Fuzzy Deep Daily Nutrients Requirements Representation

Over the years, the optimal regime problem has been the focus of many researchers whose contributions vary only in the objective functions considered. Stigler & Danzig introduced the first optimization model in which the objective function is the cost of the diet while the constraints represent the requirements on the good balance of the diet [13]. In the model proposed in Ref. [14], the objective function makes a trade-off between the different meals using the penalty technique. In this context, the authors consider the three usual meals, a snack and a portion of fruit. The proposal of Masset et al. [15] aims at controlling the difference between the actual and recommended intake according to the nutritional requirements. Additional investigations have proposed complementary diets and nutritional menus at minimal cost to children [16]. To control different objectives at the same time, other authors have used multi-objective mathematical models [16, 17]. Van Mierlo et al. [18] considered almost the same situation by substituting the cost of the diet with the minimization of fossil fuel depletion. Cholesterol intake and glycemic load are known to be the main factors contributing to childhood obesity and were the subject of the two objective functions of the multi-objective model proposed by Taniguchi [19].

Generally, in order to model the problem of optimal diet, one needs to know the individual's daily requirements of favorable and unfavorable nutrients. Several favorable and unfavorable nutrients exist but almost all research studies solve the optimal diet problem based on Calories (2000 kcal), Protein (91 g), Carbohydrate (271 g), Potassium (4044 mg), Magnesium (380 mg), Calcium (1316 mg), Iron (18 mg), Phosphorus (1740 mg), Zinc (14 mg), Vitamin b6 (2.4 mg), Vitamin b12 (8.3 µg), Vitamin C (155 mg), Vitamin A (1052 µg), Vitamin E (9.5 mg), Satured fat (17 g), Sodium (1779 mg), Total fat (65 g), and Cholesterol (230 mg) [5, 17, 19]. In parentheses, we give the estimates of favorable and unfavorable nutrient requirements adopted in Refs. [5-20].

To point out the daily nutrient’s requirements, consider, for example, the favorable nutrient Potassium. In 2019, the committee of the Academies of Science, Engineering, and Medicine (NASEM) updated the Dietary Reference Intakes (DRIs) for potassium (and sodium) [21]. The committee found the data insufficient to derive an estimated average requirement (EAR) for potassium. Therefore, it established AIs for all ages based on the highest median potassium intakes in healthy children and adults, and on estimates of potassium intakes from breast milk and complementary foods in infants. Table 1 presents the current potassium AIs for healthy individuals.

Table 1. Adequate Intakes (AIs) for Potassium

So, according to this table, the value 4044 mg [5] is a very exaggerated over-estimate and does not take into account the age, gender, pregency etc.

The idea proposed in this work is to generate a database on Table 1 and on the notion of triangular fuzzy numbers. A part of this database is used to educate an auto-encoder neural network. Then, given an individual (age, gender, ...), never seen before, our auto-encoder will give an adequate estimation of the needs of our individual in terms of Potassium. This process will be generalized to other nutrients (favorable or unfavorable), which will allow to automate and personalize the needs in a precise way.

It should be noted that the models proposed in the literature to solve the optimal diet problem are based on nominal (for example average) estimations of favorable and unfavorable nutrient requirements. In this sense, it is the first time in the literature that a technique based on auto-encoding is implemented to predict the requirements of different nutrients.

The auto-encoder is widely used to compress data [6]. We use this multilayers neural network to transform different TN to crisp values. First, we start with an auto-encoder with random architecture (random number of layers and number of units in each layer) (Figure 2). In this figure, the auto-encoder is a deep neural network that contains two principles parts encoder and decoder, the middle layer is formed by a single neuron that produces the coded information or that produces the crisp value which represents the nutrient prediction. The encoder contains three layers: one output layer of six neurons whom 4 represent the TN components and 2 for gender and age, two hidden layers or treatment layers. The decoder contains three layers: the output layer that will produce the coded information from the crisp value and the treatment layers. At the biggening we don’t know the optimal architecture of the auto-encoder that permit to produce predictions of the nutrient with minimum loss of information. To solve this problem, we propose a new regulation function and we use fuzzy genetic algorithm to select the optimal auto-encoder architecture problem basing on the generated data.

2.png

Figure 2. Auto-encoder with random architecture; the middle layer produces crisp values

Fitting the auto-encoder parameters Ω consists of minimizing some cost function E(Ω) whose principal terms are the mean square error, the regulation term, and the sparsity term.

$E(\Omega)=M S E(\Omega)+\sigma * \Lambda(\Omega)+\mu * \phi(\Omega)$      (1)

The first term is the global reconstruction error, the second term is the regulation function (norm of the matrix formed by the auto-encoder parameters) [22], and the last term is the sparsity term [23]. When training a sparse autoencoder, it is possible to make the sparsity regulariser small by increasing the values of the weights and decreasing the values of the compression obtained by the auto-encoder. Adding a regularization term on the weights to the cost function prevents it from happening. This term is called the regularization term and is defined by $\sum_{i}\left\|\Omega_{i}\right\|^{2}$ where $\left(\Omega_{i}\right)$ are the parameters of the auto-encoder. Unfortunately, this formulation doesn't distinguish between different units and doesn't control the number of neurons. In this work, we introduce an original regulation term given by the Eqns. (2), (3), and (4).

$\Lambda(\Omega)=\Lambda 1(\Omega)+\Lambda 2(\Omega)$     (2)

where,

$\Lambda 1(\Omega)=\sum_{l, i} P_{E n c, l, i}\left\|\Omega_{E n c, l, i}\right\|^{2}$       (3)

$\Lambda 2(\Omega)=\sum_{l, i} P_{\text {Dec }, l, i}\left\|\Omega_{\text {Dec }, l, i}\right\|^{2}$          (4)

We comment the Eq. (3) and the equation (4) is obtained by analogies. $P_{D e c, l, i}$  is the penalty coefficient that permits to control the weighted term. Here, we penalize each of the parameter’s neuron with specific penalty coefficients:

If the neurons $i$ and $j$ are in the same hidden layer 1 , and if $\mathrm{i}<\mathrm{j}$, then $P_{E n c, l, i}<P_{E n c, l, j}$ ; compared to the neuron $\mathrm{j}$, the neuron i has a very high chance of remaining in the layer 1 .

For two given hidden layers 1 and $\mathrm{k}$, if $1<\mathrm{k}$, then $\max _{i} P_{E n c, l, i}<\min _{j} P_{E n c, k, j}$；compared to the layer $\mathrm{k}$, the layer 1 has high chance of remaining in the auto-encoder.

To find a local minimum for the function E, one can use a recurrent network called continuous Hopfield network [24] and [25]. As we can use an evolutionary heuristic method such as: firefly optimization based [26], genetic algorithm [27], particle swarm algorithm, stochastic fractal search, and moth swarm algorithm with specific operators [20].

To benefit from the evolutionary strategy of producing good quality solutions from the current solutions thanks to the operators of crossing and mutation and the ability of fuzzy logic of reason in stochastic environments, we use, in this work, fuzzy genetic algorithm to minimize the function E to obtain an auto-encoder with optimal architecture.

To determine a personalized optimal diet basing on mathematical modeling, one needs to know the individual's daily requirements of favorable and unfavorable nutrients. giving an estimate of daily nutrient requirement in terms of a single value for all ages and genders is a restrictive approach that is subject to over- or underestimation. In this paper, we propose a personalized representation of nutrients expert knowledge based on fuzzy trapezoidal numbers, an optimal auto-encoder based on a new regulation function, and on fuzzy genetic algorithm. Thanks to our regulation function and the deep learning of the optimal architecture auto-encoder educated on the basis of intelligently generated data set, the proposed system is able to predict the positive and negative nutrient requirements of each individual once the age and gender of that individual is provided. In a practical setting, our system can be used by a dietician to accurately determine the daily nutrient requirements of a given individual. In addition, the different mathematical models can use the different predictions to automatically solve the optimal diet problem. The resulted system can be used by a dietician to select a set of foods from a predetermined set of foods that meet the daily nutrient requirements while setting an economic function that depends on the targeted disease.

Regardless of the contributions of the model and the good results obtained in this work, the proposed system and its application have some limitations. The proposed system uses the same autoencoder to predict the requirements of the 18 nutrients, which may lead to erroneous predictions in some borderline cases. For example, vitamin requirements are very low while calorie requirements are very high. In addition, positive nutrients should not be treated in the same way as negative nutrients. At least three auto-encoders should be considered: one for unfavorable nutrients, one for small favorable nutrients and large favorable nutrients.

We will use our system to dress optimal diets for different patients that suffer from chronic diseases. And to take into account of the falsity, one can use intuitionist logic in genetic algorithm instead of classical logic that considers the trueness only.