by a mechanistic model. Meanwhile, the mechanistic model is able to explain the interrelation in the black box of ANN model. Therefore, hybrid model is proved to be superior to both ANN model and mechanistic model.[14] [15] [16]

The aim of this study was to model the relationship between electrical conductivity and milk constituents and temperature by using hybrid neural modelling. The new model was expected to provide more explicit physical significance than the previously established ANN model.[12]

The experimental data used in this study were taken from Hutabarat,[11] who conducted the experiments by a 2×4×4×4 (fat×protein×lactose×temperature) factorial design. A total of 32 samples of recombined milk were prepared. The fat level of the samples was at 3% and 6%. The protein level was at 1%, 2%, 3%, and 4%. The lactose level was at 4%, 6%, 8%, and 10%. The electrical conductivity of the samples was measured by a Microprocessor 900C electrical conductivity meter at 50°C, 55°C, 60°C, and 65°C. Consequently, 128 experimental data sets were produced. Each set consisted of fat content, protein content, lactose content, temperature, and electrical conductivity value.

The rest of this paper is organised as follows. Firstly, the structure of hybrid neural model will be given in the next section. Then the results of modelling including mechanistic modelling and hybrid neural modelling are presented together with discussions. Conclusions are provided at the end.

STRUCTURE OF HYBRID NEURAL MODELS

There are two typical structures for hybrid neural models, namely serial structure and parallel structure. In the serial structure, an ANN model produces its output that becomes the input of a mechanistic model. In the parallel structure, an ANN model and a mechanistic model produce their own outputs. The combination of the outputs is the final result.[14] The structure of the hybrid neural model for this study is shown in Figure 1. Input data including %protein, %lactose, %fat and temperature were introduced to both mechanistic part and ANN part of the hybrid model. The difference between the modelled electrical conductivity by the mechanistic model and the actual measured electrical conductivity was taken as the target output for training the ANN model. As a result, the ANN model provided the estimated difference, which was added to the estimated electrical conductivity by the mechanistic model. The mechanistic model provided physical insight to the major relationship between the electrical conductivity and milk constituents and temperature. The ANN model provided an estimation of the complicated and subtle interactions among the variables, which were not accounted by the mechanistic model. The electrical conductivity estimated by the hybrid model was compared to the actual electrical conductivity to verify the model performance in terms of SSE and correlation coefficient.

Structure of hybrid neural model.

MECHANISTIC MODELS

Two mechanistic models were developed, based on different electro-chemical theories.

The First Mechanistic Model

According to Nernst-Einstein Equation,[2] the influence of ion concentration on electrical conductivity is linear in fashion. Each ionic component in milk contributes to electrical conductivity at constant temperature as follows:

where K _i is specific conductivity, c _i is ion concentration, D _i is diffusion coefficient, z _i is charge number, F is the Faraday constant, R is the ideal gas constant, and T is temperature.

Taking (D _i z _i ² F ²)/RT as a temperature dependent coefficient, we have:

Let K _P , K _L , K _F , and K _O represent the contribution of protein, lactose, fat and other components to the electrical conductivity respectively, we have:

where A, B, C, and D are coefficients. Then the total conductivity becomes:

or

The 128 experimental data sets from[11] were used to calculate the coefficients by the least square regression (LSR) method. At each constant temperature, the coefficients A, B, and C were calculated for the molar concentration of protein, lactose and fat respectively. The constant D, which represents the electrical conductivity contributed by other components in milk, was also calculated. Results are presented in Table 1.

Electrical Conductivity at Different Temperatures in the First Mechanistic Model

T	Equation	R	SSE
50°C	K 50=791.0c P −2.6c L −1.5c F +2.5	0.9849	0.1712
55°C	K 55=851.9c P −2.9c L −1.6c F +2.7	0.9864	0.1792
60°C	K 60=918.2c P −3.1c L −2.2c F +3.0	0.9849	0.2317
65°C	K 65=976.3c P −3.2c L −2.1c F +3.1	0.9814	0.3221

In all equations in Table 1, the protein coefficients were positive while the lactose and fat coefficients were negative. This proves that protein can contribute to electrical conductivity whereas lactose and fat are only ion dillutants because they cannot conduct current. The constant coefficients were also positive. This term includes the conductivity contributed by other constituents including salts, the major conductors in milk. Due to the small SSEs and high correlation coefficients in all equations at various temperatures, the relationship between electrical conductivity and concentration of each component is reasonably concluded to be linear.

In fact, the coefficients embedded several parameters such as temperature that influences the activity of ions. As a result, with the same concentration, the electrical conductivity was different at different temperatures. This coincides with the claim in[2] that electrical conductivity is influenced by the activity of component, not its concentration. The influence of temperature on the concentration coefficients was proposed to follow Arrhenius type of equation as follows:

Again, LSR method was used for computing the values of the parameters. Results are given in Table 2.

Concentration Coefficients in the First Mechanistic Model

Component	Equation	R	SSE
Protein	A=+9.87×10^4exp(−1.54×10^3/T)	0.9994	7329.5
Lactose	B=−2.00×10^2exp(−1.40×10^3/T)	0.9501	0.0220
Fat	C=−1.99×10^4exp(−3.09×10^3/T)	0.8987	0.0941
Constant	D=+6.02×10^2exp(−1.77×10^3/T)	0.9969	0.0019

Substituting all equations in Table 2 into equation8, the first mechanistic model for the electrical conductivity of recombined milk was obtained as follows:

The corresponding R and SSE were 0.9878 and 1.337.

Based on this mechanistic model, the relationship between electrical conductivity and milk component concentration was linear whereas the relationship between electrical conductivity and temperature was non-linear. However, this model produced a much higher SSE and a much lower correlation coefficient than the ANN model in the previous study [12]. Therefore the model's accuracy is not acceptable and a hybrid model incorporating ANN will be developed to address this issue.

The Second Mechanistic Model

Kohlrausch Equation [2] describes the relationship between molar conductivity and ion concentration for weak electrolyte. It can be modified to express the relationship between electrical conductivity and ion concentration as follows:

where k is dissociation constant, Λ _c is molar conductivity at finite dilution, and Λ ₀ is molar conductivity at infinite dilution.

Taking Λ _c =K/c, equation14 can be rearranged to:

Therefore, the relationship between molar concentration and electrical conductivity is quadratic. For simplicity, we assume kΛ ₀≈2, equation15 becomes:

Taking 2Λ ₀ as a coefficient, we have:

Thus each milk component contributes to the electrical conductivity as follows:

The total conductivity becomes:

LSR was again used for solving the coefficients in the above non-linear equation. At constant temperature, the coefficients A, B, C, and D were calculated. Results are presented in Table 3.

Similar to the first mechanistic model, the protein coefficient was positive whereas the lactose and fat coefficients were negative at any temperature. Again, this strongly proves that protein can contribute to electrical conductivity whereas lactose and fat cannot. In fact, they diluted ion concentration in milk resulting in a decreased conductivity with increasing either fat or lactose concentration or both. The constant D was also positive. This is because other constituents also contribute to the electrical conductivity in milk. Compared to the linear equations in the first model, these non-linear equations provided slightly higher SSEs and lower correlation coefficients. However, the results were still reasonable.

Electrical Conductivity at Different Temperatures in the Second Mechanistic Model

T	Equation	R	SSE
50°C	(K ^2+2K)50=5978.3c P −20.0c L −11.2c F +11.2	0.9794	0.2346
55°C	(K ^2+2K)55=6852.3c P −23.7c L −12.9c F +13.0	0.9811	0.2511
60°C	(K ^2+2K)60=7868.8c P −26.7c L −18.7c F +14.8	0.9774	0.3505
65°C	(K ^2+2K)65=8879.6c P −28.9c L −19.4c F +16.4	0.9731	0.4688

The influence of temperature on the coefficients was again proposed to follow Arrhenius type of equation. New parameters were computed by LSR and given in Table 4.

Concentration Coefficients in the Second Mechanistic Model

Component	Equation	R	SSE
Protein	A=+4.86×10^7exp(−2.89×10^3/T)	0.9998	2170.6
Lactose	B=−8.08×10^4exp(−2.68×10^3/T)	0.9882	1.0748
Fat	C=−9.82×10^6exp(−4.42×10^3/T)	0.9459	5.3777
Constant	D=+5.99×10^4exp(−2.78×10^3/T)	0.9975	0.0764

Substituting all equations in Table 4 into equation22, the second mechanistic model for the electrical conductivity of recombined milk was obtained:

Solving the above second order algebraic equation yields:

where SQRT stands for square root function. This model presented a non-linear relationship between electrical conductivity and milk component concentration as well as a non-linear relationship between electrical conductivity and temperature. It produced a correlation coefficient of 0.9822 between the actual electrical conductivity and the modelled electrical conductivity with a SSE of 1.5656. The model performance was not significantly different from that of the first model. Therefore, the relationship between electrical conductivity and milk component concentration could be mechanistically described as being either linear or non-linear for the recombined milk studied (1–4% protein, 4–10% lactose, and 3–6% fat). The relationship between electrical conductivity and temperature was found to be nonlinear from both mechanistic models and can be described by an Arrhenius type of equation. However, both mechanistic models produced significantly higher SSE and lower correlation coefficient compared to the ANN models in[12].

HYBRID NEURAL MODELS

As the first and second mechanistic models produced comparable model performance, both models were selected to combine with ANN to produce hybrid models. The difference (i.e. the gap) between the actual electrical conductivity and the modelled electrical conductivity by the mechanistic models was calculated and used as the target output for training ANN, as shown in Figure 1. Therefore, for an ANN model, the inputs consisted of %protein, %lactose, %fat and temperature, while the output was only the difference between the actual electrical conductivity and the modelled electrical conductivity by the mechanistic model. Such an ANN model does not directly predict electrical conductivity but only the gap. It will be part of a final hybrid neural model.

The 128 experimental data from[11] were divided into two sets for developing ANN models, 90% for training and 10% for verification. The training set contained 115 pairs of input and output vectors and they were normalised to a dimensionless range from zero to one. Normalising data to a dimensionless range between zero and one generally helps to improve the training speed of an ANN. The training set was then used to train 4-layer feed-forward ANN models with back propagation algorithm. The 4-layer ANNs had four neurons in the input layer, four neurons in the first hidden layer, four neurons in the second hidden layers and one neuron in the output layer. All neurons had a bias and a log-sigmoid transfer function. Two ANN models were developed, one for each of the two mechanistic models. Table 5 shows the performance of the two ANN models after being trained with back propagation algorithm for 1×10⁶ epochs with an error criteria of 0.001. The model performance for verification is also shown in Table 5.

Performance of the Two ANN Models for Hybrid Neural Modelling

Models	Training	Verification
ANN model to be combined with the first mechanistic model (4-4-4-1 structure with log-sigmoid transfer function)	SSE=0.4826	SSE=0.5397
	MSE=0.0042	MSE=0.0415
	R=0.9278	R=0.9069
ANN model to be combined with the second mechanistic model (4-4-4-1 structure with log-sigmoid transfer function)	SSE=0.6308	SSE=0.0479
	MSE=0.0055	MSE=0.0037
	R=0.8988	R=0.9420

As shown in Figure 1, the first and second mechanistic models were then combined with the first and second ANN models, resulting in two hybrid neural models. The model performance of the two hybrid neural models is given in Table 6. Also given in Table 6 are the model performance of the two mechanistic models and the model performance of the ANN model in [12]. It is clear from this table that the hybrid neural models are much better than both the mechanistic models and the ANN model, as they have smaller sum square error (SSE), smaller mean square error (MSE), and higher correlation coefficient (R) between the actual electrical conductivity and the modelled electrical conductivity. Comparisons between the actual electrical conductivity and the modelled electrical conductivity by the first and the second hybrid neural models are shown in Figures 2 and 3 respectively.

Model Performance of Various EC Models

Performance	ANN Model[12]	The First Mechanistic Model	The Second Mechanistic Model	The First Hybrid Neural Model^1	The Second Hybrid Neural Model^2
SSE	0.4864	1.3376	1.5656	0.1410	0.2747
MSE	0.0038	0.0104	0.0122	0.0011	0.0021
R	0.9937	0.9878	0.9822	0.9982	0.9964

¹Combining the first mechanistic model and the first ANN model in Table 5.

²Combining the second mechanistic model and the second ANN model in Table 5.

Figure 2.Correlation between the actual EC and the modelled EC by the 1st hybrid neural model.

Figure 3. Correlation between the actual EC and the modelled EC by the 2nd hybrid neural model.

Among the three types of models in Table 6 (ANN model,[12] mechanistic models, and hybrid neural models), the mechanistic models produced the worst performance. The ANN model proved to be very effective, which demonstrated ANN's modelling potential over the conventional techniques, because it can learn from experience and can predict from recognised data patterns.[12] However, the hybrid models showed even better performance than the ANN model. Furthermore, the interrelation between electrical conductivity and concentration of milk constituents at different temperatures could be explained physically in the hybrid models. All these substantiated the great potential of hybrid neural modelling technique.

Compared to the second hybrid neural model, the mathematical structure of the first hybrid neural model was simpler while its performance was better. Consequently, the best model for the electrical conductivity of recombined milk was the first hybrid neural model. It described the major linear relationship between electrical conductivity and concentration as well as the non-linear relationship between electrical conductivity and temperature with the highest correlation coefficient (0.9982) and the smallest SSE (0.0011).

CONCLUSIONS

Relationship that accounts for the effects of milk composition and temperature on the electrical conductivity of recombined milk has been established by hybrid neural modelling. The hybrid neural model was obtained by combining a mechanistic model with a 4-layer ANN.

Within the hybrid neural model, its mechanistic model part can physically explain that the molar concentration of milk constituents affects the electrical conductivity in a linear fashion. Increasing the protein concentration increases the electrical conductivity whereas increasing the lactose and fat concentration decreases the electrical conductivity. Moreover, the influence of temperature is described by a non-linear function. The electrical conductivity increases as temperature increases.

The ANN model part within the hybrid neural model accounts for the difference between the measured electrical conductivity value and the value by the mechanistic model. This difference is due to many factors that are not covered by the mechanistic model. Some of these factors include the non-linear effects of milk composition, interactions among the milk constituents and temperature and measurement noise.

It has been shown that the hybrid neural model is superior to other models developed by mechanistic modelling only or by ANN modelling only. This research has demonstrated the great potential of hybrid neural modelling technique.

ACKNOWLEDGEMENT

Financial support to N. Therdthai from the Australian Agency for International Development (AusAID) is gratefully acknowledged.