Research Article  Open Access
Ferdinando Di Martino, Salvatore Sessa, "A Method Based on Extended Fuzzy Transforms to Approximate Fuzzy Numbers in Mamdani Fuzzy RuleBased System", Advances in Fuzzy Systems, vol. 2018, Article ID 8458916, 16 pages, 2018. https://doi.org/10.1155/2018/8458916
A Method Based on Extended Fuzzy Transforms to Approximate Fuzzy Numbers in Mamdani Fuzzy RuleBased System
Abstract
We propose a new Mamdani fuzzy rulebased system in which the fuzzy sets in the antecedents and consequents are assigned in a discrete set of points and approximated by using the extended inverse fuzzy transforms, whose components are calculated by verifying that the dataset is sufficiently dense with respect to the uniform fuzzy partition. We test our system in the problem of spatial analysis consisting in the evaluation of the livability of residential housings in all the municipalities of the district of Naples (Italy). Comparisons are done with the results obtained by using trapezoidal fuzzy numbers in the fuzzy rules.
1. Introduction
A fuzzy number (FN) is a fuzzy set with membership function A: Reals → defined as where a ≤ c ≤ d ≤ b, A^{}: [a,c] → is a not decreasing continuous function with A^{−}(a) = 0, A^{−} (c) = 1 and A^{+}: [b,d] → is a not increasing continuous function with A^{+}(d) = 1, A^{+}(b) = 0. A^{} and A^{+} are called left side and right side of A, respectively.
Complicated leftside and rightside functions can generate serious computational difficulties when imprecise information is modeled by FNs. In order to overcome this problem, the original FN can be approximated with other easier functions. The simplest FNs used in fuzzy modeling, fuzzy control, and fuzzy decisionmaking are the trapezoidal and triangular FNs. In a trapezoidal FN the functions A^{} and A^{+} are linear; for instance, A^{} (x) = (xa)/(ca) and A^{+}(x) = (bx)/(bd) with a≤b≤c≤d, a≠c, b≠d. In a triangular FN it is assumed that d=c. Other simple FNs widely used are the degenerated left (resp., right) size semitrapezoidal FNs with a = c < d < b (resp., a < c < d = b). In many problems trapezoidal, triangular, or semitrapezoidal approximations of FNs could give a loss of information not negligible and this can significantly affect the reliability of the results.
Furthermore, the membership functions of FNs used in applications are not generally known, for example, when they are obtained as relative frequencies of measured occurrences in a discrete set of points or in collaborative applications in which a set of stakeholders evaluate separately the membership degrees of a FN and the function is assigned as an average of these membership degrees. For making understandable this idea, in the example of Figure 1, the membership degree of the fuzzy set “daily temperature T” (measured in °C) for a discrete set of 100 points is the average of the membership degrees evaluated separately by many stakeholders.
Recently many methods are proposed in order to approximate FNs with easier FNs using a suitable metric (see, e.g., [1–6]). Some authors investigate approximations by adding some restrictions to preserve properties of a FN as core [7], ambiguity [8–10], expected interval, translation invariance, and scale invariance [11, 12]. As pointed out in [13], by using a trapezoidal FN as approximation function by, only a limited number of characteristics can be preserved since a trapezoidal FN depends only on four parameters, and the best approach to preserve multiple characteristics is to use sequences of FNs. In [14] a new method is proposed based on the inverse fuzzy transform (Ftransform) [15] in order to construct sequence of FNs which converge uniformly to a FN, preserving properties as its support, core, ambiguity, quasiconcavity, and expected interval (Algorithm 1). The Ftransform method was already used in image analysis (see, e.g., [15–18]) and data analysis applications (see, e.g., [19, 20]). In [21] the bidimensional Ftransform is used to approximate type 2 FNs. In [13], the extended iFtransform method, proposed in [15], is applied to approximate FNs preserving the support and the quasiconcavity property. The main advantage of this method is to reach the desired approximation with a linear rate of uniform convergence. However, when the membership function is given in a discrete set of points, it is necessary to verify that this dataset is sufficiently dense with respect to the uniform fuzzy partition of the support of the FN. More specifically, the Ftransform method divides the interval [a,b] in n subintervals of width h = (n1)/(ba). The points x_{1}= a, x_{2}= a+h,…, = a+(i1)h,…, = b are called nodes: a uniform fuzzy partition of [a,b] is created by assigning n fuzzy sets with continuous membership functions A_{1},…,: [a,b] → , called basic functions, where (x) = 0 if x(,), i = 1,…,n. When the input data form a dataset of points in [a,b], it is necessary to control that this set is dense with respect to the uniform fuzzy partition; namely, we must verify that at least one data point with nonzero membership degree falls within a subinterval () for i=1,…,n. In Figure 2 we show an example of dataset not sufficiently dense with respect to the fuzzy partition: no data is included in ().

The FNs are largely used in fuzzy reasoning systems, particularly in fuzzy rulebased inference systems in which fuzzy rules are applied in an inferential process. In a fuzzy rulebased inference system [22] the fuzzy rule set is composed of fuzzy rules, called “compositional rules of inference”: each antecedent in a fuzzy rule is a fuzzy relation in which the min operator is applied for the conjunction and the max operator is applied for the disjunction of fuzzy sets. The max operator is applied for the aggregation of the rules as well. The discrete Center of Gravity (CoG) method is applied in the defuzzification process to obtain the final crisp value of the output variable.
We apply the iFtransform method for constructing the FN modeling the input variables in the antecedent and the output variables in the consequent of fuzzy rules in a Mamdani fuzzy inference system.
The paper is organized as follows: Section 2 contains the basic notions of fuzzy number and Ftransform and in Section 3 we introduce the extended iFtransform method which in Section 4 is applied to a Fuzzy RuleBased Systems (FRBS). In Section 5 we give the results of our tests, and final considerations are reported in Section 6.
1.1. Preliminaries
As already shown in [17], the extended iFtransform method, proposed in [13], approximates a function assigned on a discrete set of points by means of an iterative process. Strictly speaking, we set initially the dimension n of the fuzzy partition to a value n_{0}; afterwards it is necessary to verify at any step that the dataset is sufficiently dense with respect to the fuzzy partition and that the approximation error is less than or equal to a prefixed threshold: in this case the process stops and the direct Ftransform components are stored; otherwise, n is set to n + 1 and the process is iterated by considering a finer fuzzy partition. Below, we schematize the pseudocode of this process.
We propose a new Mamdani FRBS in which we use the extended iFtransform to approximate FNs and we apply the above process for constructing the input fuzzy sets in the antecedent and the output fuzzy sets.
The extended iFtransform method for approximation of the FNs is used to fuzzify the crisp input data. The min and max operators are applied as AND and OR connectives in the antecedent of the fuzzy rules to calculate the strength of any rule. The defuzzification process of the output fuzzy set is carried out via the discrete Center of Gravity (CoG) method. For example, we consider a system formed by two fuzzy rules in the following form:where A_{1} and A_{2} are two FNs for the linguistic input variable x, B_{1} and B_{2} are two FNs for the input linguistic variable y, and C_{1} and C_{2} are two FNs for the output variable z. Applying the extended iFtransforms to evaluate each fuzzy set, we suppose that , , , With max (resp., min) operator as connective OR (resp., AND), we obtain the value of the two rules: r_{1} = max(0.4, 0.7) = 0.7 and r_{2} = min(0.7, 0,3) = 0.3. In the defuzzification process we reconstruct the output fuzzy set aswhere s_{1} and s_{2} are suitable thresholds prefixed a priori (Figure 3).
The CoG method is useful for obtaining the final crisp value of the output variable as where Nc is the number of rules and z_{1}< z_{2} <⋯< are points of the support of C. In Figure 3 we give an example.
2. Fuzzy Numbers and FTransforms
2.1. Fuzzy Numbers
Given a value α∈, we denote with A_{α}, called αcut of a FN A, the crisp set containing the elements xR with a membership degree greater than or equal to α. We also use the interval whereFor α = 1, is called the core of the FN and denoted by core(A). Note that for α = 0, = . The support of a fuzzy set is given by the closure of the crisp set Given two arbitrary FNs, A and B, two metrics are considered in [23, 24]:
the Chebyshev distanceand the extension of the Euclidean metric given by
Two properties of A are given in [25] called ambiguity and value, defined asandrespectively, where r: → is a not decreasing function called reducing function with r(0) = 0 and r = 1. Another important propriety is the expected interval of A, introduced in [3, 24], defined as follows:We have EI(A) = [(a+c)/2, (d+b)/2] for a trapezoidal FN A.
2.2. Direct and Inverse FTransforms
Following the definitions and notations of [15], let n ≥ 2 and P = , x_{2}, …, be a set of points of [a,b], called nodes, such that x_{1} = a < x_{2} <⋯< = b. Let ,…, be an assigned family of fuzzy sets with membership functions A_{1}(x),…,(x): [a,b] → , called basic functions. We say that it constitutes a fuzzy partition of [a,b] if the following properties hold:(1)() =1 for every i =1,2,…,n(2)(x) = 0 if x (,) for i=2,…,n1(3)(x) is a continuous function on [a,b](4)(x) strictly increases on [,] for i = 2, …, n and strictly decreases on [,] for i = 1,…, n1(5) for every x[a,b]
Furthermore, we say that the fuzzy sets ,…, form an huniform fuzzy partition of [a,b] if(6)n ≥ 3 and = a + h·(i1), where h = (ba)/(n1) and i = 1, 2, …, n (that is, the nodes are equidistant)(7)( – x) = ( + x) for every x and i = 2,…, n1(8) (x) = (x  h) for every x[,] and i = 1,2,…, n1
Let be a continuous function on [a,b]. The quantityfor i = 1, …, n, is the ith component of the direct Ftransform , F_{2}, …, of f with respect to the family of basic functions , A_{2}, …, . If this fuzzy partition is huniform, the components are as follows [26]:The function where x[a,b], is defined as the iFtransform of f with respect to , A_{2}, …, and it approximates in the sense of the following theorem [26].
Theorem 1. Let f(x) be a continuous function on [a,b]. For every ε > 0, then there exist an integer n(ε) and a fuzzy partition , A_{2}, …, of a,b such that  with respect to the existing fuzzy partition.
In the discrete case we know that the function assumes assigned values in the points p_{1},…, of [a,b]. If the set ,…, is sufficiently dense with respect to the fixed partition , A_{2}, …, , that is, for each i = 1,…,n, there exists an index j,…, such that () > 0, we can define the ntuple , F_{2},…, as the discrete direct Ftransform of f with respect to , A_{2}, …, , where each is given byfor i=1,…,n. Similarly we define the discrete iFtransform of f with respect to the , A_{2}, …, by settingfor every j,…,. We have the following theorem [15].
Theorem 2. Let be a function assigned on a set of points ,…, . Then, for every ε > 0, there exist an integer n(ε) and a related fuzzy partition , A_{2}, …, of such that ,…, is sufficiently dense with respect to the existing fuzzy partition and for every , j = 1,…,m, the inequalityremains true.
3. The Extended iFTransform and Fuzzy Numbers
In [15] the extended iFtransform of a continuous function is introduced in order to preserve the monotonicity as follows. For an huniform fuzzy partition , A_{2}, …, , the function is extended to [ah,b+h] as follows: Then the following basic functions are defined asThen the ith component of the extended direct Ftransform of with respect to the family of basic functions , A_{2}, …, is given byHence the extended iFtransform of is given byBy [13, Lemma 9], we obtain that Let S be a fuzzy number with a continuous membership function and supp(S) = [a,b]. We consider an huniform fuzzy partition , A_{2}, …, of [a,b] with n ≥ 3 and let be the extended iFtransform of S. We obtain that [13, Prop. 11]where is the ith component of the direct Ftransform of S (cfr., formulae (15)). Theorem 13 of [13] provides the approximation property of the extended iFtransform as follows.
Theorem 3. Let S be a FN having a continuous membership function and supp = . Let a fuzzy partition , A_{2}, …, of be huniform with n ≥ 3 and be the extended iFtransform of S calculated by (23). Then the following inequality holds:where is the modulus of continuity of S given by
Another important theorem [13, Th. 14] is the following.
Theorem 4. Let S be a FN having a continuous membership function, supp = , and core = , a < c < d < b. Let a fuzzy partition , A_{2}, …, of be huniform with n ≥ 3 and a fuzzy set T such that T = calculated by (23) in and T=0 if x. If h = ba/n1 is such that h ≤ mindc)/4, ca, b, then T is a FN for which the following hold:1.supp = supp2.If core = ,, then c ≤ ≤ ≤ d,  ≤ 2h,  ≤ 2h3.4.If S^{} strictly increases on a,c, then T strictly increases on a,5.If S^{+} strictly decreases on d,b, then T strictly decreases on ,b The preservation of the properties “ambiguity” and “value” of a FN and their approximation with an extended iFtransform is given by the following theorem in [13, Theorem 27]:
Theorem 5. Let S be a FN having a continuous membership function with supp = a,b and core = c,d, a < c < d < b. Let a fuzzy partition , A_{2}, …, of a,b be huniform with n ≥ 3 and a fuzzy set T such that T = given by (23) in a,b and T=0 if xa,b. Let core = with ≤ . By putting , we obtain thatwhere and .
In order to apply the extended iFtransform to approximate a FN S with oneelement core, in [13] the concept of regular huniform partition of [a,b] is introduced as an huniform partition of [a,b] such that A_{1} is differentiable in [a,x_{2}], is differentiable in [,] for i = 2,…,n1, and is differentiable in [,b]. Thus, we can define the normalized extended iFtransform given asA theorem similar to Theorem 5 is given in [13, Theorem 29] as follows.
Theorem 6. Let S be a FN having a continuous membership function with supp = a,b and core = , a < c < b. Let , A_{2}, …, be a regular huniform partition of a,b and T = a fuzzy set given by (30) in a,b and T=0 if xa,b. Let core = , with ≤ and . Then the following properties hold:where and .
Now we suppose that the membership values of a FN S in form (1) are assigned on a discrete set of m points a = p_{1} < p_{2} <⋯< < = b. We consider an huniform fuzzy partition , A_{2}, …, of [a,b]. If the set of points are sufficiently dense with respect to the fuzzy partition, i.e., if then the extended iFtransform of is defined for any x[a,b] as follows [13]:where , , and is the ith component of the direct Ftransform of S in [a,b] for i=1,…,n. Similarly, it can be proved that all the above properties of the extended iFtransform of a FN with continuous membership function apply in the discrete case as well.
4. Extended iFTransform and Fuzzy RuleBased System
Let the expert knowledge be formed by a set of fuzzy rules in a linguistic fuzzy model:where x_{1}, x_{2},…, are input variables, y is the output variable, , ,…, , are fuzzy sets and the operator (i=1,…,n) is an AND or an OR operator. We construct a fuzzy rule set considering only AND connectives, splitting rules in which there are OR connectives in the antecedent. This fact can be also represented via a fuzzy relation equation.
We propose a FRBS in which the FNs of the fuzzy rule set are approximated by using extended iFtransforms. We suppose that the fuzzy sets in the antecedent and consequent of each rule are given by FNs whose membership functions are assigned in a discrete set of points p_{1} = a < p_{2} <⋯< < = b. An example of this case occurs when, in a collaborative project, the membership values of a fuzzy set are given over a discrete set of points by means of averages of membership values assigned by different stakeholders.
Let [a,b] be the core and [c,d] be the support of this FN. We approximate the membership function of it by the extended iFtransform calculated with (34). As already said above in Section 3, we find a fuzzy partition such that the set of points is sufficiently dense with respect to it and we apply the iterative process given in Section 1.1. For each FN in the antecedents and in the consequents of the fuzzy rules, we calculate the discrete extended direct Ftransform storing them in the fuzzy rule set. The crisp input data are fuzzified via (34) by using the stored direct Ftransform components of the FNs. The inference engine applies to the maxmin Mamdani inference model to calculate the strength of each rule and to obtain the final fuzzy set aggregating the output fuzzy sets. The crisp output value is obtained by applying the CoG method. The FRBS is schematized in Figure 4.
The extended iFtransform approximates each fuzzy number by considering the set of points in which its membership function is assigned. This function creates an huniform fuzzy partition of the support of the fuzzy set and verifies that the set of points is sufficiently dense with respect to the fuzzy partition. Initially n is set to a value n_{0} (for example, n_{0} = 3). If the set of points is not sufficiently dense with respect to the fuzzy partition, the Ftransform approximation method cannot be applied; otherwise, the extended direct Ftransform components and the approximation error are calculated.
If this error is less than a defined threshold, the process stops and the extended direct Ftransform components are stored; otherwise, n is increased by 1 and the process is iterated.
If the set of points is not sufficiently dense with respect to the fuzzy partition, the process stops with an error and the previous extended direct Ftransform components are stored.
In this last case, the best possible approximation of the FN is obtained, even if the approximation error is higher than the threshold. In order to create an huniform fuzzy partition of [a,b], the following basic functions are used: The approximation error is given by the Root Mean Square Error (RMSE) defined asThe threshold for the RMSE is set as a positive value much smaller than 1. The extended iFtransform method is schematized in Algorithm 2.

The fuzzification reads the input data and calculates the membership degree of each fuzzy set related to the input variable using (34). The strength of each rule is obtained via the min connective. If is the approximated membership degree of the input variable , the strength of the kth rule is as follows:The output fuzzy set is constructed as follows:where is the approximated membership function of the output variable to the fuzzy set in the consequent of the kth rule. The defuzzification function implements the CoG algorithm for converting the fuzzy output in a crisp number. We partition the support of the output fuzzy set in intervals with equal width. Let be the value of the midpoint of the ith interval. The output crisp value is as follows:We test our FRBS to a spatial decision problem in Section 5.
5. Experimental Results: The Livability in Residential Housings
We apply the extended Ftransform in a FRBS based on a set of census data of the 92 municipalities of the district of Naples (Italy), related to the residential housing. Our aim is to evaluate their livability whose crisp output variable is evaluated in percentage on the basis of a set of fuzzy rules extracted by experts in which the following six linguistic input variables are considered: x_{1} = average surface of the housings in m^{2}, x_{2} = percentage of housings with six or more rooms, x_{3} = percentage of residential buildings built since 2000, x_{4} = percentage of housings with centralized or autonomous heating system, x_{5} = percentage of housings with two or more showers or bathtubs, and x_{6} = percentage of housings with two or more restrooms. The crisp input data are extracted from the ISTAT dataset. The crisp value of the variable x_{1} is given by the total surface of the housings in the municipality dividing by the number of housings. The crisp values of the variables x_{2}, …, x_{6} are obtained dividing the corresponding absolute value recorded in the dataset by the total number of housings in the municipality. The domain of any variable is partitioned in 5 fuzzy sets labeled as “Low”, “Mean Low”, “Mean”, “Mean High”, and “High”. The fuzzy rule set contains the 62 fuzzy rules in Table 1 constructed by a set of twenty experts.

In the preprocessing phase we apply the extended Ftransform based algorithm to approximate the five FNs associated with each variable. Each FN is obtained as average of the membership values assigned by the experts in 200 points.
In Figure 5 we show some FNs and their approximations obtained by applying the extended Ftransform. We set the threshold to 0.01, so having a RMSE less than 0.01 for every FN.
The FNs (x_{1} = Low) and (x_{4} = High) have a degenerated side. In Tables 2(a)–2(f) we show the parameters a, c, d, b of each FN i = 1, 2, 3, 4, 5, 6 and the RMSE, respectively.
(a) Parameters and RMSE of the approximation for fuzzy sets of x_{1}  
 
(b) Parameters and RMSE of the approximation for fuzzy sets of x_{2}  
 
(c) Parameters and RMSE of the approximation for fuzzy sets of x_{3}  
 
(d) Parameters and RMSE of the approximation for fuzzy sets of x_{4}  
 
(e) Parameters and RMSE of the approximation for fuzzy sets of x_{5}  
 
(f) Parameters and RMSE of the approximation for fuzzy sets of x_{6}  

In Table 3 we show the parameters a, c, d, b of the FNs used for the output variable y and the RMSE obtained applying the extended Ftransform.

At the end of the preprocessing phase, the fuzzification of the input data is performed as well. In Figures 6(a)–6(f) we show the thematic maps (in a Geographic Information System environment) of the six input variables (i = 1,2, 3, 4, 5, 6), respectively, in the municipalities of the district of Naples. In each map the municipality is classified with the linguistic label of the fuzzy set with the highest approximated membership value.
(a) Thematic map for the input variable x1
(b) Thematic map for input variable x2
(c) Thematic map for input variable x3
(d) Thematic map for input variable x4
(e) Thematic map for input variable x5
(f) Thematic map for input variable x6
The defuzzified final values of livability in the residential housings (calculated in percentage) for every municipality are in Table 4.

In Figure 7 we show a thematic map of the index of livability in the residential housings: the label of output variable fuzzy set with the greatest membership degree is assigned for every municipality.
We compare these results with the ones obtained by approximating the input and output variables fuzzy sets with trapezoidal FNs, by using the approximation method in [12] (Table 5). We apply the inference system to the residential housing dataset again, by using the approximated trapezoidal FNs as fuzzy sets in the antecedents and consequents of the rule set. Then we calculate the RMSE and we calculate the number and the percentage of municipalities classified with a livability linguistic label different by the one contained in Figure 7.

The mean RMSE index obtained by using the trapezoidal FN is 6.3×10^{−2}: this value is greater than the threshold 1×10^{−2} set by applying the extended Ftransform. The mean difference in absolute value between the crisp livability obtained by using the trapezoidal approximation of the input and output FNs with respect to the ones obtained by using the extended iFtransform approximation overcomes 5%: this difference is generated by the greater error obtained by the approximation with trapezoidal FNs. The percentage of 7.61% of the municipalities is classified differently in the final map of livability underlining the effective improvement of the final results obtained with the extended iFtransform method. The seven municipalities with different livability class are given in Table 6.

We can appropriately select the RMSE threshold in order to increase the reliability of the final results; however, we point out that the choice of a very small threshold can lead to a fuzzy uniform partition too finer for which the dataset of the corresponding values is not sufficiently dense.
6. Conclusions
We present a new method based on the extended Ftransform to approximate FNs. We apply this method in a fuzzy rulebased system of Mamdani type related to a spatial analysis problem consisting in the evaluation of the livability of residential housings in the municipality of the district of Naples. In many spatial analysis problems, decisionmaking systems based on expert rules are used in order to extract thematic maps of a final index. A finer approximation of the membership functions of the fuzzy sets in the antecedents and in the consequence of the fuzzy rules is necessary to guarantee a good reliability of the final thematic maps. In many cases, for example, in participatory contexts in which knowledge is provided by different experts, these FNs are assigned on a discrete set of points. In future we propose to apply the extended Ftransform method to the approximation of FNs in multicriteria fuzzy decisionmaking problems. Data analysis shall be another field of investigation, mainly for establishing linear dependency of attributes from other attributes in large datasets via a fuzzy number: this is useful for the reduction of the size of these datasets.
Data Availability
The data are the census data extracted from the Italian Statistical Institute (ISTAT, Istituto nazionale di STATistica) at the website: the crisp input data are extracted from the ISTAT dataset (http://daticensimentopopolazione.istat.it).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright
Copyright © 2018 Ferdinando Di Martino and Salvatore Sessa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.