FUZZY AVERAGE TO ESTIMATE A SITUATION OF A COMPLEX SYSTEM OF BIOAVAILABILITY
REGINA SERRÃO LANZILLOTTI
Institute of Mathematics and Statistics
HAYDÉE SERRÃO LANZILLOTTI
MONICA DUTRA BAPTISTA BORGES
Institute of Nutrition
University of Estado do Rio de Janeiro
São Francisco Xavier 254, Rio de Janeiro, RJ. CEP 205550-013
BRASIL.
Abstract
The objective at this paper was to delineate a diagnosis system that translates the interactive effect of the calcium and of the ascorbic acid intake on nonheme-iron absorption from a meal. It was used the fuzzy arithmetic mean, an extension of the arithmetic mean, to a cluster algorithm by using the concept of fuzzy numbers and the extension principle. The results allowed building a diagnosis system that presents two procedures: on non-parametric and another parametric. It was concluded that the model could help the experts in Nutrition in analysis about the interactive effect of the calcium and of the ascorbic acid intake on nonheme-iron absorption from a meal.
Key-Words: Nonheme-iron availability, Calcium, Ascorbic acid, Fuzzy sets
1 Introduction
The bioavailability of a nutrient is a complex system where there is an interaction between nutritional and anti-nutritional factors that could stimulate or inhibit the absorption by organism.
The assessment of iron status at individual and populational levels has been the target of many studies because some issues still need to be defined. Moreover, some advantages and limitations should be taken into consideration for the choice of the appropriate model. [1]
In that way, the application of fuzzy sets could be a tool to evaluate the interaction among nutrients by translating the available of nonheme-iron.
Many studies in animals and human beings showed that calcium inhibit the absorption of the nonheme-iron and the ascorbic acid is a stimulator in its absorption [2,3,4,5,6,7,8].
2 Objective:
The objective was to delineate a diagnosis system that translates the interactive effect of the calcium and of the ascorbic acid intake on nonheme-iron absorption from a meal.
3 Methods:
A triangular fuzzy number (TFN) is a fuzzy number denoted by (a1,a2,a3) (a1£a2£a3) if its membership function mA is given by
mA (u)=(u-a1)/(a2-a1), uÎ[a1, a2] (1)
mA (u)=(u-a3)/(a2-a3), uÎ[a2, a3] (2)
mA (u)= 0 otherwise (3)
A support of a fuzzy number is an interval on real-valued R denoted by
supp A = {u½mA(u)0, uÎ R} (4)
If its membership function is continuous on real-valued, and is called a mean value of the fuzzy number, if and only if mA(u)=1. [9]
A linguist variable X is characterized by quintuple [x, T(x), U,G,M] in which x is the name of the variable; T(x) denotes the term set of x, that is, the set of names of linguistic values of x with each value being a Fuzzy variable denoted generically by x and ranging over a universe of discourse U which is associated with the base u; G is syntactic rule for generating the name; M is a semantic rule for associating with each X its meaning, M(x) is a Fuzzy subset of U. The purpose of defining the fuzzy arithmetic mean is to describe a average of these different M belonging to the linguistic variable, and decide a certain output on U. When inputs on U corresponding to each different M(x) are given.
Suppose there is two M(xi) to the X, and denote it as M1(xi) e M2(xi). Which correspond numbers fuzzy Ai (i=1...n) and Bj (j=1...m) defined on U, where Ai and Bj satisfy:
supp Ai Ç Supp Aj ¹f (empty set)
= f( j¹i; i+1) (5)
(j=i; i+1)
supp Bi Ç Supp Bj ¹f (empty set)
= f( j¹i; i+1) (6)
( j=i; i+1)
by being mAi and mBj membership function of the Aj and Bi, respectively.
The fuzzy average, fuzzy arithmetic mean, concerning u1 and u2 is defined as
Uf(nm)(u1,u2)=Si=1..nSj=1..mwij[mAi(u1),mBj(u2))]rij
(7)
where [u1Î supp Ai,] and [u2Î supp Bj ]
and [wij (mAi(u1), mBj(u2)] is called a weighted matrix that satisfies both:
0£[wij (mAi(u1), mBj(u2)] £1 and
Si=1..nSj=1..m [wij (mAi(u1), mBj(u2)]=1.
(8)
The fuzzy average is a convex combination of elements of the consequence matrix rij, especially when rij satisfies
rij= (a2i+b2j)/2 (i=1...n; j=1m)
(9)
In general, wij[mAi(u1), mBj(u2)] is defined by
T[mAi(u1),mBj(u2)]/Si=1..nSj=1..mT[mAi(u1), mBj(u2)]
(i=1...n; j=1.....m) (10)
where T[mAi(u1),mBj(u2)] is a t-norm [9] concerning [mAi(u1), mBj(u2)].
The t-norm refers to a class of averaging operators and for the intersection of fuzzy sets can be used the bounded-sum operator, defined by [10]
:mAÅB(x)=min[1,mA+mB)]
(11)
4 Results
It was got dietetics measurements of the calcium and ascorbic acid from 69 lunches that were consumed by 24 university students of the Nutrition Course.
The association of these two nutrients was displayed trough a scatterplots 2D
Fig.1 Association of both nutrients: inhibitor and stimulator
By implementing the model it was taken the interval to two deviations standard (2s). This allowed delimiting the minimum intake and the maximum intake. This interval corresponds to 97% of whole distribution. In that way it was eliminated the off lines intakes[11].
The fuzzy number for the calcium (inhibitor) and the ascorbic acid (stimulator) were delineated by two experts, one in Nutrition another in Fuzzy. The fuzzy sets constitute the abaci that translate the calcium and ascorbic acid intakes in values of pertinence (0,1).
In this way, the pertinence functions, mAi(u1), and mBj(u2), were calculated to both calcium and ascorbic acid, respectively.
Fig 2: Abacus of the effect of the calcium intake on nonheme –iron absorption from a meal
Fig 3 Abacus of effect of the Ascorbic Acid intake on nonheme –iron absorption from a meal
The weighted matrix, wij (10) used the value of pertinence for the median points of support for both nutrients and it was applied the bounded-sum operator and the result was standardized according to the conditions in (8)
The consequence matrix was obtained through of rij (9) it gets the feature
Each cell was obtained like the average of the median points of support that correspond to the maximum value of pertinence of the number fuzzy for each nutrient (Calcium and Ascorbic Acid).
Starting from both the matrixes, the fuzzy arithmetic average concerning to Calcium and Ascorbic Acid was calculated by (7) and it reached 108,39 mg that corresponds to an interactive average intake of both the nutrients.
Discriminating groups it was used the bounded-sum operator for each lunches according two nutrients, then it was made a scatterplots 3D, Fig 4.
Fig 4: Clustering by interactive effect of Calcium and the Ascorbic Acid intakes on nonheme –iron absorption from a meal
The graph allowed to discriminate 5 categories of possibility to available the absorption of nonheme-iron in these meals.
The table 1 is showing the possibilities of the combinations that arise to translate the interactions among nutrients nonheme-iron, calcium and ascorbic acid. That was possible through of the semantic rules on the universe of discourse
Table 1 Possibility to available the nonheme-iron in the meal
Categories / Calcium / Ascorbic Acidm / mg / m / mg
Low / Low / High / Low / Low
Tendency / Medium / Medium / Low / Low
Medium / Medium / Medium / Medium / Medium
High / Low / High / High / High
Upper / + High / + Low / + High / + High
By applying these categories on the 69 lunches it was possible to find the following perceptual distribution:
Table 2 Distribution of the lunches according to semantic rules
The method will can be considered as an
Appropriate discriminator because it showed that 39% among the lunches has presented low possibility in the availability of the iron in the meals. On the other hand, 35% with upper possibility. The remainder lunches, 26%, was classificated.between the extremes.
Moreover, the fuzzy arithmetic mean can be taken as a point of cut to value the available nonheme-iron in the meals.
The scatterplots 3D to values of pertinence displays the clusters that were arisen from the classification of lunches
By confronting the results between average of nutrients (calcium and ascorbic acid) to each lunch and its fuzzy arithmetic mean, it was possible verify that 42 lunches were over fuzzy mean and 27 were lunches bellow. The lunches that were bellow of fuzzy arithmetic mean coincided with the classification “upper” from scatterplots.3D. Seven among 27 lunches did not agree with former classification.
Although there has been a divergence in these classifications, the fuzzy arithmetic mean could play a role the frontier marker to value the possibility of the availability nonheme-iron.
According to propose model when the values of the average of nutrients (calcium and ascorbic acid) are bellow the fuzzy arithmetic mean, there is a hard possibility of the nonheme-iron of the meal is not able to be available and absorption by organism.
4.1 Application
A patient consumed in a lunch 97,8 mg of calcium and 43,04 mg of ascorbic acid. How can we diagnostic the organism possibility available the nonheme-iron in the meal?
Step 1: The expert in Nutrition finds in the abaci the values of the pertinence (Fig. 2 and Fig 3).
Step 2: By applying these values of pertinence on the axes of the nutrients, it can be found the pertinence of bounded-sum (join interactive pertinence). Here, the values reached: calcium=0,35 and ascorbic acid = 0,86. The intersection of the axes allows classifying the possibility too available the nonheme-iron in the meal. In this case, the classification gets “Tendency”, it indicates that there is a tendency of nonheme-iron to be available in the target meal.
Step 3: Must be still verified if the average of these nutrients (calcium and ascorbic acid) is over or below the fuzzy arithmetic mean (108,39 mg). In the present case, the average got 140,84mg (140,84>108,39). Then, the combination among nutrient lead to think that there is a possibility of the nonheme-iron of the lunch to be available.
5 Conclusion
The diagnosis system presents two procedures: one non-parametric and another parametric.
The non-parametric consists of cluster meals according to interactive effect of the calcium and of the ascorbic acid intake on nonheme-iron absorption.
The parametric consists of a cut of point taken as the fuzzy arithmetic mean.
The model can help the experts in Nutrition in analysis about the interactive effect of the calcium and of the ascorbic acid intake on nonheme-iron absorption from a meal.
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