Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 8458916, 16 pages
https://doi.org/10.1155/2018/8458916

Research Article

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

Ferdinando Di Martino1,2 and Salvatore Sessa1,2

¹Università degli Studi di Napoli Federico II, Dipartimento di Architettura, Via Toledo 402, 80134 Napoli, Italy
²Università degli Studi di Napoli Federico II, Centro Interdipartimentale di Ricerca A. Calza Bini, Via Toledo 402, 80134 Napoli, Italy

Correspondence should be addressed to Ferdinando Di Martino; fdimarti@unina.it

Received 9 April 2018; Accepted 25 July 2018; Published 12 September 2018

Academic Editor: Yasar Becerikli

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.

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.

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 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 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.

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,…, = a+(i-1)h,…, = b are called nodes: a uniform fuzzy partition of [a,b] is created by assigning n fuzzy sets with continuous membership functions A₁,…,: [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 ().

n≔n₀

Create the fuzzy partition

Calculate the direct F-transform components

WHILE the dataset is sufficiently dense with respect to the fuzzy

partition

Calculate the approximation error

IF (approximation error ≤ threshold) THEN

Store the direct F-transform components

RETURN  “SUCCESS”

END IF

n≔n+1

Calculate the extended direct F-transform components

END WHILE

RETURN  “ERROR: Dataset not sufficiently dense”

END

Algorithm 1: Approximation of a set of data by using the extended inverse F-transform.

Figure 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: Section 2 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:

where 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 , , , 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

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 value of the output variable as

where Nc is the number of rules and z₁< z₂ <⋯< 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 value α∈, we denote with Aα, called α-cut 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

where

For 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 distance

and the extension of the Euclidean metric given by

Two properties of A are given in [25] called ambiguity and value, defined as

and

respectively, 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 F-Transforms

Following the definitions and notations of [15], let n ≥ 2 and P = , x₂, …, be a set of points of [a,b], called nodes, such that x₁ = a < x₂ <⋯< = b. Let ,…, be an assigned family of fuzzy sets with membership functions A₁(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,…,n-1(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,…, n-1(5) for every x[a,b]

Furthermore, we say that the fuzzy sets ,…, form an h-uniform fuzzy partition of [a,b] if

(6)n ≥ 3 and = a + h·(i-1), where h = (b-a)/(n-1) and i = 1, 2, …, n (that is, the nodes are equidistant)(7)( – x) = ( + x) for every x and i = 2,…, n-1(8) (x) = (x - h) for every x[,] and i = 1,2,…, n-1

Let be a continuous function on [a,b]. The quantity

for i = 1, …, n, is the ith component of the direct F-transform , F₂, …, of f with respect to the family of basic functions , A₂, …, . If this fuzzy partition is h-uniform, the components are as follows [26]:

The function

where x[a,b], is defined as the iF-transform of f with respect to , A₂, …, 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 , A₂, …, 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 p₁,…, of [a,b]. If the set ,…, is sufficiently dense with respect to the fixed partition , A₂, …, , that is, for each i = 1,…,n, there exists an index j,…, such that () > 0, we can define the n-tuple , F₂,…, as the discrete direct F-transform of f with respect to , A₂, …, , where each is given by

for i=1,…,n. Similarly we define the discrete iF-transform of f with respect to the , A₂, …, by setting

for 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 , A₂, …, of such that ,…, is sufficiently dense with respect to the existing fuzzy partition and for every ,  j = 1,…,m, the inequality

remains true.

3. The Extended iF-Transform and Fuzzy Numbers

In [15] the extended iF-transform of a continuous function is introduced in order to preserve the monotonicity as follows. For an h-uniform fuzzy partition , A₂, …, , the function is extended to [a-h,b+h] as follows:

Then the following basic functions are defined as

Then the ith component of the extended direct F-transform of with respect to the family of basic functions , A₂, …, is given by

Hence the extended iF-transform of is given by

By [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 h-uniform fuzzy partition , A₂, …, of [a,b] with n ≥ 3 and let be the extended iF-transform of S. We obtain that [13, Prop. 11]

where 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 = . Let a fuzzy partition , A₂, …, of be h-uniform with n ≥ 3 and be the extended iF-transform 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 , A₂, …, of be h-uniform with n ≥ 3 and a fuzzy set T such that T = calculated by (23) in and T=0 if x. If h = b-a/n-1 is such that h ≤ mind-c)/4, c-a, 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 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 = a,b and core = c,d, a < c < d < b. Let a fuzzy partition , A₂, …, of a,b be h-uniform 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 that

where and .

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], is differentiable in [,] for i = 2,…,n-1, and is differentiable in [,b]. Thus, we can define the normalized extended iF-transform given as

A 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 , A₂, …, be a regular h-uniform 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 = p₁ < p₂ <⋯< < = b. We consider an h-uniform fuzzy partition , A₂, …, of [a,b]. If the set of points are sufficiently dense with respect to the fuzzy partition, i.e., if

then the extended iF-transform of is defined for any x[a,b] as follows [13]:

where , , and 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:

where x₁, x₂,…, 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 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 <⋯< < = 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:

The approximation error is given by the Root Mean Square Error (RMSE) defined as

The threshold for the RMSE is set as a positive value much smaller than 1. The extended iF-transform method is schematized in Algorithm 2.

Description:  Approximate a fuzzy number with an extended iF-

transform

Input:  Initial fuzzy partition size n₀

Threshold parameter

A set of m points and their membership function value

Output:  RMSE error

Extended Direct F-transform components

n≔ n₀

Read the dataset of points

Create a h-uniform fuzzy partition by using the basic functions (36)

Calculate the extended direct F-transform components

WHILE the dataset is sufficiently dense with respect to the fuzzy partition

Calculate the RMSE approximation error (37)

IF (RMSE approximation error ≤ threshold) THEN

Store the extended direct F-transform components and the RSME error

RETURN “Success”

END IF

n≔n+1

Create a h-uniform fuzzy partition by using the basic functions (36)

Calculate the extended direct F-transform components

END WHILE

Store the extended direct previous F-transform components (n = n-1) and the

RMSE error

RETURN “ERROR: Dataset non sufficiently dense”

END

Algorithm 2: Extended F-transform approximation.

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 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 = Low

In 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 Figure 5 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   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

x₁ (m²)

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

x₂

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

x₃

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

x₄

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

x₅

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

x₆

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
9.28×10⁻³

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 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
9.31×10⁻³

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

Figure 6

The defuzzified final values of livability in the residential housings (calculated in percentage) for every municipality are in Table 4.

Table 4: Defuzzified values obtained for livability of residential housings.

Municipality

Municipality

Municipality

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
39.43

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.

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
High

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 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.

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://dati-censimentopopolazione.istat.it).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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