# A Method Based on Extended Fuzzy Transforms to Approximate Fuzzy Numbers in Mamdani Fuzzy Rule-Based System

Authors: Ferdinando Di Martino; Salvatore Sessa
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

We propose a new Mamdani fuzzy rule-based 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.


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## Body

## 1. Introduction

A fuzzy number (FN) is a fuzzy set with membership function A: Reals →[0,1] defined as (1)Ax=0IFx<aA-xIFa≤x<c1IFc≤x≤dA+xIFd<x≤b0IFx>bwhere a ≤ c ≤ d ≤ b,A-: [a,c] →[0,1] is a not decreasing continuous function withA−(a) = 0,A− (c) = 1 andA+: [b,d] →[0,1] is a not increasing continuous function withA+(d) = 1, A+(b) = 0. A- and A+ are calledleft side andright side of A, respectively.Complicated left-side and right-side 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 decision-making are the trapezoidal and triangular FNs. In a trapezoidal FN the functions A- and A+ are linear; for instance, A- (x) = (x-a)/(c-a) and A+(x) = (b-x)/(b-d) 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 Figure1, the membership degree f(T) 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.Figure 1
Example of FN constructed for a discrete set of points and approximated with a trapezoidal membership function.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 (F-transform) [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 F-transform 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 F-transform is used to approximate type 2 FNs. In [13], the extended iF-transform 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 F-transform method divides the interval [a,b] in n subintervals of width h = (n-1)/(b-a). The points x1= a, x2= a+h,…, xi= a+(i-1)h,…, xn= b are called nodes: a uniform fuzzy partition of [a,b] is created by assigning n fuzzy sets with continuous membership functions A1,…,An: [a,b] → [0,1], called basic functions, where Ai(x) = 0 if x∉(xi-1,xi+1),  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 (xi-1,xi+1) 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 (xi-1,xi+1).Algorithm 1:Approximation of a set of data by using the extended inverse F-transform.
( 1 ) n≔n0 ( 2 )  Create the fuzzy partition ( 3 )  Calculate the direct F-transform components ( 4 )  WHILE the dataset is sufficiently dense with respect to the fuzzy partition ( 5 )  Calculatethe approximation error ( 6 )  IF (approximation error ≤ threshold) THEN ( 7 )  Store the direct F-transform components ( 8 )  RETURN  “SUCCESS” ( 9 )  END IF ( 10 ) n≔n+1 ( 11 )  Calculate the extended direct F-transform components ( 12 )  END WHILE ( 13 )  RETURN  “ERROR: Dataset not sufficiently dense” ( 14 )  ENDFigure 2
Example of input dataset nonsufficiently dense with respect to the fuzzy partition.The FNs are largely used in fuzzy reasoning systems, particularly in fuzzy rule-based inference systems in which fuzzy rules are applied in an inferential process. In a fuzzy rule-based 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 iF-transform 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: Section2 contains the basic notions of fuzzy number and F-transform and in Section 3 we introduce the extended iF-transform method which in Section 4 is applied to a Fuzzy Rule-Based 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 iF-transform 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 n0; 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 F-transform 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 iF-transform 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 iF-transform 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:(2)r1:xisA1ORyisB1→zisC1r2:xisA2ANDyisB2→zisC2where A1 and A2 are two FNs for the linguistic input variable x, B1 and B2 are two FNs for the input linguistic variable y, and C1 and C2 are two FNs for the output variable z. Applying the extended iF-transforms to evaluate each fuzzy set, we suppose that A1-x=0.4, A1+x=0.7, B1-x=0.7, B2+x=0.3. With max (resp., min) operator as connective OR (resp., AND), we obtain the value of the two rules: r1 = max(0.4, 0.7) = 0.7 and r2 = min(0.7, 0,3) = 0.3. In the defuzzification process we reconstruct the output fuzzy set as(3)Cz=maxminC1z,s1,minC2z,s2,where s1 and s2 are suitable thresholds prefixed a priori (Figure 3).Figure 3
Defuzzification of the output fuzzy set.The CoG method is useful for obtaining the final crisp valuez^ of the output variable as (4)z^=∑i=1NcCzi·zi∑i=1NcCziwhere Nc is the number of rules and z1< z2 <⋯< zNc are points of the support of C. In Figure 3 we give an example.


## 1.1. Preliminaries

As already shown in [17], the extended iF-transform 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 n0; 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 F-transform 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 iF-transform 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 iF-transform 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:(2)r1:xisA1ORyisB1→zisC1r2:xisA2ANDyisB2→zisC2where A1 and A2 are two FNs for the linguistic input variable x, B1 and B2 are two FNs for the input linguistic variable y, and C1 and C2 are two FNs for the output variable z. Applying the extended iF-transforms to evaluate each fuzzy set, we suppose that A1-x=0.4, A1+x=0.7, B1-x=0.7, B2+x=0.3. With max (resp., min) operator as connective OR (resp., AND), we obtain the value of the two rules: r1 = max(0.4, 0.7) = 0.7 and r2 = min(0.7, 0,3) = 0.3. In the defuzzification process we reconstruct the output fuzzy set as(3)Cz=maxminC1z,s1,minC2z,s2,where s1 and s2 are suitable thresholds prefixed a priori (Figure 3).Figure 3
Defuzzification of the output fuzzy set.The CoG method is useful for obtaining the final crisp valuez^ of the output variable as (4)z^=∑i=1NcCzi·zi∑i=1NcCziwhere Nc is the number of rules and z1< z2 <⋯< zNc are points of the support of C. In Figure 3 we give an example.


## 2. Fuzzy Numbers and F-Transforms

### 2.1. Fuzzy Numbers

Given a valueα∈[0,1], we denote with Aα, calledα-cut of a FN A, the crisp set containing the elements x∈R with a membership degree greater than or equal to α. We also use the interval (5)Aα=a1α,a2αwhere(6)a1α=infx∈R:Ax≥α(7)a2α=supx∈R:Ax≥αFor α = 1, [A]1=[a1(1),a2(1)] is called thecore of the FN and denoted bycore(A). Note that for α = 0, [A]0=[a1(0),a2(0)]=-∞,+∞= R. The support of a fuzzy set is given by the closure of the crisp set (8)suppA=x∈R∣Ax>0Given two arbitrary FNs, A and B, two metrics are considered in [23, 24]:theChebyshev distance(9)dA,B=supx∈R:Ax-Bxand the extension of the Euclidean metric given by (10)dA,B=∫01a1α-b1α2dα+∫01a2α-b2α2dαTwo properties of A are given in [25] called ambiguity and value, defined as(11)AmbrA=∫01rα·a2α-a1αdαand(12)ValrA=∫01rα·a2α+a1αdα,respectively, where r: [0,1]→[0,1] is a not decreasing function called reducing function with r(0) = 0 and r(1) = 1. Another important propriety is the expected interval of A, introduced in [3, 24], defined as follows:(13)EIA=∫01a1αdα,∫01a2αdαWe have EI(A) = [(a+c)/2, (d+b)/2] for a trapezoidal FN A.


### 2.2. Direct and Inverse F-Transforms

Following the definitions and notations of [15], let n ≥ 2 andP = {x1, x2, …, xn} be a set of points of [a,b], called nodes, such that x1 = a < x2 <⋯< xn = b. Let {A1,…,An} be an assigned family of fuzzy sets with membership functions A1(x),…,An(x): [a,b] →  [0,1], called basic functions. We say that it constitutes a fuzzy partition of [a,b] if the following properties hold:(1)
Ai(xi) =1 for every i =1,2,…,n(2)
Ai(x) = 0 if x∉ (xi-1,xi+1) for i=2,…,n-1(3)
Ai(x) is a continuous function on [a,b](4)
Ai(x) strictly increases on [xi-1,xi] for i = 2, …, n and strictly decreases on [xi,xi+1] for i = 1,…, n-1(5)
∑i=1nAi(x)=1 for every x∈[a,b]Furthermore, we say that the fuzzy sets{A1,…,An} form an h-uniform fuzzy partition of [a,b] if(6)
n ≥ 3 andxi = a + h·(i-1), where h = (b-a)/(n-1) and i = 1, 2, …, n (that is, the nodes are equidistant)(7)
Ai(xi – x) = Ai(xi + x) for every x∈[0,h] and i = 2,…, n-1(8)
Ai+1 (x) = Ai(x - h) for every x∈[xi,xi+1] and i = 1,2,…, n-1Letf(x) be a continuous function on [a,b]. The quantity(14)Fi=∫abfxAix∫abAix,for i = 1, …, n, is the ith component of the direct F-transform {F1, F2, …, Fn} of f with respect to the family of basic functions {A1, A2, …, An}. If this fuzzy partition is h-uniform, the components are as follows [26]:(15)
F
i
=
2
h
-
1
∫
x
1
x
2
f
x
A
1
x
d
x
i
f
i
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1
h
-
1
∫
x
i
-
1
x
i
f
x
A
i
x
d
x
i
f
i
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2
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…
,
n
-
1
2
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x
n
-
1
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x
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n
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d
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f
i
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nThe function (16)fF,nx=∑i=1nFiAix,where x∈[a,b], is defined as the iF-transform off with respect to {A1, A2, …, An} and it approximates f in the sense of the following theorem [26].Theorem 1.
Let f(x) be a continuous function on [a,b]. For everyε > 0, then there exist an integer n(ε) and a fuzzy partition {A1, A2, …, An(ε)} of [a,b] such that |f(x) - fF,n(ε)|<ε with respect to the existing fuzzy partition.In the discrete case we know that the functionf assumes assigned values in the points p1,…,pm of [a,b]. If the set {p1,…,pm} is sufficiently dense with respect to the fixed partition {A1, A2, …, An}, that is, for each i = 1,…,n, there exists an index j∈{1,…,m} such that Ai(pj) > 0, we can define the n-tuple {F1, F2,…, Fn} as the discrete direct F-transform off with respect to {A1, A2, …, An}, where each Fi is given by(17)Fi=∑j=1mfpj·Aipj∑j=1mAipjfor i=1,…,n. Similarly we define the discrete iF-transform off with respect to the {A1, A2, …, An} by setting(18)fF,npj=∑i=1nFiAipjfor every j∈{1,…,m}. We have the following theorem [15].Theorem 2.
Letf(x) be a function assigned on a set of points {p1,…, pm}⊆[a,b]. Then, for every ε > 0, there exist an integer n(ε) and a related fuzzy partition {A1, A2, …, An(ε)} of [a,b] such that {p1,…,pm} is sufficiently dense with respect to the existing fuzzy partition and for every pj∈[a,b],  j = 1,…,m, the inequality(19)fpj-fF,nεpj<εremains true.


## 2.1. Fuzzy Numbers

Given a valueα∈[0,1], we denote with Aα, calledα-cut of a FN A, the crisp set containing the elements x∈R with a membership degree greater than or equal to α. We also use the interval (5)Aα=a1α,a2αwhere(6)a1α=infx∈R:Ax≥α(7)a2α=supx∈R:Ax≥αFor α = 1, [A]1=[a1(1),a2(1)] is called thecore of the FN and denoted bycore(A). Note that for α = 0, [A]0=[a1(0),a2(0)]=-∞,+∞= R. The support of a fuzzy set is given by the closure of the crisp set (8)suppA=x∈R∣Ax>0Given two arbitrary FNs, A and B, two metrics are considered in [23, 24]:theChebyshev distance(9)dA,B=supx∈R:Ax-Bxand the extension of the Euclidean metric given by (10)dA,B=∫01a1α-b1α2dα+∫01a2α-b2α2dαTwo properties of A are given in [25] called ambiguity and value, defined as(11)AmbrA=∫01rα·a2α-a1αdαand(12)ValrA=∫01rα·a2α+a1αdα,respectively, where r: [0,1]→[0,1] is a not decreasing function called reducing function with r(0) = 0 and r(1) = 1. Another important propriety is the expected interval of A, introduced in [3, 24], defined as follows:(13)EIA=∫01a1αdα,∫01a2αdαWe have EI(A) = [(a+c)/2, (d+b)/2] for a trapezoidal FN A.


## 2.2. Direct and Inverse F-Transforms

Following the definitions and notations of [15], let n ≥ 2 andP = {x1, x2, …, xn} be a set of points of [a,b], called nodes, such that x1 = a < x2 <⋯< xn = b. Let {A1,…,An} be an assigned family of fuzzy sets with membership functions A1(x),…,An(x): [a,b] →  [0,1], called basic functions. We say that it constitutes a fuzzy partition of [a,b] if the following properties hold:(1)
Ai(xi) =1 for every i =1,2,…,n(2)
Ai(x) = 0 if x∉ (xi-1,xi+1) for i=2,…,n-1(3)
Ai(x) is a continuous function on [a,b](4)
Ai(x) strictly increases on [xi-1,xi] for i = 2, …, n and strictly decreases on [xi,xi+1] for i = 1,…, n-1(5)
∑i=1nAi(x)=1 for every x∈[a,b]Furthermore, we say that the fuzzy sets{A1,…,An} form an h-uniform fuzzy partition of [a,b] if(6)
n ≥ 3 andxi = a + h·(i-1), where h = (b-a)/(n-1) and i = 1, 2, …, n (that is, the nodes are equidistant)(7)
Ai(xi – x) = Ai(xi + x) for every x∈[0,h] and i = 2,…, n-1(8)
Ai+1 (x) = Ai(x - h) for every x∈[xi,xi+1] and i = 1,2,…, n-1Letf(x) be a continuous function on [a,b]. The quantity(14)Fi=∫abfxAix∫abAix,for i = 1, …, n, is the ith component of the direct F-transform {F1, F2, …, Fn} of f with respect to the family of basic functions {A1, A2, …, An}. If this fuzzy partition is h-uniform, the components are as follows [26]:(15)
F
i
=
2
h
-
1
∫
x
1
x
2
f
x
A
1
x
d
x
i
f
i
=
1
h
-
1
∫
x
i
-
1
x
i
f
x
A
i
x
d
x
i
f
i
=
2
,
…
,
n
-
1
2
h
-
1
∫
x
n
-
1
x
n
f
x
A
n
x
d
x
i
f
i
=
nThe function (16)fF,nx=∑i=1nFiAix,where x∈[a,b], is defined as the iF-transform off with respect to {A1, A2, …, An} and it approximates f in the sense of the following theorem [26].Theorem 1.
Let f(x) be a continuous function on [a,b]. For everyε > 0, then there exist an integer n(ε) and a fuzzy partition {A1, A2, …, An(ε)} of [a,b] such that |f(x) - fF,n(ε)|<ε with respect to the existing fuzzy partition.In the discrete case we know that the functionf assumes assigned values in the points p1,…,pm of [a,b]. If the set {p1,…,pm} is sufficiently dense with respect to the fixed partition {A1, A2, …, An}, that is, for each i = 1,…,n, there exists an index j∈{1,…,m} such that Ai(pj) > 0, we can define the n-tuple {F1, F2,…, Fn} as the discrete direct F-transform off with respect to {A1, A2, …, An}, where each Fi is given by(17)Fi=∑j=1mfpj·Aipj∑j=1mAipjfor i=1,…,n. Similarly we define the discrete iF-transform off with respect to the {A1, A2, …, An} by setting(18)fF,npj=∑i=1nFiAipjfor every j∈{1,…,m}. We have the following theorem [15].Theorem 2.
Letf(x) be a function assigned on a set of points {p1,…, pm}⊆[a,b]. Then, for every ε > 0, there exist an integer n(ε) and a related fuzzy partition {A1, A2, …, An(ε)} of [a,b] such that {p1,…,pm} is sufficiently dense with respect to the existing fuzzy partition and for every pj∈[a,b],  j = 1,…,m, the inequality(19)fpj-fF,nεpj<εremains true.


## 3. The Extended iF-Transform and Fuzzy Numbers

In [15] the extended iF-transform of a continuous function f is introduced in order to preserve the monotonicity as follows. For an h-uniform fuzzy partition {A1, A2, …, An}, the function f is extended to [a-h,b+h] as follows: (20)f¯x=2fa-f2a-xifx∈a-h,afxifx∈a,b2fb-f2b-xifx∈b,b+hThen the following basic functions are defined as(21)A¯1x=A12a-xifx∈a-h,aA1xifx∈a,a+hA¯ix=Aixfori=2,…,n-1A¯nx=Anxifx∈b-h,bAn2b-xifx∈b,b+hThen the ith component F¯i of the extended direct F-transform of f with respect to the family of basic functions {A1, A2, …, An} is given by(22)F¯1=1h∫a-ha+hf¯xA1¯xdx,F¯ix=Fixi=2,…,n-1F¯n=1h∫b-hb+hf¯xAn¯xdxHence the extended iF-transform of f is given by(23)f¯F,nx=F1¯A¯1x+∑i=2n-1FiAix+Fn¯An¯xx∈a-h,b+hBy [13, Lemma 9], we obtain that (24)fF,n¯a=F¯1=fafF,n¯b=F¯n=fbLet S be a fuzzy number with a continuous membership function and supp(S) = [a,b]. We consider an h-uniform fuzzy partition {A1, A2, …, An} of [a,b] with n ≥ 3 and let S¯F,n(x) be the extended iF-transform of S. We obtain that [13, Prop. 11](25)iS¯F,na=S¯F,nb=0iiS¯F,nx>0∀x∈a,biiiS¯F,nx=∑i=2n-1SiAixwhere Si is the ith component of the direct F-transform of S (cfr., formulae (15)). Theorem 13 of [13] provides the approximation property of the extended iF-transform as follows.Theorem 3.
Let S be a FN having a continuous membership function and supp(S) = [a,b]. Let a fuzzy partition {A1, A2, …, An} of [a,b] be h-uniform with n ≥ 3 and S¯F,n(x) be the extended iF-transform of S calculated by (23). Then the following inequality holds:(26)supx∈a,bS¯F,nx-Sx≤2ωS,hwhere ω(S,h) is the modulus of continuity of S given by(27)ωS,h=supx,y∈a,b:a-b≤h⁡Sx-SyAnother important theorem [13, Th. 14] is the following.Theorem 4.
Let S be a FN having a continuous membership function, supp(S) = [a,b], and core(S) = [c,d], a < c < d < b. Let a fuzzy partition {A1, A2, …, An} of [a,b] be h-uniform with n ≥ 3 and a fuzzy set T such that T(x) = S¯F,n(x) calculated by (23) in [a,b] and T(x)=0 if x∉[a,b]. If h = (b-a)/(n-1) is such that h ≤ min{(d-c)/4, c-a, b-d}, then T is a FN for which the following hold:1.
supp(T) = supp(S)2.
If core(T) = [c′,d′], then c ≤ c′ ≤ d′ ≤ d, |c-c′| ≤ 2h, |d-d′| ≤ 2h3.
supx∈[a,b]T(x)-S(x)≤4ω(S,h)4.
If S- strictly increases on [a,c], then T strictly increases on [a,c′]5.
If S+ strictly decreases on [d,b], then T strictly decreases on [d′,b]
The preservation of the properties “ambiguity” and “value” of a FN and their approximation with an extended iF-transform is given by the following theorem in [13, Theorem 27]:Theorem 5.
Let S be a FN having a continuous membership function with supp(S) = [a,b] and core(S) = [c,d], a < c < d < b. Let a fuzzy partition {A1, A2, …, An} of [a,b] be h-uniform with n ≥ 3 and a fuzzy set T such that T(x) = S¯F,n(x) given by (23) in [a,b] and T(x)=0 if x∉[a,b]. Let core(T) = [c′,d′] with c′ ≤ d′. By putting δh=2ω(f,h), we obtain that(28)AmbrS-AmbrT≤K~h,1S+K~h,2Sδh(29)ValrS-ValrT≤K~h,1S+K~h,2Sδhwhere K~h,1(S)=c-a+c+4h and K~h,2(S)=b-d+b+4h.In order to apply the extended iF-transform to approximate a FN S with one-element core, in [13] the concept of regular h-uniform partition of [a,b] is introduced as an h-uniform partition of [a,b] such that A1 is differentiable in [a,x2], Ai is differentiable in [xi-1,xi+1] for i = 2,…,n-1, and An is differentiable in [xn-1,b]. Thus, we can define the normalized extended iF-transform given as(30)S¯¯F,nx=S¯F,nxmaxx∈a,bS¯F,nxx∈a-h,b+hA 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(S) = [a,b] and core(S) = {c}, a < c < b. Let {A1, A2, …, An} be a regular h-uniform partition of [a,b] and T(x) = S¯¯F,n(x) a fuzzy set given by (30) in [a,b] and T(x)=0 if x∉[a,b]. Let core(T) = [c′,d′] with c′ ≤ d′ and δh=8/(1-4ω(f,h))ω(f,h). Then the following properties hold:(31)AmbrS-AmbrT≤K~1S+K~2Sδh(32)ValrS-ValrT≤K~1S+K~2Sδhwhere K~1(S)=c-a+3c+2max⁡(|a|,|b|) and K~2(S)=b-c+3c+2max⁡(|a|,|b|).Now we suppose that the membership values of a FN S in form (1) are assigned on a discrete set of m points a = p1 < p2 <⋯< pm-1 < pm = b. We consider an h-uniform fuzzy partition {A1, A2, …, An} of [a,b]. If the set of points are sufficiently dense with respect to the fuzzy partition, i.e., if (33)∑j=1mAipj>0i=2,…,n-1,then the extended iF-transform of S is defined for any x∈[a,b] as follows [13]:(34)S¯F,nx=S¯1A1x+∑i=2n-1SiAix+S¯nAnxwhere S¯1=S(a), S¯n=S(b), and Si is the ith component of the direct F-transform of S in [a,b] for i=1,…,n. Similarly, it can be proved that all the above properties of the extended iF-transform of a FN with continuous membership function apply in the discrete case as well.


## 4. Extended iF-Transform and Fuzzy Rule-Based System

Let the expert knowledge be formed by a set of fuzzy rules in a linguistic fuzzy model:(35)Rk:IFx1=X1kΔ1x2=X2kΔ2⋯Δnxn=XnkTHENy=Yiwhere x1, x2,…, xn are input variables, y is the output variable, X1i, X2i,…, Xni, Yi are fuzzy sets and the operator Δi (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 iF-transforms. 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 p1 = a < p2 <⋯< pm-1 < pm = 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 iF-transform 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 F-transform storing them in the fuzzy rule set. The crisp input data are fuzzified via (34) by using the stored direct F-transform components of the FNs. The inference engine applies to the max-min 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.Figure 4
Schema of the proposed FRBS.The extended iF-transform approximates each fuzzy number by considering the set of points in which its membership function is assigned. This function creates an h-uniform 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 n0 (for example, n0 = 3). If the set of points is not sufficiently dense with respect to the fuzzy partition, the F-transform approximation method cannot be applied; otherwise, the extended direct F-transform components and the approximation error are calculated.If this error is less than a defined threshold, the process stops and the extended direct F-transform 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 F-transform 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 h-uniform fuzzy partition of [a,b], the following basic functions are used:(36)A1x=0.5cos⁡πhx-a+1ifx∈a,x20otherwiseAix=0.5cos⁡πhx-xi+1ifi∈xi-1,xi+10otherwisei=2,…,n-1Anx=0.5cos⁡πhx-xn-1+1ifi∈xn-1,b0otherwiseThe approximation error is given by the Root Mean Square Error (RMSE) defined as(37)RMSE=∑j=1nS¯F,npj-Spj2nThe threshold for the RMSE is set as a positive value much smaller than 1. The extended iF-transform method is schematized in Algorithm 2.Algorithm 2:Extended F-transform approximation.
Description:  Approximate a fuzzy number with an extended iF- transform Input:  Initial fuzzy partition sizen0 Threshold parameter A set of m points and their membership function value ( p 1 , f ( p 1 ) ) , … , ( p n , f ( p n ) ) Output:  RMSE error Extended Direct F-transform components ( 1 ) n≔ n0 ( 2 )  Read the dataset of points ( 3 )  Create a h-uniform fuzzy partition by using the basic functions (36) ( 4 )  Calculate the extended direct F-transform components ( 5 )  WHILE the dataset is sufficiently dense with respect to the fuzzy partition ( 6 )  Calculate the RMSE approximation error (37) ( 7 )  IF (RMSE approximation error ≤ threshold) THEN ( 8 )  Store the extended direct F-transform components and the RSME error ( 9 )  RETURN “Success” ( 10 )  END IF ( 11 ) n≔n+1 ( 12 )  Create a h-uniform fuzzy partition by using the basic functions (36) ( 13 )  Calculate the extended direct F-transform components ( 14 )  END WHILE ( 15 )  Store the extended direct previous F-transform components (n = n-1) and the RMSE error ( 16 )  RETURN “ERROR: Dataset non sufficiently dense” ( 17 )  ENDThe 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 fXhk′(xk) is the approximated membership degree of the input variable xk, the strength of the kth rule is as follows:(38)Sk:minfXh1′x1,fXh2′x2,…,fXhk′xkThe output fuzzy set is constructed as follows:(39)fBy=maxminfY1′y,s1,minfY2′y,s2,…,minfYr′y,srwhere fB′(y) 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 NB intervals with equal width. Let yi be the value of the midpoint of the ith interval. The output crisp value y^ is as follows:(40)y^=∑i=1NBfByi·yi∑i=1NBfByiWe test our FRBS to a spatial decision problem in Section 5.


## 5. Experimental Results: The Livability in Residential Housings

We apply the extended F-transform 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: x1 = average surface of the housings in m2, x2 = percentage of housings with six or more rooms, x3 = percentage of residential buildings built since 2000, x4 = percentage of housings with centralized or autonomous heating system, x5 = percentage of housings with two or more showers or bathtubs, and x6 = percentage of housings with two or more restrooms. The crisp input data are extracted from the ISTAT dataset. The crisp value of the variable x1 is given by the total surface of the housings in the municipality dividing by the number of housings. The crisp values of the variables x2, …, x6 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.Table 1
The fuzzy rule set used for evaluating the livability in residential housings.
ID Rule r1 IF (x1 = High) AND (x2 = High) AND (x3 = High) THEN y = High r2 IF (x1 = High) AND (x2 = Mean High) AND (x4 = Mean High) THEN y = Mean High r3 IF (x1 = High) AND (x3 = High) THEN y = High r4 IF (x1 = High) AND (x4 = High) THEN y = High r5 IF (x1 = High) AND (x3 = Mean High) AND (x5 = High) THEN y = High r6 IF (x1 = High) AND (x3 = Mean High) AND (x6 = High) THEN y = High r7 IF (x1 = High) AND (x3 = Mean High) AND (x5 = Mean High) THEN y = Mean High r8 IF (x1 = High) AND (x3 = Mean High) AND (x6 = Mean High) THEN y = Mean High r9 IF (x1 = High) AND (x4 = Mean High) AND (x5 = High) THEN y = High r10 IF (x1 = High) AND (x4 = Mean High) AND (x6 = High) THEN y = High r11 IF (x1 = High) AND (x4 = Mean High) AND (x5 = Mean High) THEN y = Mean High r12 IF (x1 = High) AND (x4 = Mean High) AND (x6 = Mean High) THEN y = Mean High r13 IF (x2 = High) AND (x3 = High) THEN y = High r14 IF (x2 = High) AND (x4 = High) THEN y = High r15 IF (x3 = High) AND (x4 = High) THEN y = High r16 IF (x3 = High) AND (x4 = Mean High) AND (x5 = High) THEN y = High r17 IF (x3 = High) AND (x4 = Mean High) AND (x5 = Mean High) THEN y = Mean High r18 IF (x3 = High) AND (x4 = Mean High) AND (x5 = Mean) THEN y = Mean High r19 IF (x3 = High) AND (x4 = Mean High) AND (x6 = High) THEN y = High r20 IF (x3 = High) AND (x4 = Mean High) AND (x6 = Mean High) THEN y = Mean High r21 IF (x4 = High) AND (x5 = High) THEN y = High r22 IF (x1 = Mean High ) AND (x3 = Mean High) THEN y = Mean High r23 IF (x1 = Mean High ) AND (x3 = Mean) THEN y = Mean High r24 IF (x1 = Mean High ) AND (x4 = Mean High) THEN y = Mean High r25 IF (x1 = Mean High ) AND (x4 = Mean) THEN y = Mean High r26 IF (x2 = Mean High) AND (x3 = High) THEN y = Mean High r27 IF (x2 = Mean High) AND (x3 = Mean High) THEN y = Mean High r28 IF (x2 = Mean High) AND (x4 = High) THEN y = Mean High r29 IF (x2 = Mean High) AND (x4 = Mean High) THEN y = Mean High r30 IF (x1 = Mean) AND (x3 = Mean High) THEN y = Mean r31 IF (x1 = Mean) AND (x3 = Mean) THEN y = Mean r32 IF (x1 = Mean) AND (x4 = Mean High) THEN y = Mean r33 IF (x1 = Mean ) AND (x4 = Mean) THEN y = Mean r36 IF (x2 = Mean) AND (x3 = Mean) THEN y = Mean r37 IF (x2 = Mean) AND (x4 = Mean) THEN y = Mean r38 IF (x3 = Mean) AND (x5 = Mean) THEN y = Mean r39 IF (x3 = Mean) AND (x6 = Mean) THEN y = Mean r40 IF (x4 = Mean) AND (x5 = Mean) THEN y = Mean r41 IF (x4 = Mean) AND (x6 = Mean) THEN y = Mean r42 IF (x1 = Mean) AND (x3 = Mean Low) THEN y = Mean Low r43 IF (x1 = Mean) AND (x4 = Mean Low) THEN y = Mean Low r44 IF (x1 = Mean Low) AND (x3 = Mean) THEN y = Mean Low r45 IF (x1 = Mean Low) AND (x4 = Mean) THEN y = Mean Low r46 IF (x2 = Mean Low) AND (x3 = Mean) THEN y = Mean Low r47 IF (x2 = Mean Low) AND (x4 = Mean) THEN y = Mean Low r48 IF (x3 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low r49 IF (x3 = Mean Low) AND (x6 = Mean Low) THEN y = Mean Low r50 IF (x4 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low r51 IF (x4 = Mean Low) AND (x6 = Mean Low) THEN y = Mean Low r52 IF (x3 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low r53 IF (x1 = Low) AND (x4 = Mean Low) THEN y = Low r54 IF (x1 = Low) AND (x4 = Low) THEN y = Low r55 IF (x2 = Low) AND (x4 = Mean Low) THEN y = Low r56 IF (x2 = Low) AND (x4 = Low) THEN y = Low r57 IF (x2 = Low) AND (x5 = Low) THEN y = Low r58 IF (x2 = Low) AND (x6 = Low) THEN y = Low r59 IF (x3 = Low) AND (x5= Low) THEN y = Low r60 IF (x3 = Low) AND (x6= Low) THEN y = Low r61 IF (x4 = Low) AND (x5= Low) THEN y = Low r62 IF (x4 = Low) AND (x6= Low) THEN y = LowIn the preprocessing phase we apply the extended F-transform 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 Figure5 we show some FNs and their approximations obtained by applying the extended F-transform. We set the threshold to 0.01, so having a RMSE less than 0.01 for every FN.Figure 5
Fuzzy numbers x1 = Low, x2 = Mean, x3 = Mean High, and x4 = High (in blue) and their extended iF-transform approximations (in red).The FNs (x1 = Low) and (x4 = High) have a degenerated side. In Tables 2(a)–2(f) we show the parameters a, c, d, b of each FN xi  i = 1, 2, 3, 4, 5, 6 and the RMSE, respectively.Table 2  (a)
Parameters and RMSE of the approximation for fuzzy sets of x1
x 1 (m 2 ) Fuzzy number a c d b RMSE Low 20 20 45 70 9.11×10-3 Mean Low 45 70 75 90 9.91×10-3 Mean 75 90 95 100 9.17×10-3 Mean High 95 100 115 125 9.76×10-3 High 110 120 150 150 9.34×10-3  (b)
Parameters and RMSE of the approximation for fuzzy sets of x2
x 2 Fuzzy number a c d b RMSE Low 0 0 1 4 9.18×10-3 Mean Low 0.5 3 6 8 9.43×10-3 Mean 2 7 12 20 9.19×10-3 Mean High 8 12 15 25 9.57×10-3 High 15 25 50 50 9.15×10-3  (c)
Parameters and RMSE of the approximation for fuzzy sets of x3
x 3 Fuzzy number a c d b RMSE Low 0 0 0.5 1 9.21×10-3 Mean Low 0.4 0.6 1 1.5 9.35×10-3 Mean 1 2 4 6 9.33×10-3 Mean High 2 4 7 10 9.02×10-3 High 6 10 30 30 9.26×10-3  (d)
Parameters and RMSE of the approximation for fuzzy sets of x4
x 4 Fuzzy number a c d b RMSE Low 0 0 30 40 9.24×10-3 Mean Low 30 50 60 70 9.29×10-3 Mean 60 65 70 80 9.49×10-3 Mean High 75 80 85 90 9.35×10-3 High 85 95 100 100 9.08×10-3  (e)
Parameters and RMSE of the approximation for fuzzy sets of x5
x 5 Fuzzy number a c d b RMSE Low 0 0 10 15 9.30×10-3 Mean Low 7 15 20 25 9.52×10-3 Mean 20 25 30 35 9.25×10-3 Mean High 30 35 40 50 9.31×10-3 High 40 50 100 100 9.37×10-3  (f)
Parameters and RMSE of the approximation for fuzzy sets of x6
x 6 Fuzzy number a c d b RMSE Low 0 0 10 15 9.32×10-3 Mean Low 7 15 25 30 9.19×10-3 Mean 22 28 32 35 9.24×10-3 Mean High 30 40 45 55 9.48×10-3 High 50 60 100 100 9.28×10-3In Table3 we show the parameters a, c, d, b of the FNs used for the output variable y and the RMSE obtained applying the extended F-transform.Table 3
Parameters and RMSE of the approximation for fuzzy sets of output y.
y Fuzzy number a c d b RMSE Low 0 0 10 20 9.67×10-3 Mean Low 10 20 30 40 9.32×10-3 Mean 30 40 60 70 9.46×10-3 Mean High 50 70 80 85 9.78×10-3 High 80 90 100 100 9.31×10-3At the end of the preprocessing phase, the fuzzification of the input data is performed as well. In Figures6(a)–6(f) we show the thematic maps (in a Geographic Information System environment) of the six input variables xi (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.Figure 6 (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 x6The defuzzified final values of livability in the residential housings (calculated in percentage) for every municipality are in Table4.Table 4
Defuzzified values obtained for livability of residential housings.
Municipality y ^ Municipality y ^ Municipality y ^ Acerra 55.18 Forio 40.83 Procida 20.68 Afragola 27.12 Frattamaggiore 55.02 Qualiano 18.36 Agerola 59.21 Frattaminore 33.8 Quarto 65.52 Anacapri 60.29 Giugliano in Campania 81.75 Roccarainola 82.01 Arzano 23.34 Gragnano 29.64 SanGennaro Vesuviano 76.18 Bacoli 24.65 Grumo Nevano 55 SanGiorgio a Cremano 23.14 Barano d'Ischia 58.36 Ischia 27.13 SanGiuseppe Vesuviano 51.03 Boscoreale 32.23 Lacco Ameno 26.69 San Paolo BelSito 63.46 Boscotrecase 48.7 Lettere 42.57 San Sebastiano al Vesuvio 82.37 Brusciano 63.37 Liveri 81.39 San Vitaliano 66.52 Caivano 44.85 Marano di Napoli 24.93 Santa Maria la Carità 56.94 Calvizzano 52.06 Mariglianella 82.39 Sant'Agnello 33.85 Camposano 50.84 Marigliano 53.68 Sant'Anastasia 64.19 Capri 47.32 Massa di Somma 36.15 Sant'Antimo 47.82 Carbonara di Nola 73.29 MassaLubrense 71.5 Sant'Antonio Abate 58.19 Cardito 47.68 Melito di Napoli 26.87 Saviano 76.84 Casalnuovo di Napoli 23.45 Meta 56.38 Scisciano 88.93 Casamarciano 92.74 Monte di Procida 32.69 Serrara Fontana 54.08 Casamicciola Terme 34.61 Mugnano di Napoli 29.14 Somma Vesuviana 52.11 Casandrino 39.26 Napoli 53.82 Sorrento 20.18 Casavatore 33.15 Nola 75.35 Striano 73.69 Casola di Napoli 38.77 Ottaviano 52.94 Terzigno 52.01 Casoria 34.02 Palma Campania 62.9 Torre Annunziata 25.12 Castellammare di Stabia 44.26 Piano di Sorrento 60.67 Torre del Greco 26.36 Castello di Cisterna 73.89 Pimonte 27 Trecase 55.8 Cercola 49.67 Poggiomarino 75 Tufino 96.44 Cicciano 67.16 Pollena Trocchia 25.13 Vico Equense 20.37 Cimitile 87.38 Pomigliano d'Arco 19.75 Villaricca 78.36 Comiziano 85.46 Pompei 17.89 Visciano 78.45 Crispano 39.07 Portici 22.51 Volla 55.83 Ercolano 28.14 Pozzuoli 39.43In Figure7 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.Figure 7
Thematic map of index of livability in residential housings.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.Table 5
Comparisons obtained approximating input and output FN with trapezoidal FN.
Comparison parameter Value Mean RMSE index for the fuzzy sets approximation with trapezoidal FNs 6.3×10-2 Mean difference of the final crisp livability values compared with the ones obtained by using the extended iF-transform method 5.58% Number of municipalities classified with different linguistic labels 7 Percentage of municipalities classified with different linguistic labels 7.61%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 F-transform. 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 iF-transform 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 iF-transform method. The seven municipalities with different livability class are given in Table 6.Table 6
Municipalities with different livability class.
Municipality Extended IFtr livability class Trapezoidal livability class Casola di Napoli Mean Mean Low Casoria Mean Low Mean Crispano Mean Mean Low Massa di Somma Mean Mean Low Pozzuoli Mean Mean Low Sant’Agnello Mean Low Mean Scisciano Mean High HighWe 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 F-transform to approximate FNs. We apply this method in a fuzzy rule-based 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, decision-making 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 F-transform method to the approximation of FNs in multicriteria fuzzy decision-making 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.



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