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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.

A fuzzy number (FN) is a fuzzy set with membership function A: Reals →^{-}: [a,c] →^{−}(a) = 0,^{−} (c) = 1 and^{+}: [b,d] →^{+}(d) = 1, A^{+}(b) = 0. A^{-} and A^{+} are called

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

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}= a, x_{2}= a+h,…, _{1},…,

_{0}

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 [

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

As already shown in [_{0}; afterwards it is necessary to verify at any step that the dataset is sufficiently dense with respect to the fuzzy partition and that the approximation error is less than or equal to a prefixed threshold: in this case the process stops and the direct 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:_{1} and A_{2} are two FNs for the linguistic input variable x, B_{1} and B_{2} are two FNs for the input linguistic variable y, and C_{1} and C_{2} are two FNs for the output variable z. Applying the extended iF-transforms to evaluate each fuzzy set, we suppose that _{1} = max(0.4, 0.7) = 0.7 and r_{2} = min(0.7, 0,3) = 0.3. In the defuzzification process we reconstruct the output fuzzy set as_{1} and s_{2} are suitable thresholds prefixed a priori (Figure

Defuzzification of the output fuzzy set.

The CoG method is useful for obtaining the final crisp value _{1}< z_{2} <⋯<

Given a value_{α}, called

the

Two properties of A are given in [

Following the definitions and notations of [_{2}, …, _{1} = a < x_{2} <⋯< _{1}(x),…,

Furthermore, we say that the fuzzy sets

n ≥ 3 and

Let _{2}, …, _{2}, …, _{2}, …,

Let f(x) be a continuous function on [a,b]. For every _{2}, …,

In the discrete case we know that the function _{1},…,_{2}, …, _{2},…, _{2}, …, _{2}, …,

Let _{2}, …,

In [_{2}, …, _{2}, …, _{2}, …,

Let S be a FN having a continuous membership function and supp_{2}, …,

Another important theorem [

Let S be a FN having a continuous membership function, supp_{2}, …,

supp

If core

If S^{-} strictly increases on

If S^{+} strictly decreases on

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 [

Let S be a FN having a continuous membership function with supp_{2}, …,

In order to apply the extended iF-transform to approximate a FN S with one-element core, in [_{1} is differentiable in [a,x_{2}],

Let S be a FN having a continuous membership function with supp_{2}, …,

Now we suppose that the membership values of a FN S in form (_{1} < p_{2} <⋯< _{2}, …,

Let the expert knowledge be formed by a set of fuzzy rules in a linguistic fuzzy model:_{1}, x_{2},…,

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 p_{1} = a < p_{2} <⋯<

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 (

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 n_{0} (for example, n_{0} = 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:

_{0}

_{0}

The fuzzification reads the input data and calculates the membership degree of each fuzzy set related to the input variable using (

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: x_{1} = average surface of the housings in m^{2}, x_{2} = percentage of housings with six or more rooms, x_{3} = percentage of residential buildings built since 2000, x_{4} = percentage of housings with centralized or autonomous heating system, x_{5} = percentage of housings with two or more showers or bathtubs, and x_{6} = percentage of housings with two or more restrooms. The crisp input data are extracted from the ISTAT dataset. The crisp value of the variable x_{1} is given by the total surface of the housings in the municipality dividing by the number of housings. The crisp values of the variables x_{2}, …, x_{6} are obtained dividing the corresponding absolute value recorded in the dataset by the total number of housings in the municipality. The domain of any variable is partitioned in 5 fuzzy sets labeled as “Low”, “Mean Low”, “Mean”, “Mean High”, and “High”. The fuzzy rule set contains the 62 fuzzy rules in Table

The fuzzy rule set used for evaluating the livability in residential housings.

| |
---|---|

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

Fuzzy numbers x_{1} = Low, x_{2} = Mean, x_{3} = Mean High, and x_{4} = High (in blue) and their extended iF-transform approximations (in red).

The FNs (x_{1} = Low) and (x_{4} = High) have a degenerated side. In Tables

Parameters and RMSE of the approximation for fuzzy sets of x_{1}

_{ 1 } ^{ 2 } | |||||
---|---|---|---|---|---|

| | | | | |

| 20 | 20 | 45 | 70 | 9.11×10^{-3} |

| 45 | 70 | 75 | 90 | 9.91×10^{-3} |

| 75 | 90 | 95 | 100 | 9.17×10^{-3} |

| 95 | 100 | 115 | 125 | 9.76×10^{-3} |

| 110 | 120 | 150 | 150 | 9.34×10^{-3} |

Parameters and RMSE of the approximation for fuzzy sets of x_{2}

_{ 2 } | |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 1 | 4 | 9.18×10^{-3} |

| 0.5 | 3 | 6 | 8 | 9.43×10^{-3} |

| 2 | 7 | 12 | 20 | 9.19×10^{-3} |

| 8 | 12 | 15 | 25 | 9.57×10^{-3} |

| 15 | 25 | 50 | 50 | 9.15×10^{-3} |

Parameters and RMSE of the approximation for fuzzy sets of x_{3}

_{ 3 } | |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 0.5 | 1 | 9.21×10^{-3} |

| 0.4 | 0.6 | 1 | 1.5 | 9.35×10^{-3} |

| 1 | 2 | 4 | 6 | 9.33×10^{-3} |

| 2 | 4 | 7 | 10 | 9.02×10^{-3} |

| 6 | 10 | 30 | 30 | 9.26×10^{-3} |

Parameters and RMSE of the approximation for fuzzy sets of x_{4}

_{ 4 } | |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 30 | 40 | 9.24×10^{-3} |

| 30 | 50 | 60 | 70 | 9.29×10^{-3} |

| 60 | 65 | 70 | 80 | 9.49×10^{-3} |

| 75 | 80 | 85 | 90 | 9.35×10^{-3} |

| 85 | 95 | 100 | 100 | 9.08×10^{-3} |

Parameters and RMSE of the approximation for fuzzy sets of x_{5}

_{ 5 } | |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 10 | 15 | 9.30×10^{-3} |

| 7 | 15 | 20 | 25 | 9.52×10^{-3} |

| 20 | 25 | 30 | 35 | 9.25×10^{-3} |

| 30 | 35 | 40 | 50 | 9.31×10^{-3} |

| 40 | 50 | 100 | 100 | 9.37×10^{-3} |

Parameters and RMSE of the approximation for fuzzy sets of x_{6}

_{ 6 } | |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 10 | 15 | 9.32×10^{-3} |

| 7 | 15 | 25 | 30 | 9.19×10^{-3} |

| 22 | 28 | 32 | 35 | 9.24×10^{-3} |

| 30 | 40 | 45 | 55 | 9.48×10^{-3} |

| 50 | 60 | 100 | 100 | 9.28×10^{-3} |

In Table

Parameters and RMSE of the approximation for fuzzy sets of output y.

| |||||
---|---|---|---|---|---|

| | | | | |

| 0 | 0 | 10 | 20 | 9.67×10^{-3} |

| 10 | 20 | 30 | 40 | 9.32×10^{-3} |

| 30 | 40 | 60 | 70 | 9.46×10^{-3} |

| 50 | 70 | 80 | 85 | 9.78×10^{-3} |

| 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

Thematic map for the input variable x_{1}

Thematic map for input variable x_{2}

Thematic map for input variable x_{3}

Thematic map for input variable x_{4}

Thematic map for input variable x_{5}

Thematic map for input variable x_{6}

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

Defuzzified values obtained for livability of residential housings.

| | | | | |
---|---|---|---|---|---|

| 55.18 | Forio | 40.83 | Procida | 20.68 |

| 27.12 | Frattamaggiore | 55.02 | Qualiano | 18.36 |

| 59.21 | Frattaminore | 33.8 | Quarto | 65.52 |

| 60.29 | Giugliano in Campania | 81.75 | Roccarainola | 82.01 |

| 23.34 | Gragnano | 29.64 | SanGennaro Vesuviano | 76.18 |

| 24.65 | Grumo Nevano | 55 | SanGiorgio a Cremano | 23.14 |

| 58.36 | Ischia | 27.13 | SanGiuseppe Vesuviano | 51.03 |

| 32.23 | Lacco Ameno | 26.69 | San Paolo BelSito | 63.46 |

| 48.7 | Lettere | 42.57 | San Sebastiano al Vesuvio | 82.37 |

| 63.37 | Liveri | 81.39 | San Vitaliano | 66.52 |

| 44.85 | Marano di Napoli | 24.93 | Santa Maria la Carità | 56.94 |

| 52.06 | Mariglianella | 82.39 | Sant'Agnello | 33.85 |

| 50.84 | Marigliano | 53.68 | Sant'Anastasia | 64.19 |

| 47.32 | Massa di Somma | 36.15 | Sant'Antimo | 47.82 |

| 73.29 | MassaLubrense | 71.5 | Sant'Antonio Abate | 58.19 |

| 47.68 | Melito di Napoli | 26.87 | Saviano | 76.84 |

| 23.45 | Meta | 56.38 | Scisciano | 88.93 |

| 92.74 | Monte di Procida | 32.69 | Serrara Fontana | 54.08 |

| 34.61 | Mugnano di Napoli | 29.14 | Somma Vesuviana | 52.11 |

| 39.26 | Napoli | 53.82 | Sorrento | 20.18 |

| 33.15 | Nola | 75.35 | Striano | 73.69 |

| 38.77 | Ottaviano | 52.94 | Terzigno | 52.01 |

| 34.02 | Palma Campania | 62.9 | Torre Annunziata | 25.12 |

| 44.26 | Piano di Sorrento | 60.67 | Torre del Greco | 26.36 |

| 73.89 | Pimonte | 27 | Trecase | 55.8 |

| 49.67 | Poggiomarino | 75 | Tufino | 96.44 |

| 67.16 | Pollena Trocchia | 25.13 | Vico Equense | 20.37 |

| 87.38 | Pomigliano d'Arco | 19.75 | Villaricca | 78.36 |

| 85.46 | Pompei | 17.89 | Visciano | 78.45 |

| 39.07 | Portici | 22.51 | Volla | 55.83 |

| 28.14 | Pozzuoli | 39.43 |

In Figure

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 [

Comparisons obtained approximating input and output FN with trapezoidal FN.

| |
---|---|

| 6.3×10^{-2} |

| 5.58% |

| 7 |

| 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

Municipalities with different livability class.

| | |
---|---|---|

| Mean | Mean Low |

| Mean Low | Mean |

| Mean | Mean Low |

| Mean | Mean Low |

| Mean | Mean Low |

| Mean Low | Mean |

| 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.

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.

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).

The authors declare that they have no conflicts of interest.