Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 8458916 | https://doi.org/10.1155/2018/8458916

Ferdinando Di Martino, Salvatore Sessa, "A Method Based on Extended Fuzzy Transforms to Approximate Fuzzy Numbers in Mamdani Fuzzy Rule-Based 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 Rule-Based System

Academic Editor: Yasar Becerikli
Received09 Apr 2018
Accepted25 Jul 2018
Published12 Sep 2018

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.

Recently many methods are proposed in order to approximate FNs with easier FNs using a suitable metric (see, e.g., [16]). Some authors investigate approximations by adding some restrictions to preserve properties of a FN as core [7], ambiguity [810], 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., [1518]) 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

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 aswhere s1 and s2 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 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 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 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 quantityfor 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 byfor i=1,…,n. Similarly we define the discrete iF-transform of f with respect to the , A2, …, 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 , A2, …, 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 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 asThen the ith component of the extended direct F-transform of with respect to the family of basic functions , A2, …, is given byHence the extended iF-transform 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 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 thatwhere 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 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 , 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.

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 asThe 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

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.


IDRule

r1IF (x1 = High) AND (x2 = High) AND (x3 = High) THEN y = High
r2IF (x1 = High) AND (x2 = Mean High) AND (x4 = Mean High) THEN y = Mean High
r3IF (x1 = High) AND (x3 = High) THEN y = High
r4IF (x1 = High) AND (x4 = High) THEN y = High
r5IF (x1 = High) AND (x3 = Mean High) AND (x5 = High) THEN y = High
r6IF (x1 = High) AND (x3 = Mean High) AND (x6 = High) THEN y = High
r7IF (x1 = High) AND (x3 = Mean High) AND (x5 = Mean High) THEN y = Mean High
r8IF (x1 = High) AND (x3 = Mean High) AND (x6 = Mean High) THEN y = Mean High
r9IF (x1 = High) AND (x4 = Mean High) AND (x5 = High) THEN y = High
r10IF (x1 = High) AND (x4 = Mean High) AND (x6 = High) THEN y = High
r11IF (x1 = High) AND (x4 = Mean High) AND (x5 = Mean High) THEN y = Mean High
r12IF (x1 = High) AND (x4 = Mean High) AND (x6 = Mean High) THEN y = Mean High
r13IF (x2 = High) AND (x3 = High) THEN y = High
r14IF (x2 = High) AND (x4 = High) THEN y = High
r15IF (x3 = High) AND (x4 = High) THEN y = High
r16IF (x3 = High) AND (x4 = Mean High) AND (x5 = High) THEN y = High
r17IF (x3 = High) AND (x4 = Mean High) AND (x5 = Mean High) THEN y = Mean High
r18IF (x3 = High) AND (x4 = Mean High) AND (x5 = Mean) THEN y = Mean High
r19IF (x3 = High) AND (x4 = Mean High) AND (x6 = High) THEN y = High
r20IF (x3 = High) AND (x4 = Mean High) AND (x6 = Mean High) THEN y = Mean High
r21IF (x4 = High) AND (x5 = High) THEN y = High
r22IF (x1 = Mean High ) AND (x3 = Mean High) THEN y = Mean High
r23IF (x1 = Mean High ) AND (x3 = Mean) THEN y = Mean High
r24IF (x1 = Mean High ) AND (x4 = Mean High) THEN y = Mean High
r25IF (x1 = Mean High ) AND (x4 = Mean) THEN y = Mean High
r26IF (x2 = Mean High) AND (x3 = High) THEN y = Mean High
r27IF (x2 = Mean High) AND (x3 = Mean High) THEN y = Mean High
r28IF (x2 = Mean High) AND (x4 = High) THEN y = Mean High
r29IF (x2 = Mean High) AND (x4 = Mean High) THEN y = Mean High
r30IF (x1 = Mean) AND (x3 = Mean High) THEN y = Mean
r31IF (x1 = Mean) AND (x3 = Mean) THEN y = Mean
r32IF (x1 = Mean) AND (x4 = Mean High) THEN y = Mean
r33IF (x1 = Mean ) AND (x4 = Mean) THEN y = Mean
r36IF (x2 = Mean) AND (x3 = Mean) THEN y = Mean
r37IF (x2 = Mean) AND (x4 = Mean) THEN y = Mean
r38IF (x3 = Mean) AND (x5 = Mean) THEN y = Mean
r39IF (x3 = Mean) AND (x6 = Mean) THEN y = Mean
r40IF (x4 = Mean) AND (x5 = Mean) THEN y = Mean
r41IF (x4 = Mean) AND (x6 = Mean) THEN y = Mean
r42IF (x1 = Mean) AND (x3 = Mean Low) THEN y = Mean Low
r43IF (x1 = Mean) AND (x4 = Mean Low) THEN y = Mean Low
r44IF (x1 = Mean Low) AND (x3 = Mean) THEN y = Mean Low
r45IF (x1 = Mean Low) AND (x4 = Mean) THEN y = Mean Low
r46IF (x2 = Mean Low) AND (x3 = Mean) THEN y = Mean Low
r47IF (x2 = Mean Low) AND (x4 = Mean) THEN y = Mean Low
r48IF (x3 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low
r49IF (x3 = Mean Low) AND (x6 = Mean Low) THEN y = Mean Low
r50IF (x4 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low
r51IF (x4 = Mean Low) AND (x6 = Mean Low) THEN y = Mean Low
r52IF (x3 = Mean Low) AND (x5 = Mean Low) THEN y = Mean Low
r53IF (x1 = Low) AND (x4 = Mean Low) THEN y = Low
r54IF (x1 = Low) AND (x4 = Low) THEN y = Low
r55IF (x2 = Low) AND (x4 = Mean Low) THEN y = Low
r56IF (x2 = Low) AND (x4 = Low) THEN y = Low
r57IF (x2 = Low) AND (x5 = Low) THEN y = Low
r58IF (x2 = Low) AND (x6 = Low) THEN y = Low
r59IF (x3 = Low) AND (x5= Low) THEN y = Low
r60IF (x3 = Low) AND (x6= Low) THEN y = Low
r61IF (x4 = Low) AND (x5= Low) THEN y = Low
r62IF (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.

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.

(a) Parameters and RMSE of the approximation for fuzzy sets of x1

x1 (m2)
Fuzzy numberacdbRMSE

Low202045709.11×10-3
Mean Low457075909.91×10-3
Mean7590951009.17×10-3
Mean High951001151259.76×10-3
High1101201501509.34×10-3

(b) Parameters and RMSE of the approximation for fuzzy sets of x2

x2
Fuzzy numberacdbRMSE

Low00149.18×10-3
Mean Low0.53689.43×10-3
Mean2712209.19×10-3
Mean High81215259.57×10-3
High152550509.15×10-3

(c) Parameters and RMSE of the approximation for fuzzy sets of x3

x3
Fuzzy numberacdbRMSE

Low000.519.21×10-3
Mean Low0.40.611.59.35×10-3
Mean12469.33×10-3
Mean High247109.02×10-3
High61030309.26×10-3

(d) Parameters and RMSE of the approximation for fuzzy sets of x4

x4
Fuzzy numberacdbRMSE

Low0030409.24×10-3
Mean Low305060709.29×10-3
Mean606570809.49×10-3
Mean High758085909.35×10-3
High85951001009.08×10-3

(e) Parameters and RMSE of the approximation for fuzzy sets of x5

x5
Fuzzy numberacdbRMSE

Low0010159.30×10-3
Mean Low71520259.52×10-3
Mean202530359.25×10-3
Mean High303540509.31×10-3
High40501001009.37×10-3

(f) Parameters and RMSE of the approximation for fuzzy sets of x6

x6
Fuzzy numberacdbRMSE

Low0010159.32×10-3
Mean Low71525309.19×10-3
Mean222832359.24×10-3
Mean High304045559.48×10-3
High50601001009.28×10-3

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.


y
Fuzzy numberacdbRMSE

Low0010209.67×10-3
Mean Low102030409.32×10-3
Mean304060709.46×10-3
Mean High507080859.78×10-3
High80901001009.31×10-3

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.

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


MunicipalityMunicipalityMunicipality

Acerra55.18Forio40.83Procida20.68
Afragola27.12Frattamaggiore55.02Qualiano18.36
Agerola59.21Frattaminore33.8Quarto65.52
Anacapri60.29Giugliano in Campania81.75Roccarainola82.01
Arzano23.34Gragnano29.64SanGennaro Vesuviano76.18
Bacoli24.65Grumo Nevano55SanGiorgio a Cremano23.14
Barano d'Ischia58.36Ischia27.13SanGiuseppe Vesuviano51.03
Boscoreale32.23Lacco Ameno26.69San Paolo BelSito63.46
Boscotrecase48.7Lettere42.57San Sebastiano al Vesuvio82.37
Brusciano63.37Liveri81.39San Vitaliano66.52
Caivano44.85Marano di Napoli24.93Santa Maria la Carità56.94
Calvizzano52.06Mariglianella82.39Sant'Agnello33.85
Camposano50.84Marigliano53.68Sant'Anastasia64.19
Capri47.32Massa di Somma36.15Sant'Antimo47.82
Carbonara di Nola73.29MassaLubrense71.5Sant'Antonio Abate58.19
Cardito47.68Melito di Napoli26.87Saviano76.84
Casalnuovo di Napoli23.45Meta56.38Scisciano88.93
Casamarciano92.74Monte di Procida32.69Serrara Fontana54.08
Casamicciola Terme34.61Mugnano di Napoli29.14Somma Vesuviana52.11
Casandrino39.26Napoli53.82Sorrento20.18
Casavatore33.15Nola75.35Striano73.69
Casola di Napoli38.77Ottaviano52.94Terzigno52.01
Casoria34.02Palma Campania62.9Torre Annunziata25.12
Castellammare di Stabia44.26Piano di Sorrento60.67Torre del Greco26.36
Castello di Cisterna73.89Pimonte27Trecase55.8
Cercola49.67Poggiomarino75Tufino96.44
Cicciano67.16Pollena Trocchia25.13Vico Equense20.37
Cimitile87.38Pomigliano d'Arco19.75Villaricca78.36
Comiziano85.46Pompei17.89Visciano78.45
Crispano39.07Portici22.51Volla55.83
Ercolano28.14Pozzuoli39.43

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.

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.


Comparison parameterValue

Mean RMSE index for the fuzzy sets approximation with trapezoidal FNs6.3×10-2
Mean difference of the final crisp livability values compared with the ones obtained by using the extended iF-transform method5.58%
Number of municipalities classified with different linguistic labels7
Percentage of municipalities classified with different linguistic labels7.61%

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.


MunicipalityExtended IFtr livability classTrapezoidal livability class

Casola di NapoliMeanMean Low
CasoriaMean LowMean
CrispanoMeanMean Low
Massa di SommaMeanMean Low
PozzuoliMeanMean Low
Sant’AgnelloMean LowMean
SciscianoMean HighHigh

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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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.


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