Evolutionary algorithm in adaptive neuro-fuzzy inference system for modeling growth of foodborne fungi

Abstract

The formation of mycotoxins and potentially allergenic spores associated with fungal growth can cause spoilage of food and animal feed. This study integrated an improved genetic algorithm (IGA) in an adaptive neuro-fuzzy inference system (ANFIS) for predicting the presence of foodborne fungi and modeling their growth. The IGA enhanced the performance of the ANFIS model in predictive microbiology. Based on temperature, pH, and water quantity, the proposed IGA-ANFIS model can accurately predict the maximum specific growth rate of the ascomycetous fungus Monascus ruber. The model uses Gaussian membership functions to minimize the root-mean-square error, which was used as a performance index. Experiments verified that the prediction accuracy of the proposed IGA-ANFIS model is higher than those of existing neural network models and neural fuzzy network models.

Keywords

Adaptive neuro-fuzzy inference system genetic algorithm food mycology

1 Introduction

An adaptive neuro-fuzzy inference system (ANFIS) can effectively solve nonlinear mapping problems by using stipulated input–output data pairs [1 –15]. Integrating an improved genetic algorithm (IGA) in ANFIS obtains even higher prediction accuracy compared to a conventional ANFIS model [16, 17]. For example, Ho et al. [18] demonstrated that, for predicting the efficacy of a vancomycin regimen, an IGA-ANFIS was more accurate than a conventional ANFIS. Thus far, the use of IGA-ANFIS for modeling the growth of foodborne fungi has not been reported yet. Because they can survive heat treatments and can grow under reduced oxygen levels, fungi are a major cause of food spoilage. For example, spoilage of fruit such as olives can result from changes in pH and from the development of a mycelial mat on the surface. Temperature, pH, and water activity are generally considered the controlling factors in fermentation and subsequent storage of table olives [19]. Therefore, regulating the combination of these factors can effectively control the growth of fungi during storage [20].

Fungi that cause food spoilage not only have negative economic effects by rendering food products unsaleable, they also have negative health effects by raising the risk of disease. Therefore, the food manufacturing industry has begun to use newly developed prediction models during product development in order to assess the risk of illness caused by foodborne microbes such as fungi and to enable rapid and cost-effective assessment of microbial growth [21, 22]. To ensure high food quality and safety, researchers in the food industry require effective tools for predicting fungal growth [23, 24].

Panagou and Kodogiannis [25] modeled the relationships among growth parameters (temperature, pH, and water activity) of foodborne Monascus ruber fungus and determined the desired output (maximum specific growth rate). Monascus, an ascomycetous fungus first identified by van Tieghem in 1884, is traditionally used to produce food coloring, fermented food, and beverages in Southeast Asia [26 –42]. Pigments from the genus Monascus are used in the food industry to produce Chinese sausage, instant noodles, and dairy products and are used in the meat industry as a substitute for nitrite salts, which are precursors of nitrasamins [40, 43].

Neural network models have demonstrated higher prediction accuracy compared to conventional polynomial models [25]. In Ho et al. [44], a neural fuzzy network model with eight fuzzy rules outperformed earlier neural network models proposed by Panagou and Kodogiannis [25].

In the IGA-ANFIS model proposed in this study, the IGA used Gaussian membership functions for simultaneously optimizing premise and consequent parameters of the ANFIS model, which directly maximized the training accuracy performance criterion (i.e., root-mean-square error (RMSE)). Experiments were then performed to compare the prediction accuracy of the IGA-ANFIS with neural network models proposed by Panagou and Kodogiannis [25] and neural fuzzy network models proposed by Ho et al. [44].

2 Materials and methods

This study used the 73 data sets obtained by Panagou and Kodogiannis [25] in their experimental studies. Sixty data sets were used for training the proposed IGA-ANFIS model, and the remaining 13 data sets were used for testing the model. The model parameters that have the largest effects on fungal growth are temperature (x₁), pH (x₂), and water activity (x₃). Figure 1 shows the architecture of the ANFIS model used in the present study. The 27 fuzzy if–then rules used in ANFIS architecture were expressed as

Fig.1

Three-input, one-output ANFIS architecture with 27 fuzzy if-then rules.

$\begin{matrix} R^{l} : IF x_{1} is A_{i} AND x_{2} is B_{j} AND x_{3} is C_{k}, \\ THEN y_{l} = p_{l} x_{1} + q_{l} x_{2} + r_{l} x_{3} + t_{l}, \end{matrix}$ (1)

where R^l (l = 1, 2, …, 27) denotes the l-th implication; A_i, B_j, and C_k (i, j, k = 1, 2, 3) are the linguistic terms of the precondition part with Gaussian membership functions μ_{A
_i} (x₁) , μ_{B
_j} (x₂) and μ_{C
_k} (x₃) , respectively; x₁, x₂ and x₃ are the temperature, the pH value, and water activity, respectively; y_l is the output variable; and p_l, q_l, r_l and t_l are the consequent parameters.

The ANFIS model output from Equation (1) is represented as $y = \sum_{l = 1}^{27} {\bar{W}}_{l} (p_{l} x_{1} + q_{l} x_{2} + r_{l} x_{3} + t_{l}),$ (2)

where ${\bar{W}}_{l} = W_{l} / - \sum_{l = 1}^{27} W_{l},$ in which W_l = μ_{A
_i} (x₁) μ_{B
_j} (x₂) μ_{C
_k} (x₃) (l = 1, 2, …, 27, and i, j, k = 1, 2, 3) .

In the precondition part, the premise parameters use Gaussian membership functions μ_{A
_i} (x₁) , μ_{B
_j} (x₂) and μ_{C
_k} (x₃). For example, a_{A
_i} and b_{A
_i} denote the center and the width of the Gaussian membership function μ_{A
_i} (x₁) , respectively. The following RMSE objective function can obtain the Gaussian membership functions: $J = {[\sum_{m = 1}^{α} \frac{{(R_{m} - y_{m})}^{2}}{α}]}^{\frac{1}{2}},$ (3)

where α is the number of training data sets, R_m is the actual experimental value, and y_m is the predicted value.

According to Equation (3), the value J can be expressed as $\begin{matrix} J = f (a_{A_{i}}, b_{A_{i}}, a_{B_{j}}, b_{B_{j}}, a_{C_{k}}, b_{C_{k}}, p_{l}, q_{l}, r_{l}, t_{l}) \\ \equiv f (g_{1}, g_{2}, \dots, g_{126}) . \end{matrix}$ (4)

Then, Equation (4) becomes the following optimization problem: $minimize J = f (g_{1}, g_{2}, \dots, g_{126}) .$ (5)

The IGA can then be used to solve the optimization problem in Equation (5), a nonlinear function with continuous variables [16].

3 Results and discussion

For a direct comparison of the proposed IGA-ANFIS method with the neural network models developed by Panagou and Kodogiannis [25] and with the neural fuzzy network models developed by Ho et al. [44], the evolutionary environment of the IGA was set to a population size of 500, a crossover rate of 0.9, a mutation rate of 0.1, and a generation number of 500. The IGA-ANFIS model was trained with ${\underline{g}}_{i} ⩽ g_{i} ⩽ {\bar{g}}_{i}$ (i = 1, 2, …, 126) with values for ${\underline{g}}_{i}$ and ${\bar{g}}_{i}$ as shown in Table 1. Table 2 shows the optimal premise and consequent parameters obtained by Gaussian membership functions. Figure 2 shows the convergence results for training accuracy performance criterion J. Figures 3–6 show the initial and final Gaussian membership functions of the three growth parameters. The final Gaussian membership functions were adjusted by IGA. Figures 4–6 show that the largest changes occur in linguistic terms A₁, B₂, and C₂ of the precondition part. Therefore, linguistic terms A₁, B₂, and C₂ may dominate Gaussian membership functions μ_{A
_i} (x₁) , μ_{B
_j} (x₂) and μ_{C
_k} (x₃) , respectively. After training, an output value was obtained by the IGA-ANFIS model.

Table 1
Ranges of the premise and consequent parameters

Parameters Range

Premise a_{A
₁}, a_{B
₁}, a_{C
₁} 0.2–0.5

parameters (g₁, g₄, g₇)

a_{A
₂}, a_{B
₂}, a_{C
₂} 0.5–0.8

(g₂, g₅, g₈)

a_{A
₃}, a_{B
₃}, a_{C
₃} 0.5–0.8

(g₃, g₆, g₉)

b_{A
_i}, b_{B
_i}, b_{C
_i} (i = 1, 2, 3) 0–0.5

(g₁₀ - g₁₈)

Consequent p_l, q_l, r_l, t_l (l = 1, 2, …, 27) –1.0–1.0

parameters (g₁₉ - g₁₂₆)

Parameters	Range
Premise	a_{A ₁}, a_{B ₁}, a_{C ₁}	0.2–0.5
parameters	(g₁, g₄, g₇)
	a_{A ₂}, a_{B ₂}, a_{C ₂}	0.5–0.8
	(g₂, g₅, g₈)
	a_{A ₃}, a_{B ₃}, a_{C ₃}	0.5–0.8
	(g₃, g₆, g₉)
	b_{A _i}, b_{B _i}, b_{C _i} (i = 1, 2, 3)	0–0.5
	(g₁₀ - g₁₈)
Consequent	p_l, q_l, r_l, t_l (l = 1, 2, …, 27)	–1.0–1.0
parameters	(g₁₉ - g₁₂₆)

Table 2

Optimal premise and consequent parameters

Input variables	Optimal premise parameters		Optimal consequent parameters
x ₁	g₁ = 0.3780	g₁₀ = 0.4614	g₁₉ = 0.8634	g₂₀ = -0.6911	g₂₁ = 0.5198	g₂₂ = 0.2169
			g₂₃ = 0.2613	g₂₄ = -0.3664	g₂₅ = -0.9658	g₂₆ = 0.3702
			g₂₇ = 0.9999	g₂₈ = 1.0000	g₂₉ = 0.3534	g₃₀ = 0.7583
	g₂ = 0.5338	g₁₁ = 0.2278	g₃₁ = 0.9100	g₃₂ = -0.4640	g₃₃ = -0.7082	g₃₄ = 0.4456
			g₃₅ = 1.0000	g₃₆ = -0.2257	g₃₇ = -0.9936	g₃₈ = -0.0920
			g₃₉ = 1.0000	g₄₀ = 0.3464	g₄₁ = 1.0000	g₄₂ = 1.0000
	g₃ = 0.7591	g₁₂ = 0.2825	g₄₃ = 0.9991	g₄₄ = -0.0463	g₄₅ = -0.8594	g₄₆ = 0.7868
			g₄₇ = 0.9273	g₄₈ = -0.3246	g₄₉ = -0.9999	g₅₀ = -0.4572
			g₅₁ = 1.0000	g₅₂ = 0.4108	g₅₃ = 1.0000	g₅₄ = 1.0000
x ₂	g₄ = 0.2867	g₁₃ = 0.2589	g₅₅ = 0.84187	g₅₆ = 0.9993	g₅₇ = 0.2035	g₅₈ = 0.8682
			g₅₉ = 0.9342	g₆₀ = -0.0441	g₆₁ = 0.2371	g₆₂ = 0.9905
			g₆₃ = -1.0000	g₆₄ = -0.9851	g₆₅ = 0.3427	g₆₆ = 0.6980
	g₅ = 0.8000	g₁₄ = 0.4775	g₆₇ = 0.2383	g₆₈ = 0.3683	g₆₉ = -0.9905	g₇₀ = 0.8116
			g₇₁ = 0.8060	g₇₂ = 0.3864	g₇₃ = 0.5770	g₇₄ = 0.9999
			g₇₅ = -1.0000	g₇₆ = -1.0000	g₇₇ = 0.8892	g₇₈ = -0.2998
	g₆ = 0.6546	g₁₅ = 0.2805	g₇₉ = 1.0000	g₈₀ = 0.9999	g₈₁ = -0.9450	g₈₂ = 1.0000
			g₈₃ = 0.8710	g₈₄ = 0.2883	g₈₅ = -0.5108	g₈₆ = 0.9746
			g₈₇ = -1.0000	g₈₈ = -0.7392	g₈₉ = 0.9867	g₉₀ = 0.4750
x ₃	g₇ = 0.3292	g₁₆ = 0.3591	g₉₁ = -0.5756	g₉₂ = -1.0000	g₉₃ = 0.9198	g₉₄ = 0.9128
			g₉₅ = -0.3719	g₉₆ = 1.0000	g₉₇ = -0.3573	g₉₈ = 0.4446
			g₉₉ = 0.5205	g₁₀₀ = 0.9990	g₁₀₁ = -0.6912	g₁₀₂ = 0.7973
	g₈ = 0.7897	g₁₇ = 0.4540	g₁₀₃ = -0.0329	g₁₀₄ = -0.6057	g₁₀₅ = 0.9427	g₁₀₆ = 0.7504
			g₁₀₇ = 0.0189	g₁₀₈ = -0.4544	g₁₀₉ = -0.6311	g₁₁₀ = 0.8528
			g₁₁₁ = 1.0000	g₁₁₂ = -0.4819	g₁₁₃ = -0.4426	g₁₁₄ = 0.3118
	g₉ = 0.6609	g₁₈ = 0.1649	g₁₁₅ = -0.5234	g₁₁₆ = -0.0394	g₁₁₇ = 0.2465	g₁₁₈ = 0.8786
			g₁₁₉ = 0.7022	g₁₂₀ = 0.4389	g₁₂₁ = -0.0037	g₁₂₂ = 0.4692
			g₁₂₃ = 0.9952	g₁₂₄ = -0.9996	g₁₂₅ = -0.9999	g₁₂₆ = -0.3739

Fig.2

Convergence results by using IGA.

Fig.3

Initial Gaussian membership functions for three growth parameters.

Fig.4

Final Gaussian membership functions for temperature (x₁).

Table 3 shows that the proposed IGA-ANFIS model had higher prediction accuracy compared to the neural fuzzy network model and both neural network models. The RMSE values obtained by IGA-ANFIS model for the training dataset, Test dataset and full dataset (0.0524, 0.0555 and 0.0593, respectively) were superior to those obtained by the neural fuzzy network model (0.0588, 0.1614 and 0.0865, respectively), the neural network with radial basis functions (0.0630, 0.1670 and 0.0920, respectively), and the neural network with multilayer perceptron (0.0880, 0.1790 and 0.1100, respectively).

Fig.5

Final Gaussian membership functions for pH (x₂).

Fig.6

Final Gaussian membership functions for water activity (x₃).

Table 3

Performance comparison of improved genetic algorithm in adaptive neuro-fuzzy inference system (IGA-ANFIS) model and other prediction models

Performance index J	Model	Data set
		Training	Test	Full
Root-mean-square error	Neural network with multilayer perceptron model	0.0880	0.1790	0.1100
	Neural network with radial basis functions model	0.0630	0.1670	0.0920
	Neural fuzzy network model	0.0588	0.1614	0.0865
	IGA-ANFIS model	0.0524	0.0555	0.0593

Fig.7

Actual and predicted values compared between the IGA-ANFIS model and the neural fuzzy network model when using the training data set.

Fig.8

Actual and predicted values compared between the IGA-ANFIS model and the neural fuzzy network model when using the test data set.

Fig.9

Actual and predicted residual values compared between the IGA-ANFIS model and neural fuzzy network model when using the training data set.

Fig.10

Actual and predicted residual values compared between the IGA-ANFIS model and the neural fuzzy network model when using the test dataset.

Figures 7 and 8 further compare predictive performance between the IGA-ANFIS model and the neural fuzzy network model. The comparisons again show that the IGA-ANFIS model generally obtains a better model fit. Figures 9 and 10 compare residual value spreads for the two models. In both models, residuals are symmetrically distributed higher and lower than 0. However, the IGA-ANFIS model has a narrower distribution of residual values, which indicates superior prediction accuracy.

4 Conclusions

The comparison results indicate that the IGA-ANFIS architectures outperform the neural network and neural fuzzy network models under varied temperature, water activity, and pH conditions. The RMSE index and graphic plots agree that the performance of the IGA-ANFIS model was superior in the training sets and in the full data sets and obtained reasonably good predictions in the test data set. The IGA-ANFIS is mainly intended for use in predictive microbiology, and the results indicate that it increases accuracy in predicting kinetic parameters of fungi. Thus, the proposed IGA-ANFIS model provides an alternative modeling approach in predictive mycology.

Footnotes

Acknowledgments

This work was partially supported by the Ministry of Science and Technology, Taiwan under grants MOST 106-2221-E-037-001, MOST 106-2622-E-037-005-CC3, MOST 106-2218-E-327-001, MOST 107-2221-E-037-006, MOST 107-2218-E-992-308, and the “Intelligent Manufacturing Research Center” (iMRC) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.

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