Automatic fuzzy rules production based on clustering and implication selection

Abstract

This paper deals with improving the approximation capability of fuzzy systems. Fuzzy negations produced via conical sections are a promising methodology towards better fuzzy implications in fuzzy rules. The linguistic variables and the fuzzy rules are induced automatically following a fuzzy equivalence relation. The uncertainty of linear or nonlinear systems is thus dealt with. In this study, the clustering is optimized without human intervention, but also the best inference mechanism for a particular dataset is prescribed. It has been found that clustering based on fuzzy equivalence relation and fuzzy inference via conical sections leads to remarkably accurate approximations. A fuzzy rule based system with fewer control parameters is proposed. An application on telecom data shows the use of the methodology, its applicability to a real problem and its performance compared to other alternatives in terms of quality.

Keywords

Fuzzy inference fuzzy negation rule based systems fuzzy clustering fuzzy equivalence relation

1 Introduction

Fuzzy rule based systems (FRBSs) are one of the most important areas of application of fuzzy set theory. Research on fuzzy modeling has a long history in soft computing [1], especially in fuzzy control [2] and expert systems [3, 4]. Early research on the development of fuzzy expert systems involved an expert providing the set of linguistic IF-THEN rules, while recent research deals with automatically derived rules via fuzzy clustering [5, 6] and fuzzy inference variations [7 –9].

Among the methods for fuzzy clustering, the fuzzy equivalence relation has recently gained momentum due to its hierarchical structure and its guarantee for a solution [10 –12]. Other widely used clustering algorithms are the fuzzy C-means and fuzzy K-means algorithms and their derivatives [13, 14]. All the above machine learning techniques broaden the application of FRBSs, while they improve the generation of linguistic variables and fuzzy rules directly from data without any human intervention.

Various attempts have been proposed to automatically extract fuzzy rules from experimental input/output data. Among these, learning methods based on clustering and recognition of patterns [15 –17]. These are computational procedures for generating fuzzy rules by learning from examples. These procedures lead to automatically derive membership functions and fuzzy if-then rules from a set of given training examples as Wang & Mendel and Hong & Lee have shown [15, 16]. The main problems in the use of such pattern-based algorithms are: 1) the clustering algorithms are heuristic in nature and 2) certain assumptions are made about the structure of the data (e.g. choice of a distribution, number of clusters, etc.). Such assumptions may be wrong when it is difficult to discern patterns in the data.

Motivated by the need to create fuzzy system models with better approximations and less risk for some erroneous assumption about the data, this paper presents a promising new prototype for the development of FRBSs. The fuzzy equivalence relation is used to cluster the data automatically.

Another design parameter of FRBSs is the type of fuzzy inference. There exist three fuzzy modelling inference systems: Mamdani models, Takagi-Sugeno models and genuine type inference fuzzy models [18]. In Mamdani fuzzy systems, both the antecedent and consequent of each rule is a fuzzy set. Takagi-Sugeno systems assume a crisp consequent for each rule, usually a first-order polynomial function of the input variables. Their fuzzy inference mechanism is the so-called engineering implication. This is different from a logical implication, so various attempts in the literature propose genuine logical implications instead, which are compatible with the Compositional Rule of Inference (CRI) of Zadeh [19]. In theory, it has been proven [20] that fuzzy inference systems function as universal approximators. However, Mamdani and Sugeno fuzzy systems perform only conjunctional (t-norm type) fuzzy implications, while genuine logical fuzzy systems are able to perform any type of fuzzy implications [21].

In this work, the fuzzy implications used are a special class of fuzzy implications derived from fuzzy negations via conical sections. The research tested various classes of implications towards choosing the best. This has led to a different unsupervised machine learning methodology. The fuzzy linguistic variables and the rules of FRBSs are derived automatically.

This paper is organized as follows. Section 2 presents terminology and notation. Section 3 explains the proposed fuzzy implications derived from fuzzy negations. Section 4 presents the proposed clustering method and linguistic variables identification. Section 5 is about the generation of fuzzy rules. Section 6 deals with the ideal number of fuzzy terms. Section 7 discusses fuzzy inference and the performance evaluation of various fuzzy inference choices. Finally, in Section 8, an application clearly shows the advantages of the proposed methodology. Discussion and conclusions follow in Sections 9 and 10.

2 Terminology

Basic notions from fuzzy equivalence relations and rule based systems are summarized below.

2.1 Fundamentals of fuzzy set theory

Let X be a universal set. Every function of the form $\tilde{A} : X \to [0, 1]$ is called a fuzzy set or a fuzzy subset of X, where $\tilde{A} (x)$ is interpreted as the membership degree of x in the fuzzy set $\tilde{A}$ . We place a tilde symbol over a fuzzy set so as to distinguish it from a classical set. Classical sets are also called crisp sets and are special cases of fuzzy sets where $\tilde{A} (x)$ is only zero or one.

A fuzzy set $\tilde{A}$ is normal if there exists x ∈ X such that $\tilde{A} (x) = 1$ , that is $sup_{x \in X} \tilde{A} (x) = 1$ .

A fuzzy set $\tilde{A}$ is convex iff

$\tilde{A} (λ x + (1 - λ) y) ⩾ \tilde{A} (x) \land \tilde{A} (y), \forall x, y \in X$ , ∀λ ∈ [0, 1] where ∧ denotes the minimum operator.

The α-cuts of a fuzzy set are defined as $^{α} \tilde{A} = {x \in X : \tilde{A} (x) ⩾ α}, α \in (0, 1],$ while its support $^{0} \tilde{A}$ is the closure in the topology of X of the union of all the α-cuts, that is $^{0} \tilde{A} = \bar{\cup_{α \in (0, 1]}^{α} \tilde{A}} = \bar{{x : \tilde{A} (x) > 0}} .$

It is known that α-cuts uniquely determine the fuzzy set $\tilde{A}$ .

$\tilde{A}$ is a fuzzy number iff $\tilde{A}$ is normal and convex on X.

A trapezoidal fuzzy number $\tilde{A}$ is a fuzzy number with piecewise linear membership function $\tilde{A} (x)$ defined by $\tilde{A} (x) = {\begin{matrix} \begin{matrix} 0, \\ (x - a_{l}) / (a_{m_{1}} - a_{l}), \\ 1 \\ (a_{r} - x) / (a_{r} - a_{m_{2}}), \\ 0, \end{matrix} & \begin{matrix} x < a_{l} \\ a_{l} ⩽ x < a_{m_{1}} \\ a_{m_{1}} ⩽ x ⩽ a_{m_{2}} \\ a_{m_{2}} < x ⩽ a_{r} \\ x > a_{r} \end{matrix} \end{matrix}$ that can be denoted as a quadruplet (a_l, a_{m
₁}, a_{m
₂}, a_r). A triangular fuzzy number is a special case of a trapezoidal fuzzy number such that a_{m
₁} = a_{m
₂}. The point x where $\tilde{A} (x) = 1$ is called pivot of the triangular fuzzy number.

2.2 Fuzzy relations on fuzzy numbers

Some notions from the theory of fuzzy relations on fuzzy numbers are presented here, which will be used in fuzzy clustering.

A fuzzy relation $\tilde{R}$ on X × Y is defined as the set of ordered pairs $\tilde{R} = {(x, y), r (x, y) | (x, y) \in X \times Y},$ where r is a function that maps X × Y → [0, 1].

A fuzzy relation on a single universe Xis also a relation from X to X. In that case, it is called a fuzzy binary relation. It is called compatible relation if the following conditions hold for $\tilde{R}$ :

$\tilde{R}$ is reflexive, if r (x, x) = 1, ∀ x ∈ X,

$\tilde{R}$ is symmetric, if r (x, y) = r (y, x) , ∀ x, y ∈ X .

2.3 Fuzzy equivalence relation

A fuzzy relation $\tilde{R}$ is a fuzzy equivalence relation if it is compatible and satisfies the following property of transitivity as well, that is $\begin{matrix} r (x, y) ⩾ max_{y \in X} min (r (x, y), r (y, z)), \\ \forall x, y, z \in X \end{matrix}$ (1)

A fuzzy equivalence relation induces a set of equivalent ordinary relations on X defined by the α-cuts of $\tilde{R}$ as follows: $^{α} \tilde{R} = {(x, y) | r (x, y) ⩾ α, \forall x, y \in X > 0},$ (2) where 0 ⩽ α ⩽ 1.

Let $\tilde{R}$ be a fuzzy compatible relation on a finite universal set X with |X| = n, then $\tilde{R}$ can be reformed into a fuzzy equivalence relation by the following algorithm in at most n - 1 steps. The number of clusters of is not fixed, but depends on the α-cut of the fuzzy equivalent relation, automatically generated.

Algorithm 1 Agglomerative hierarchical clustering
Step 1. ${\tilde{R}}^{'} = \tilde{R} \cup (\tilde{R} \circ \tilde{R})$
Step 2. If ${\tilde{R}}^{'} \neq \tilde{R}$ , make $\tilde{R} = {\tilde{R}}^{'}$ and go to Step 1.
Step 3. Stop: ${\tilde{R}}^{'} = {\tilde{R}}_{T}$ .

∪ denotes the max operator for the set union and ∘ denotes the max-min composition. ${\tilde{R}}_{T}$ is called the transitive closure of $\tilde{R}$ because it is transitive, it contains $\tilde{R}$ and has the smallest possible membership grades.

2.4 Partitions of fuzzy equivalence relation

Each α-cut of ${\tilde{R}}_{T}$ induces a partition, say ^αS, which is defined as a family of disjoint subsets {^αS₁, ^αS₂, … , ^αS_n }, such that the union of these subsets coincides with the entire set ^αS, that is ^αS₁ ∪ ^αS₂ ∪ ⋯ ∪ ^αS_n = ^αS and ^αS_i ∩ ^αS_j ≠ Ø , ∀ i ≠ j.

2.5 Fuzzy rule based inference systems

Linguistic variables are essential to rule based systems, because they are used in the determination of the truth of fuzzy rule antecedents and in approximate reasoning. Formally, a linguistic variable is a quintuple (x, T (x) , U, G, M) in which x is the name of the variable; T (x) is the term set of x, that is, the set of names of names of linguistic values of x with each value being a fuzzy number defined on U; G is a syntactic rule for generating the names of values of x; and M is a semantic rule for associating with each value its meaning. Informally speaking, a linguistic variable is a variable whose values are words or sentences rather than numerical entities [22]. For example, “height” can be a linguistic variable with values “short”, “medium”, “tall”. In this paper, the use of fuzzy term is an abbreviation for the fuzzy number associated with each linguistic term.

A fuzzy rule is an IF-THEN statement of the form:

R^l: IF x₁ is $A_{1}^{l}$ and … and x_n is $A_{n}^{l}$ THEN y is B^l

where R^lrepresents the l-th rule (l = 1 … M) and the $A_{i}^{l}$ , B^l are linguistic terms characterized by membership functions ${\tilde{A}}_{i}^{l} (x)$ and ${\tilde{B}}^{l} (y)$ respectively. Each rule R^l can be viewed as a fuzzy relation ${\tilde{R}}^{l}$ of the fuzzy sets ${\tilde{A}}_{i}^{l} \subset X_{i}$ and $\tilde{B} \subset Y$ , called fuzzy implication, defined in X₁ × X₂ × ⋯ × X_n → Y as follows: ${\tilde{R}}_{\tilde{A} \to \tilde{B}}^{l} (x, y) = {\tilde{A}}^{l} (x) \otimes {\tilde{B}}^{l} (y) .$

The operator ⊗ is usually the min (Mamdani) or prod (Sugeno) implication operator, or it can be a logical fuzzy implication derived from the Compositional Rule of Inference (CRI) of Zadeh [19].

Based on CRI, the fuzzy syllogism is defined as follows. Given a fuzzy set $\tilde{A}$ on a domain X and a fuzzy implication relation $\tilde{R}$ on the domain X × Y, then CRI gives the fuzzy set $\tilde{B}$ on Y which is given by ${\tilde{B}}^{l} (y) = sup_{x \in X} [{\tilde{A}}^{l} (x) \nabla {\tilde{R}}_{A \to B}^{l} (x, y)],$ (3) where ∇ can be any operator in the class of t-norms [22].

The output of the fuzzy rule based system is obtained by aggregation of the M fuzzy results by union ${\tilde{B}}^{'} = \cup_{l = 1}^{M} {\tilde{B}}_{l}^{'}$

It must be recalled that fuzzy rule based systems (FRBSs) are composed of a knowledge base (fuzzy rules and information provided by the user), a fuzzification interface, that transforms the crisp values of the input variables into fuzzy sets; an inference engine that uses the fuzzy values from the fuzzification interface and the fuzzy rules to perform the reasoning process; and a defuzzification interface which takes the fuzzy result from the inference and performs a mapping from the fuzzy set of the control variable to a crisp result. The defuzzification strategies used in this investigation are the Center of Gravity and the Mean of Maxima [23].

3 The proposed fuzzy implications based on fuzzy negations from conical section

The production of fuzzy implications from fuzzy negations is presented in this section. The new strong fuzzy negations via conical sections presented in [24, 25] are combined with f-generated implications I (x, y) = f^-1 (x · f (y)), where f is a decreasing function [26]. The result is the Equation 4 used in the proposed methodology. A short theoretical background is provided here for better understanding the formula at the end of this section.

3.1 Genuine type fuzzy implications

Definition 1. A function I : [0, 1] × [0, 1] → [0, 1]is called a fuzzy implication if for all x, x₁, x₂, y, y₁, y₂ ∈ [0, 1]the following conditions are satisfied:

(I1) x₁ ⩽ x₂ then I (x₁, y) ⩾ I (x₂, y), i.e., I (· , y) is decreasing,

(I2) y₁ ⩽ y₂ then I (x, y₁) ⩽ I (x, y₂), i.e., I (x, ·) is increasing,

(I3) I (0, 0) = 1,

(I4) I (1, 1) = 1,

(I5) I (1, 0) = 0 .

The set of all fuzzy implications will be denoted by FI.This study aims to determine which fuzzy implications best suit the application under investigation. Some fuzzy implication formulas examined are in Table 1.

Table 1
Some fuzzy implication formulas by various authors

Name Formula implication

Lukasiewicz I_LK (x, y) = min{ 1, 1 - x + y }

Gödel $I_{GD} (x, y) = \begin{matrix} {\begin{matrix} 1 \\ y \end{matrix} & \begin{matrix} if x ⩽ y \\ if x > y \end{matrix} \end{matrix}$

Reichenbach I_RC (x, y) = 1 - x + xy

Kleene-Dienes I_KD (x, y) = max(1 - x, y)

Dubois- Prade $I_{DP} (x, y) = \begin{matrix} {\begin{matrix} 1 - x \\ y \\ 1 \end{matrix} & \begin{matrix} if y = 0 \\ if x = 1 \\ otherwise \end{matrix} \end{matrix}$

Goguen $I_{G} (x, y) = \begin{matrix} {\begin{matrix} 1 \\ \frac{y}{x} \end{matrix} & \begin{matrix} if x < y \\ otherwise \end{matrix} \end{matrix}$

Name	Formula implication
Lukasiewicz	I_LK (x, y) = min{ 1, 1 - x + y }
Gödel	$I_{GD} (x, y) = \begin{matrix} {\begin{matrix} 1 \\ y \end{matrix} & \begin{matrix} if x ⩽ y \\ if x > y \end{matrix} \end{matrix}$
Reichenbach	I_RC (x, y) = 1 - x + xy
Kleene-Dienes	I_KD (x, y) = max(1 - x, y)
Dubois- Prade	$I_{DP} (x, y) = \begin{matrix} {\begin{matrix} 1 - x \\ y \\ 1 \end{matrix} & \begin{matrix} if y = 0 \\ if x = 1 \\ otherwise \end{matrix} \end{matrix}$
Goguen	$I_{G} (x, y) = \begin{matrix} {\begin{matrix} 1 \\ \frac{y}{x} \end{matrix} & \begin{matrix} if x < y \\ otherwise \end{matrix} \end{matrix}$

Definition 2. Let f : [0.1] → [0.1]be a strictly decreasing and continuous function with f (1) = 0. The function I : [0.1] ² → [0.1]defined by $I (x, y) = f^{- 1} (x \cdot f (y)), x, y \in [0, 1]$ with the understanding 0 · ∞ =0 is called an f- generated implication and is denoted I_f. The function fitself is called an f- generator.

Proposition 1. If f is an f- generator, then I_f ∈ FI.

The special class of fuzzy implications derived from fuzzy negation

A fuzzy negation N is a generalization of the classical complement or negation ¬. Fuzzy negation truth table consists of the two conditions: ¬1 ≡0 and ¬0 ≡1 .

Definition 3. A function N : [0, 1] → [0, 1] is called a fuzzy negation if

N (0) = 1, N (1) = 0, (N1)

N is decreasing. (N2)

Definition 4.

A fuzzy negation N is called strict if, in addition,

N is strictly decreasing, (N3)

N is continuous, (N4)

A fuzzy negation N is called strong if the following property is met,

N (N (x)) = x, x ∈ [0, 1]. (N5)

In this paper the strong negation will be denoted by

N_S (x) x ∈ [0, 1].

Examples for fuzzy negations are given in the Table 2 below:

Table 2

Examples of fuzzy negations with properties

Formula	Formula properties
N_K (x) = 1 - x²	N1 to N4 strict
$N_{R} (x) = 1 - \sqrt{x}$	N1 to N4 strict
Sugeno class $N^{λ} (x) = \frac{1 - x}{1 + λ x}, λ \in (- 1, + \infty)$	N1 to N5 strong
Yager class $N^{w} (x) = {(1 - x^{w})}^{\frac{1}{w}}, w \in (0, + \infty)$	N1 to N5 strong

The paper [24] proves a new family of strong fuzzy negations, which is produced by conical sections and is given from the following formula $N (x) = \sqrt{(m^{2} - 1) x^{2} + 1} + mx, x \in [0, 1], m ⩽ 0$

According to the Definition 2, the above strong negations are f- generators so if in the following formula N is used instead of f, $I (x, y) = f^{- 1} (x \cdot f (y)), x, y \in [0, 1],$ then an algorithm is formed for producing fuzzy implications via conical sections whose final formula is $\begin{matrix} I (x, y) = N (x, N (y)) = \sqrt{(m^{2} - 1) {xN}^{2} (y) + 1} + mxN (y) \Leftrightarrow \\ I (x, y) = \sqrt{(m^{2} - 1) x {(\sqrt{(m^{2} - 1) y + 1} + my)}^{2} + 1} + \\ + mx (\sqrt{(m^{2} - 1) y + 1} + my), x, y \in [0, 1], m ⩽ 0 \end{matrix}$ (4)

4 Clustering and linguistic variables

Clustering is necessary in order to group a large number of data and thus reduce the number of cases. Fuzzy clustering is a preliminary step for linguistic variables determination [27] and the subsequent extraction of expert rules from data. There exist various clustering algorithms in the literature [5 , 14]. The proposed clustering method in FRBS is based on fuzzy equivalence relation [10 –12]. The algorithm does not require any human intervention, thus making it suitable for machine learning and automatically generated fuzzy rules. In the following subsections is the description in detail of how the various system interfaces function. Figure 1 shows the proposed FRBS structure.

Fig. 1

Structure of a fuzzy rule based system integrating fuzzy clustering and genuine fuzzy implications from conical sections.

4.1 The proposed fuzzy clustering method

At first the creation of a fuzzy equivalence relation matrix from the data is shown. This ultimately leads to group the possible solutions for linguistic variables by means of a hierarchical group of partitions of the available dataset. To achieve this, one may take advantage of the basic property of fuzzy equivalence relation to partition a set X into a family of disjoint subsets {^αS₁, ^αS₂, …, ^αS_k } , 0 ⩽ α ⩽ 1.

The procedure starts by forming a fuzzy compatible relation from a given data sequence {x₁, x₂, … x_n }. The numerical values that characterize a fuzzy relation can be derived by a similarity measure [28]. In this paper, the min-max method of the similarity was used. First, a compatible matrix C = [c_ij] _n×n (symmetric and reflexive) is formed where $c_{ij} = d (x_{i}, x_{j})$ and d (x_i, x_j) is the metric distance, in our example Euclidean. Then, the fuzzy compatible relation matrix $\tilde{R} = {[r_{ij}]}_{n \times n}$ is formed as follows: $r_{i, j} = \frac{\sum_{k = 1}^{m} min (x_{ik}, x_{jk})}{\sum_{k = 1}^{m} max (x_{ik}, x_{jk})}$ (5) $\tilde{R}$ is usually reflexive and symmetric, but not transitive. To ensure the property of transitivity, the transitive closure ${\tilde{R}}_{T}$ of $\tilde{R}$ is computed by the iterative Algorithm 1 of Section 2. ${\tilde{R}}_{T}$ is called a fuzzy equivalence relation being reflexive, symmetric and transitive.

The α-cuts of fuzzy equivalence relation are equivalent ordinary relations. Different α-cuts values of ${\tilde{R}}_{T}$ give different partitions {S₁, S₂, … , S_k } over the initial data sequence X₁, X₂, … X_n, where k is the number of partitions (clusters). An agglomerative (bottom-up) hierarchical clustering procedure is used to decide which values of X belong to which partitions, based on the α-cuts of the fuzzy equivalence relation (see Algorithm 2).

Algorithm 2 Agglomerative hierarchical clustering
Fill the main diagonal of $\tilde{R} = [r_{ij}]_{n \times n}$ with 0
Select a desired α-cut, 0 ⩽ α ⩽ 1
Initialize partition vector S ={ S₁, …, S_k }
Start with k = 0 partitions
Compute the maximums of $\tilde{R}$ along the vertical axis, $r_{max}^{(j)}$
For each j do:
If $r_{max}^{(j)} ⩾ α$ :
For each i, find
$r_{min}^{(i)} = min_{j = 1, \dots, n} {r_{ij} \| r_{ij} > α}$
If $r_{min}^{(i)}$ already exists in
S_l ∈ S, l = 1, …, k:
Add X_j to existing S_l
Else:
Increase partition counter by 1
Create new partition S_k and
add X_j to S_k
EndIf
EndFor
Else:
Increase partition counter by 1
Create new partition S_l and
add X_j to S_l
EndIf
EndFor

The linguistic variables are formed automatically in such a way that the membership functions of the linguistic terms share a portion of their space with their neighboring clusters in order that every input to be able to fire more than one fuzzy rules. The ideal number of clusters depends on the relative importance of the specific problem and is decided automatically using cluster validity described in Section 4.3.

4.2 Linguistic variables identification

The partitions derived from the fuzzy equivalence relation are used to generate the fuzzy terms of the linguistic variables. Suppose that it is given a set of input-output data pairs: $(x_{1}^{(1)}, x_{1}^{(1)}; y^{(1)}), (x_{1}^{(2)}, x_{1}^{(2)}; y^{(2)}), \dots$ where x₁ and x₂ are the inputs and y is the output. This is a two-input one-output case for illustrative purposes. Since the IF-THEN rules are to be recognized linguistically, let us denote by ${\tilde{L}}_{X_{1}}^{(1)}, {\tilde{L}}_{X_{1}}^{(2)}, \dots, {\tilde{L}}_{X_{1}}^{(k_{1})}$ , ${\tilde{L}}_{X_{2}}^{(1)}, {\tilde{L}}_{X_{2}}^{(2)}, \dots, {\tilde{L}}_{X_{2}}^{(k_{2})}$ and ${\tilde{L}}_{Y}^{(1)}, {\tilde{L}}_{Y}^{(2)}, \dots, {\tilde{L}}_{Y}^{(k)}$ the fuzzy subsets of the linguistic variables X₁, X₂ and Y. The shape of membership functions of the fuzzy terms is not restrictive, as long as there is appropriate overlapping among the neighboring fuzzy regions. But in this paper, triangular and trapezoidal fuzzy numbers are used. The support (base) of fuzzy terms can vary, extending to the length of two adjacent cluster centers or even more. Figure 2 shows an example of layout of input and output spaces into fuzzy regions. The ideal overlapping of the fuzzy subsets will be discussed in detail in Section 6.

Fig. 2

Partitioning of the inputs spaces X₁, X₂ and output space Y into fuzzy regions and their corresponding membership functions.

4.3 Optimal number of clusters

A problem one faces when clustering is to decide the optimal partitioning into clusters. In the case of large multi-feature data sets, visualization of data of more than three dimensions is impossible. There must be another way to form clusters when one cannot visualize. A procedure for evaluating the results of clustering known as Dunn’s index of cluster validity [29] has been chosen. Cluster validity consists of a set of techniques for finding the set of clusters that best fits natural partitions without any a priori class information. The outcome of the clustering process is validated by the designated cluster validity index. The index is defined in Equation 6: $D_{k} = min_{i = 1, \dots, k} {min_{j = i + 1, \dots, k} (\frac{d (C_{i}, C_{j})}{max_{r = 1, \dots, k} diam (C_{r})})}$ (6) where $d (C_{i}, C_{j}) = min_{x \in C_{i}, y \in C_{j}} d ((x, y))$ is the dissimilarity between clusters C_i and C_j, and $diam (C) = max_{x, y \in C} d (x, y)$ may be viewed as a measure of dispersion of C. If the k clusters of the data set are compact and well separated, then the index D_k will be large, since the in-between cluster distance is expected to be large and the diameter of clusters is expected to be small. Thus, it can be concluded that, based on Dunn’s index, the optimum choice for the number of clusters k is where the plot of D_k versus the number of clusters reaches its maximum. It is an indication that the clusters better fit the data. Alternative techniques with high computational demands, like Monte Carlo methodology, would be a major drawback that has been avoided. More cluster validation approaches can be found in literature [30] with comparative results.

5 Generation of fuzzy rules

In this section the method to generate fuzzy rules from data is presented. A fuzzy rule is of the form

If X₁ is ${\tilde{L}}_{X_{1}}^{(i)}$ and X₂ is ${\tilde{L}}_{X_{2}}^{(i)}$ ... and X_d is ${\tilde{L}}_{X_{d}}^{(i)}$ then Y is ${\tilde{L}}_{Y}^{(i)}$

where ${\tilde{L}}_{X_{1}}^{(i)}, {\tilde{L}}_{X_{2}}^{(i)}, \dots, {\tilde{L}}_{X_{d}}^{(i)}$ and ${\tilde{L}}_{Y}^{(i)}$ are fuzzy terms of the input and output linguistic variables, respectively. In that sense, a fuzzy system, say fs, acts as a function approximator. The function to be approximated is usually a real valued function f : R^d → R on a finite set of distinct input points X = { x₁, …, x_n } ⊂ R^d, where d ⩾ 1 is the dimension of the input space R^d. First, the case of d = 1 is examined (one input one output) in order to emphasize the basic idea of the new approach. To make fs approximate f, fuzzy rules are generated based on the partition results of clustering. The proposed approach consists of the following steps.

5.1 Creation of output subsets for each partition derived from clustering

Suppose S₁, S₂, . . . S_l are partitions of X derived from clustering, where l is the number of partitions. The image of S_j under function fs : X → Y is the subset of Y that consists of the images of the elements of S_j. That is $\begin{matrix} Y_{j} = fs (S_{j}) = {y | \exists x \in S_{j} (y = fs (S_{j}))} \\ or {fs (x) | x \in S_{j}}, j = 1, \dots, l . \end{matrix}$

In case of multi-dimensional datasets, the above equation becomes $\begin{matrix} Y_{j} = fs (S_{1 j_{1}}, \dots, S_{{dj}_{d}}) \\ = {y | \exists x_{1 j_{1}}, \dots, x_{j_{d}} \in S_{1 j_{1}}, \dots, S_{{dj}_{d}} (y = fs (S_{1 j_{1}}, \dots, S_{{dj}_{d}}))} \end{matrix}$ for j = 1, …, l₁ · l₂ · … · l_m and l_m is the number of partitions of the m th input.

5.2 Generation of fuzzy numbers on the Y_j partitions

The elements of set Y_j, namely y_j1, y_j2, . . . , y_{jc
_j} are considered to represent statistical samples. The mean of each sample ${\bar{y}}_{j} = \frac{\sum Y_{j}}{c_{j}}$ is the pivot of a triangular fuzzy number, denoted ${\tilde{Y}}_{j}$ , having as support the nearest linguistic term base of Y. The nearest base is found by the minimum Euclidean distance between the sample mean ${\bar{y}}_{j}$ and the pivots of the fuzzy subsets ${\tilde{L}}_{Y}^{(1)}, {\tilde{L}}_{Y}^{(2)}, \dots, {\tilde{L}}_{Y}^{(k)}$ of the linguistic terms of Y.

5.3 Fuzzy comparison between the created fuzzy numbers ${\tilde{Y}}_{j}$ and the fuzzy terms ${\tilde{L}}_{Y}^{(i)}$

A proximity criterion is used to find the nearest fuzzy distance between ${\tilde{Y}}_{j}$ formed from Step 2 and the fuzzy terms ${\tilde{L}}_{Y}^{(1)}, {\tilde{L}}_{Y}^{(2)}, \dots, {\tilde{L}}_{Y}^{(k)}$ of Y. The proposed fuzzy proximity measure is based on the overlapping areas of the fuzzy numbers [31]. For two fuzzy numbers $\tilde{θ}, \tilde{ρ}$ to be compared when $^{0} \tilde{θ} = [θ_{1}, θ_{2}],^{0} \tilde{ρ} = [ρ_{1}, ρ_{2}]$ are their support levels, the proximity index is: ${PI}_{\tilde{θ}}^{\tilde{ρ}} = {\begin{matrix} \frac{A_{R}^{\tilde{ρ}}}{A_{T}^{\tilde{ρ}}}, if pivot (\tilde{θ}) < if pivot (\tilde{ρ}) \\ \frac{A_{R}^{\tilde{θ}}}{A_{R}^{\tilde{θ}}}, if pivot (\tilde{θ}) > if pivot (\tilde{ρ}) \\ 0, otherwise \end{matrix}$ (7) $A_{T}^{\tilde{ρ}} = \int_{ρ_{1}}^{ρ_{2}} \tilde{ρ} (x) dx$ is the total area under the fuzzy number $\tilde{ρ}$ ,

$A_{R}^{\tilde{ρ}} {= A}_{T}^{\tilde{ρ}} - \int_{ρ_{1}}^{τ} \tilde{ρ} (x) dx - \int_{τ}^{θ_{2}} \tilde{θ} (x) dx$ is the area under the graph of $\tilde{ρ}$ but to the right of $\tilde{θ}$ , and $τ = {\tilde{θ}}^{- 1} (α_{τ}) = {\tilde{ρ}}^{- 1} (α_{τ})$ , $α_{τ} = sup_{x \in X} min {\tilde{θ} (x), \tilde{ρ} (x)}$ . $A_{T}^{\tilde{θ}}, A_{R}^{\tilde{θ}}$ are analogous. The lower the ${PI}_{\tilde{θ}}^{\tilde{ρ}}$ , the higher the proximity of the two fuzzy numbers. When PI = 1 the fuzzy numbers are completely separated (not overlapping at all).

5.4 Creation of a fuzzy proximity matrix out of all the proximity indices

The goal is to associate the fuzzy linguistic terms that represent the input variables with the fuzzy linguistic terms of the output variable. By performing k · l fuzzy comparison between ${\tilde{Y}}_{j}$ and ${\tilde{L}}_{Y}^{(1)}, {\tilde{L}}_{Y}^{(2)}, \dots, {\tilde{L}}_{Y}^{(k)}$ , a fuzzy proximity matrix is obtained: ${[{PI}_{{\tilde{L}}_{Y}^{(i)}}^{{\tilde{Y}}_{j}}]}_{k, l} = [\begin{matrix} {PI}_{{\tilde{L}}_{Y}^{(1)}}^{{\tilde{Y}}_{1}} & {PI}_{{\tilde{L}}_{Y}^{(1)}}^{{\tilde{Y}}_{2}} & \dots & {PI}_{{\tilde{L}}_{Y}^{(1)}}^{{\tilde{Y}}_{l}} \\ {PI}_{{\tilde{L}}_{Y}^{(2)}}^{{\tilde{Y}}_{1}} & {PI}_{{\tilde{L}}_{Y}^{(2)}}^{{\tilde{Y}}_{2}} & \dots & {PI}_{{\tilde{L}}_{Y_{k}}}^{{\tilde{L}}_{Y_{2}}} \\ ⋮ & ⋮ & ⋮ \\ {PI}_{{\tilde{L}}_{Y}^{(k)}}^{{\tilde{Y}}_{1}} & {PI}_{{\tilde{L}}_{Y}^{(k)}}^{{\tilde{Y}}_{2}} & \dots & {PI}_{{\tilde{L}}_{Y}^{(k)}}^{{\tilde{Y}}_{l}} \end{matrix}]$ (8)

The elements of the proximity matrix with the lowest value per column j give the desired fuzzy rules as ${\tilde{L}}_{X}^{(j)} \to {\tilde{L}}_{Y}^{(i)}$ . Thus, given the l clusters of X a number of l fuzzy rules have been created for one feature dataset or l₁ · l₂ · … · l_m fuzzy rules for multi-feature datasets. The discussion about the control rules along with fuzzy inference is in Section 7.

6 The ideal overlapping of fuzzy terms

As it was mentioned earlier, the number of fuzzy terms in the proposed method comes from the optimum number of clusters obtained as shown in Section 4.3. The necessary condition in forming fuzzy rules is that for every x_i ∈ X_i, there is at least one antecedent for each rule, which is true over x_i at least to a fixed degree a > 0. Formally,

$If \forall x_{i} \in X_{i}, i \dots k, \exists (l \in {1 \dots n}) : F_{i}^{l} (x_{i}) ⩾ a > 0$ then ${F_{i}^{l}} (l = 1 \dots n)$ represents an a-cover of X_i. The fuzzy terms usually have an overlap of 50 percent. A larger overlap would lead to firing more fuzzy rules (as more rule antecedents contribute to the final result). A smaller overlap has the danger of missing important contributors. To automate the process of the ideal number of contributors, the fuzzy comparison Formula 7 is used. The proximity comparisons are between neighboring fuzzy terms for various base widths. Thus a k × k fuzzy comparison matrix is formed. ${[{PI}_{{\tilde{L}}_{Y_{i}}}^{{\tilde{L}}_{Y_{j}}}]}_{k, k} = [\begin{matrix} 0 & {PI}_{{\tilde{L}}_{Y_{2}}}^{{\tilde{L}}_{Y_{1}}} & \dots & {PI}_{{\tilde{L}}_{Y_{k}}}^{{\tilde{L}}_{Y_{1}}} \\ {PI}_{{\tilde{L}}_{Y_{1}}}^{{\tilde{L}}_{Y_{2}}} & 0 & \dots & {PI}_{{\tilde{L}}_{Y_{k}}}^{{\tilde{L}}_{Y_{2}}} \\ ⋮ & ⋮ & 0 & ⋮ \\ {PI}_{{\tilde{L}}_{Y_{1}}}^{{\tilde{L}}_{Y_{k}}} & {PI}_{{\tilde{L}}_{Y_{2}}}^{{\tilde{L}}_{Y_{k}}} & \dots & 0 \end{matrix}]$ (9)

The ideal overlapping is found by the following formula: $PI = (\frac{1}{k} \sum_{i = 1, \dots, k} {\begin{matrix} 1, i = 1 \\ {PI}_{{\tilde{L}}_{Yi}}^{{\tilde{L}}_{Y_{i - 1}}}, 1 < i < k \\ {PI}_{{\tilde{L}}_{Yi}}^{{\tilde{L}}_{Y_{i - 1}}}, i = k \end{matrix} + \frac{1}{k} \sum_{i = 1, \dots, k} {\begin{matrix} {PI}_{{\tilde{L}}_{Yi}}^{{\tilde{L}}_{Y_{i + 1}}}, i = 1 \\ {PI}_{{\tilde{L}}_{Yi}}^{{\tilde{L}}_{Y_{i + 1}}}, 1 < i < k \\ 1, i = k \end{matrix}) / 2$ (10)

Varying the base width of fuzzy terms, different PI are computed. Best is the one closer to 0.5 as an ideal value of overlapping.

7 Fuzzy control rules and fuzzy inference

7.1 Fuzzy inference choices

Using the fuzzy rules obtained from the procedures above, a fuzzy inference mechanism is needed to perform the actual mapping from the input variables to the output variable. There exist more than one choices for the fuzzy inference.

To explain those choices, let us suppose that the problem involves a two-feature dataset (two inputs one output). Then, a fuzzy control rule system is of the form: $\begin{matrix} R_{1} : if X_{1} is {\tilde{L}}_{X_{1}}^{(1)} X_{2} is {\tilde{L}}_{X_{2}}^{(1)} then Y is {\tilde{L}}_{Y}^{(1)} \\ R_{2} : if X_{1} is {\tilde{L}}_{X_{1}}^{(2)} and X_{2} is {\tilde{L}}_{X_{2}}^{(2)} then Y is {\tilde{L}}_{Y}^{(2)} \\ \dots \dots \dots \dots \\ \dots \dots \dots \dots \end{matrix}$ R_n : if X₁ is ${\tilde{L}}_{X_{1}}^{(n)}$ and X₂ is ${\tilde{L}}_{X_{2}}^{(n)}$ then Y is ${\tilde{L}}_{Y}^{(n)}$ where ${\tilde{L}}_{X_{1}}^{(i)}, {\tilde{L}}_{X_{2}}^{(i)}$ and ${\tilde{L}}_{Y}^{(i)}$ are fuzzy subsets of the input and output linguistic variables, respectively. Each fuzzy rule R_i is a fuzzy implication defined as follows: $μ_{R_{i}} = μ_{({\tilde{L}}_{X_{1}}^{(i)} and {\tilde{L}}_{X_{2}}^{(i)} \to {\tilde{L}}_{Y}^{(i)})} (x_{1}, x_{2}, y)$ or $μ_{R_{i}} = [{\tilde{L}}_{X_{1}}^{(i)} (x_{1}) \land {\tilde{L}}_{X_{2}}^{(i)} (x_{2})] \to {\tilde{L}}_{Y}^{(i)} (y)$

The inference process is performed according to Equation 3 as follows: ${\tilde{L}}_{Y}^{' (i)} (y) = sup_{x_{1} \in X_{1}, x_{2} \in X_{2}} [({\tilde{L}}_{X_{1}}^{' (i)} (x_{1}) \land {\tilde{L}}_{X_{2}}^{' (i)} (x_{2})) \land μ_{R_{i}} (x_{1}, x_{2}, y)]$ (11) where ∧ is any t-norm combining the antecedents of the fuzzy rules. Our choices about fuzzy inference do not restrict on conjunctional fuzzy implications (Mamdani and Sugeno). Genuine type logical fuzzy implications may be used as well. Table 3 illustrates the differences of the two types of fuzzy inference.

Table 3

Types of FRBS according to their inference

Type of FRB systems	Fuzzy Inference	Aggregation Method	Defuzzification method
Conjunctional type (Mamdani, Sugeno)	T-norm implication	Sum or Max	COF
Genuine type	Genuine implication	Min	MOM

The aggregation method applied is either the max or sum of the fuzzy results ${\tilde{L}}_{Y}^{' (i)} (y), i = 1, \dots, n$ , and the defuzzification strategy is the Center of Gravity, while the aggregation method of the genuine type fuzzy implications is the Min and the defuzzification strategy is the Mean of Maxima [23].

Final approximations of the FRBS are obtained by both types of systems and the results after applying fuzzy clustering and fuzzy inference are compared with the original output values using performance metrics.

7.2 Fuzzy implications from conical sections

A generalization of fuzzy implications was presented in Section 3. Fuzzy implications from conical sections provide a much broader spectrum of choices for the genuine type fuzzy implications. The proposed approach, being an integration of fuzzy clustering with automatic fuzzy rule generation, benefits from the flexibility of genuine type fuzzy implications, because the produced FRBS has a small set of fuzzy rules with well overlapped fuzzy terms of the linguistic variables. This freedom allows us to find an implication that better represents the available dataset.

7.3 Performance evaluation

To evaluate the approximation performance, three measures are usually applied. The normalized mean squared error (NMSE), the root mean squared error (RMSE) and the mean absolute error (MAE).

Normalized Mean Squared Error, NMSE

Given a set of output values and their approximated ones ${y_{t}, {\hat{y}}_{t}}_{t = 1}^{N}$ , NMSE is defined as $NMSE = \frac{1}{N} \frac{\sum_{t = 1}^{N} {(y_{t} - {\hat{y}}_{t})}^{2}}{var (y_{t})}$ where var (·) denotes variance. Recall that $var (y) = \frac{1}{N - 1} \sum_{t = 1}^{N} {(y_{t} - μ_{y})}^{2}$ . NMSE is a mean squared error (MSE) normalized by variance.

Root Mean Squared Error, RMSE

RMSE is the square root of MSE, defined as follows: $RMSE = \sqrt{\frac{1}{N} \sum_{t = 1}^{N} {(y_{t} - {\hat{y}}_{t})}^{2}}$

Mean Absolute Error, MAE

MAE is defined as follows: $MAE = \frac{1}{N} \sum_{t = 1}^{N} | y_{t} - {\hat{y}}_{t} |$

Note that the last metric (MAE) does not penalize huge errors as MSE does. Thus, it’s not sensitive to outliers compared to the mean square error. For more interpretations, see [32].

8 Case study

To illustrate the proposed methodology, an application concerning cell phone activity data is examined. The data were offered by the Telecom Italia Big Data Challenge 2014 [33], which is a rich source on telecommunications data for major Italian cities. We processed the raw data for the city of Milan and formed the data set in Table 4 comprised of 24 hours of cell phone activity for a particular day. Variable X represents the hours of day and Y is a measure of telephone activity (received/sent SMS, incoming/outgoing calls and internet activity). One can immediately observe the low cell phone activity during night hours and the high activity during the midday. The goal of this example application is to create a FRBS that better approximates the non-linear behavior of the underlying system. Finally, a fuzzy implication that better suits the problem is selected.

Table 4
Case study dataset

Timezone (X) Cell Phone Activity (Y) Timezone (X) Cell Phone Activity (Y)

0 11293.03 12 54814.25

1 7025.77 13 37153.44

2 4404.80 14 36227.98

3 3155.37 15 39083.14

4 2415.17 16 45028.43

5 2768.56 17 53499.63

6 3474.97 18 51209.40

7 7106.24 19 43814.48

8 18525.08 20 30649.55

9 39435.78 21 21646.51

10 55688.50 22 13488.44

11 62832.15 23 9333.44

Timezone (X)	Cell Phone Activity (Y)	Timezone (X)	Cell Phone Activity (Y)
0	11293.03	12	54814.25
1	7025.77	13	37153.44
2	4404.80	14	36227.98
3	3155.37	15	39083.14
4	2415.17	16	45028.43
5	2768.56	17	53499.63
6	3474.97	18	51209.40
7	7106.24	19	43814.48
8	18525.08	20	30649.55
9	39435.78	21	21646.51
10	55688.50	22	13488.44
11	62832.15	23	9333.44

The proposed methodology is presented step by step below.

Step 1: Creation of a compatible symmetric and reflexive matrix C = [c_ij] _n×n from the dataset using a metric distance c_i,j = d (y_i, y_j). Euclidean distance was used only for the Y values. The X values do not need clustering in the particular case study.

Step 2: Using Formula 5, a fuzzy relation $\tilde{R} = {[r_{ij}]}_{n \times n}$ is constructed. Fuzzy clustering for Y is derived by the transitive closure of $\tilde{R}$ , named ${\tilde{R}}_{T}$ , which is the equivalence relation matrix. ${\tilde{R}}_{T}$ is obtained by the iterative procedure of Algorithm 1. The values of ${\tilde{R}}_{T}$ can be viewed in Table 5.

Table 5

Transitive closure ${\tilde{R}}_{T}$ for Y

1	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.91
.9	1	.89	.89	.89	.89	.89	1	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.9
.89	.89	1	.96	.96	.96	.96	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	1	.98	.98	.99	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.98	1	.99	.98	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.98	.99	1	.98	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.99	.98	.98	1	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.9	1	.89	.89	.89	.89	.89	1	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.9
.77	.77	.77	.77	.77	.77	.77	.77	1	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.85	.77	.77
.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77	.77	.91	.91	.98	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.8	.97	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.8	1	.8	.77	.77	.77	.77	.8	.8	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.97	.8	1	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	1	.95	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	.95	1	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.98	.77	.77	.77	.91	.91	1	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	1	.77	.77	.95	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.95	.8	.95	.77	.77	.77	.77	1	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.92	.8	.92	.77	.77	.77	.77	.92	1	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	.95	.77	.77	1	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	1	.61	.61	.61
.77	.77	.77	.77	.77	.77	.77	.77	.85	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77
.9	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.9
.91	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	1

Step 3: Calculation of the partitions induced by ${\tilde{R}}_{T}$ for various α-cut levels, using Algorithm 2. Figure 3. shows the hierarchical tree structure of partitions. The integer numbers inside the squares represent the indices of Y values. Table 6 shows the grouping of the actual Y values for 4,5,6,7 and 8 clusters.

Fig. 3

Hierarchical structure of partitions created by ${\tilde{R}}_{T}$ .

Table 6

Cluster values of Y for various α-cuts

α-cut = 0,77
Cluster 1	11293.03	7025.76	4404.8	3155.37	2415.17	2768.56	3474.97	7106.24	13488.44	9333.44
Cluster 2	18525.08	21646.51
Cluster 3	39435.78	55688.5	62832.15	54814.25	37153.44	36227.98	39083.13	45028.43	53499.63	51209.4	43814.48
Cluster 4	30649.55
α-cut = 0,79
Cluster 1	11293.03	7025.76	4404.8	3155.37	2415.17	2768.56	3474.97	7106.24	13488.44	9333.44
Cluster 2	18525.08	21646.51
Cluster 3	39435.78	37153.44	36227.98	39083.13	45028.43	43814.48
Cluster 4	55688.5	62832.15	54814.25	53499.63	51209.4
Cluster 5	30649.55
α-cut = 0,805
Cluster 1	11293.03	7025.76	4404.8	3155.37	2415.17	2768.56	3474.97	7106.24	13488.44	9333.44
Cluster 2	18525.08	21646.51
Cluster 3	39435.78	37153.44	36227.98	39083.13	45028.43	43814.48
Cluster 4	55688.5	54814.25	53499.63	51209.4
Cluster 5	62832.15
Cluster 6	30649.55
α-cut = 0,83
Cluster 1	11293.03	7025.76	4404.8	3155.37	2415.17	2768.56	3474.97	7106.24	13488.44	9333.44
Cluster 2	18525.08	21646.51
Cluster 3	39435.78	37153.44	36227.98	39083.13
Cluster 4	55688.5	54814.25	53499.63	51209.4
Cluster 5	62832.15
Cluster 6	45028.43	43814.48
Cluster 7	30649.55
α-cut = 0,88
Cluster 1	11293.03	7025.76	4404.8	3155.37	2415.17	2768.56	3474.97	7106.24	13488.44	9333.44
Cluster 2	18525.08
Cluster 3	39435.78	37153.44	36227.98	39083.13
Cluster 4	55688.5	54814.25	53499.63	51209.4
Cluster 5	62832.15
Cluster 6	45028.43	43814.48
Cluster 7	30649.55
Cluster 8	21646.51

Step 4: Calculation of Dunn’s cluster validity index D_k for each cluster number k. The optimal number of clusters was found to be k = 6 corresponding to the highest value D₆ = 0.455 as shown in Table 7.

Table 7

Cluster validity values for various numbers of clusters

Number of clusters (k)	Dunn’s cluster validity index (D_k)
4	0.189
5	0.433
6	0.455
7	0.395
8	0.282

Step 5: Creation of fuzzy variables by assigning a fuzzy number to each cluster center. Triangular fuzzy numbers were chosen for the middle clusters and trapezoidal fuzzy numbers for the borders. For variable Y, the linguistic terms ${\tilde{L}}_{Y}^{(1)}, \dots, {\tilde{L}}_{Y}^{(6)}$ are shown in Fig. 4. As stated earlier, in this case study there is no need to apply clustering for X because the X data are already separated. The creation of the linguistic variable for X was done by dividing the domain of X into equally spaced partitions. In this example, 24 linguistic terms for X were used, one for each partition.

Fig. 4

Final fuzzy variables induced from data after fuzzy clustering and optimization.

Step 6: The ideal overlapping of ${\tilde{L}}_{Y}^{(1)}, \dots, {\tilde{L}}_{Y}^{(6)}$ and ${\tilde{L}}_{X}^{(1)}, \dots, {\tilde{L}}_{X}^{(24)}$ is obtained by applying Equation 10. The cases of fuzzy terms extending to 1, 2 or 3 neighboring clusters were examined. The results of the calculations are shown in Table 8. An overlap factor PI close to 0.5 concludes that the overlapping of 2 neighboring clusters is preferable for both Y and X.

Table 8

PI values for fuzzy terms of X and Y

Number of neighboring clusters	Overlap factor PI for X	Overlap factor PI for Y
1	0.7415	0.7585
2	0.5628	0.4542
3	0.4325	0.3214

Step 7: Creation of the fuzzy proximity matrix defined by Equation 8. Table 9 shows the results of the calculations. The lowest values of the proximity matrix per column give the generated fuzzy rules $R_{i} : {\tilde{L}}_{X}^{(i)} \to {\tilde{L}}_{Y}^{(i)}$ , in total 24 fuzzy rules were derived. Table 10 shows the fuzzy expert rules in standard fuzzy control language specification IEC 61131-7, along with the linguistic variables definitions.

Table 9

The fuzzy proximity matrix for the creation of the fuzzy rule base

	${\tilde{L}}_{X}^{(1)}$	${\tilde{L}}_{X}^{(2)}$	${\tilde{L}}_{X}^{(3)}$	${\tilde{L}}_{X}^{(4)}$	${\tilde{L}}_{X}^{(5)}$	${\tilde{L}}_{X}^{(6)}$	${\tilde{L}}_{X}^{(7)}$	${\tilde{L}}_{X}^{(8)}$	${\tilde{L}}_{X}^{(9)}$	${\tilde{L}}_{X}^{(10)}$	${\tilde{L}}_{X}^{(11)}$	${\tilde{L}}_{X}^{(12)}$
${\tilde{L}}_{Y}^{(1)}$	0.22	0.03	0.09	0.15	0.19	0.17	0.14	0.04	0.72	1.00	1.00	1.00
${\tilde{L}}_{Y}^{(2)}$	0.69	0.73	0.75	0.76	0.77	0.76	0.76	0.73	0.04	0.76	0.94	1.00
${\tilde{L}}_{Y}^{(3)}$	0.88	0.89	0.90	0.90	0.91	0.91	0.90	0.89	0.48	0.37	0.65	1.00
${\tilde{L}}_{Y}^{(4)}$	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.77	0.02	0.33	0.72
${\tilde{L}}_{Y}^{(5)}$	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.94	0.31	0.06	0.50
${\tilde{L}}_{Y}^{(6)}$	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.72	0.44	0.00
	${\tilde{L}}_{X}^{(13)}$	${\tilde{L}}_{X}^{(14)}$	${\tilde{L}}_{X}^{(15)}$	${\tilde{L}}_{X}^{(16)}$	${\tilde{L}}_{X}^{(17)}$	${\tilde{L}}_{X}^{(18)}$	${\tilde{L}}_{X}^{(19)}$	${\tilde{L}}_{X}^{(20)}$	${\tilde{L}}_{X}^{(21)}$	${\tilde{L}}_{X}^{(22)}$	${\tilde{L}}_{X}^{(23)}$	${\tilde{L}}_{X}^{(24)}$
${\tilde{L}}_{Y}^{(1)}$	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.90	0.75	0.66	0.14
${\tilde{L}}_{Y}^{(2)}$	0.94	0.75	0.74	0.76	0.79	0.93	0.93	0.79	0.46	0.04	0.15	0.71
${\tilde{L}}_{Y}^{(3)}$	0.65	0.34	0.32	0.37	0.45	0.64	0.62	0.43	0.00	0.44	0.53	0.89
${\tilde{L}}_{Y}^{(4)}$	0.31	0.06	0.08	0.02	0.10	0.29	0.26	0.08	0.38	0.76	0.80	1.00
${\tilde{L}}_{Y}^{(5)}$	0.03	0.34	0.35	0.31	0.21	0.01	0.07	0.24	0.64	0.93	0.94	1.00
${\tilde{L}}_{Y}^{(6)}$	0.47	0.74	0.75	0.72	0.66	0.51	0.56	0.68	1.00	1.00	1.00	1.00

Table 10

Fuzzy linguistic variables and expert rules in fuzzy control language specification IEC 61131-7

#FUZZY LINGUISTIC VARIABLES

fuzzy PHONE_ACTIVITY

lefttrapezoid LY1 (2415.17 6446.58 20085.79)

triangle LY2 (2415.17 20085.79 40123.87)

triangle LY3 (6446.58 30649.55 53802.94)

triangle LY4 (20085.79 40123.87 62832.15)

triangle LY5 (30649.55 53802.94 62832.15)

righttrapezoid LY6 (53802.94 62832.15 62832.15)

end

fuzzy TIMEZONE

lefttriangle LX1 (0 4.0)

triangle LX2 (0 2.0 6.0)

triangle LX3 (0.0 4.0 8.0)

triangle LX4 (2.0 6.0 10.0)

triangle LX5 (4.0 8.0 12.0)

triangle LX6 (6.0 10.0 14.0)

triangle LX7 (8.0 12.0 16.0)

triangle LX8 (10.0 14.0 18.0)

triangle LX9 (12.0 16.0 20.0)

triangle LX10 (14.0 18.0 22.0)

triangle LX11 (16.0 20.0 24)

righttrapezoid LX12 (18.0 22.0 24)

end

#FUZZY RULE BASE

RULE 1: IF (TIMEZONE is LX1) THEN (PHONE_ACTIVITY is LY1)

RULE 2: IF (TIMEZONE is LX2) THEN (PHONE_ACTIVITY is LY1)

RULE 3: IF (TIMEZONE is LX3) THEN (PHONE_ACTIVITY is LY1)

RULE 4: IF (TIMEZONE is LX4) THEN (PHONE_ACTIVITY is LY1)

RULE 5: IF (TIMEZONE is LX5) THEN (PHONE_ACTIVITY is LY1)

RULE 6: IF (TIMEZONE is LX6) THEN (PHONE_ACTIVITY is LY1)

RULE 7: IF (TIMEZONE is LX7) THEN (PHONE_ACTIVITY is LY1)

RULE 8: IF (TIMEZONE is LX8) THEN (PHONE_ACTIVITY is LY1)

RULE 9: IF (TIMEZONE is LX9) THEN (PHONE_ACTIVITY is LY2)

RULE 10: IF (TIMEZONE is LX10) THEN (PHONE_ACTIVITY is LY4)

RULE 11: IF (TIMEZONE is LX11) THEN (PHONE_ACTIVITY is LY5)

RULE 12: IF (TIMEZONE is LX12) THEN (PHONE_ACTIVITY is LY6)

RULE 13: IF (TIMEZONE is LX13) THEN (PHONE_ACTIVITY is LY5)

RULE 14: IF (TIMEZONE is LX14) THEN (PHONE_ACTIVITY is LY4)

RULE 15: IF (TIMEZONE is LX15) THEN (PHONE_ACTIVITY is LY4)

RULE 16: IF (TIMEZONE is LX16) THEN (PHONE_ACTIVITY is LY4)

RULE 17: IF (TIMEZONE is LX17) THEN (PHONE_ACTIVITY is LY4)

RULE 18: IF (TIMEZONE is LX18) THEN (PHONE_ACTIVITY is LY5)

RULE 19: IF (TIMEZONE is LX19) THEN (PHONE_ACTIVITY is LY5)

RULE 20: IF (TIMEZONE is LX20) THEN (PHONE_ACTIVITY is LY4)

RULE 21: IF (TIMEZONE is LX21) THEN (PHONE_ACTIVITY is LY3)

RULE 22: IF (TIMEZONE is LX22) THEN (PHONE_ACTIVITY is LY2)

RULE 23: IF (TIMEZONE is LX23) THEN (PHONE_ACTIVITY is LY2)

RULE 24: IF (TIMEZONE is LX24) THEN (PHONE_ACTIVITY is LY1)

Step 8: Use of the Compositional Rule of Inference (Equation 11) for application of approximate reasoning. The results are compared by using NMSE, RMSE and MAE performance metrics. Sixteen different fuzzy implication methods were compared. The results are shown in Table 11. Column 1 is hours, column 2 is the number of telephone calls, the rest of the columns represent the fuzzy system responses by using the classical Mamdani and Sugeno implications. Next to them are the results based on fuzzy implications out of various conical sections. For completeness of the study, we added eight more fuzzy implications from the literature, the last one that of Zadeh.

Table 11

Fuzzy system responses for various fuzzy implications

X	Y values	MAMDANI	SUGENO	CONIC – 1	CONIC – 0,5	CONIC 0	CONIC 0.1	CONIC 0.15	CONIC 0.2
0	11293.03	11968.16	11968.16	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
1	7025.76	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
2	4404.80	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
3	3155.37	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
4	2415.17	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
5	2768.56	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
6	3474.97	11994.54	11994.54	4408.93	4408.93	4408.93	4408.93	4408.93	4408.93
7	7106.24	21108.11	18114.83	12021.47	11356.88	9181.87	8275.62	2415.17	2415.17
8	18525.08	28539.03	26513.31	20540.26	20540.26	20540.26	20540.26	20540.26	20540.26
9	39435.78	34645.80	35465.59	36973.68	36973.68	36973.68	37940.35	38544.52	39269.53
10	55688.50	40640.39	43385.54	51473.76	51473.76	51473.76	51655.01	52259.18	52923.77
11	62832.15	47303.24	50070.90	57817.54	58300.88	60113.39	61019.64	62832.15	62832.15
12	54814.25	40640.42	43383.16	51473.76	51473.76	51473.76	51655.01	52259.18	52923.77
13	37153.44	37672.60	36845.98	43559.13	43196.63	41807.04	41263.29	40900.79	40477.87
14	36227.98	36467.53	36467.53	40115.37	40115.37	40115.37	40115.37	40115.37	40115.37
15	39083.13	36467.53	36467.53	40115.37	40115.37	40115.37	40115.37	40115.37	40115.37
16	45028.43	37672.60	36845.98	43559.13	43196.63	41807.04	41263.29	40900.79	40477.87
17	53499.63	39789.54	42653.87	47969.57	48513.33	50748.75	51655.01	52259.18	52923.77
18	51209.40	39789.54	42653.87	47969.57	48513.33	50748.75	51655.01	52259.18	52923.77
19	43814.48	35620.55	36496.41	41988.29	41988.29	41807.04	41263.29	40900.79	40477.87
20	30649.55	31887.36	31965.62	30146.56	30146.56	30146.56	30146.56	30146.56	30448.65
21	21646.51	29262.93	26845.44	24044.45	23742.36	22292.36	21506.94	21084.02	20479.85
22	13488.44	23845.52	23937.62	16552.74	16915.25	18425.67	18969.42	19271.51	19694.43
23	9333.44	21108.11	18114.83	12021.47	11356.88	9181.87	8275.62	7792.28	2415.17
NMSE		0.2116	0.1564	0.0274	0.0249	0.0196	0.0186	0.0200	0.0247
RMSE		9465.64	8137.82	3404.04	3246.78	2880.85	2808.73	2910.29	3232.51
MAE		8249.76	7194.58	2888.88	2770.56	2350.20	2242.89	2304.04	2490.32
X	LUKASIEWICZ	KLEENEDIENE	QL_IMPLICATION		REICHENBACH	DUBOISPRADE	GOGUEN	GODEL	EARLY ZADEH
0	4408.93	4408.93	4408.93		4408.93	4408.93	4408.93	4408.93	4408.93
1	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
2	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
3	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
4	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
5	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
6	4408.93	4408.93	10450.63		4408.93	4408.93	4408.93	4408.93	10450.63
7	9242.29	14377.73	10450.63		12021.47	4439.14	10027.71	11296.47	10450.63
8	20540.26	24104.87	24074.66		20540.26	20117.35	20540.26	20540.26	24074.66
9	38544.52	37396.60	37396.60		36973.68	40115.37	38000.77	36973.68	37396.60
10	52440.43	47637.28	47637.28		51473.76	53830.02	52077.93	51473.76	47637.28
11	59569.63	56367.53	57183.16		57817.54	62771.73	59025.88	58300.88	57183.16
12	52440.43	47637.28	47637.28		51473.76	53830.02	52077.93	51473.76	47637.28
13	40115.37	45613.31	37396.60		43559.13	40115.37	40115.37	40115.37	37396.60
14	40115.37	40115.37	37396.60		40115.37	40115.37	40115.37	40115.37	37396.60
15	40115.37	40115.37	37396.60		40115.37	40115.37	40115.37	40115.37	37396.60
16	40115.37	45613.31	37396.60		43559.13	40115.37	40115.37	40115.37	37396.60
17	52440.43	45613.31	47637.28		47969.57	53830.02	52077.93	51413.34	47637.28
18	52440.43	45613.31	47637.28		47969.57	53830.02	52077.93	51413.34	47637.28
19	40115.37	41988.29	37396.60		41988.29	40115.37	40115.37	40115.37	37396.60
20	30629.90	30146.56	31626.78		30146.56	30629.90	30629.90	30629.90	31626.78
21	20117.35	25856.96	24135.07		24044.45	20117.35	20117.35	20117.35	24135.07
22	19452.76	14377.73	24135.07		16552.74	20117.35	19211.09	18486.09	24135.07
23	9302.71	14377.73	10480.84		12021.47	4439.14	10027.71	11296.47	10480.84
NMSE	0.0200	0.0530	0.0776		0.0274	0.0214	0.0211	0.0235	0.0776
RMSE	2907.73	4738.09	5730.64		3404.04	3012.95	2987.00	3154.49	5730.64
MAE	2315.89	3826.25	4999.83		2888.88	2320.23	2441.72	2640.59	4999.83

The best approximation was obtained with fuzzy implications derived from conical sections with conic parameter m = 0.1.

Figure 5 shows the result which is 72% improvement compared to Mamdani inference.

Fig. 5

Comparison of inference implementations.

In general, genuine logical fuzzy inference performed better than non-genuine fuzzy inference, with Lukasiewicz implication being also a good choice. However, the flexibility of conical sections and the single control parameter m, allows us to achieve better approximations. The proposed method reached the highest approximation accuracy with value m = 0.14. Figure 6 shows how the normalized mean squared error drops. In conclusion, this methodology gives a superior rule based system.

Fig. 6

Normalized Mean Squared Error (NMSE) curve.

9 Discussion of results

The traditional FRBSs require appropriate clustering to derive a relatively small set of fuzzy rules directly from the data. Still, it is difficult to automatically induce the linguistic variables and fuzzy rules when the data structure is not obvious. Other clustering methods assume an intuitive choice of the initial number of clusters. This may be subjective (heuristic) and not always effective.

The proposed method uses a fuzzy equivalence relation to cluster the data. For each variable, a set of optimal fuzzy linguistic terms are identified. Both the ideal number of linguistic terms and their degree of overlapping is decided algorithmically using cluster validity and the fuzzy proximity index respectively. Then, a fuzzy rule base is derived which is easy to interpret and applicable to any fuzzy inference engine.

In order to study the selection of the most appropriate fuzzy implication, a class of fuzzy implications derived from conical sections was examined. Generally genuine fuzzy implications performed better over well overlapped membership functions. From this case study, it was demonstrated that the proposed integration of clustering, rule generation and conical fuzzy implications improved the approximation accuracy of FRBS, plus with fewer control parameters. So, any system designer has now the ability to select the best fuzzy implication corresponding to the particular application.

10 Conclusions

This paper contributes in showing that the selection of fuzzy implications produced via conical sections improves the approximation capability of fuzzy systems. It also presents a new method for fully automatic production of fuzzy rules based on equivalent relation clustering. The parameters of clustering are decided and optimized by the algorithm and not assumed by humans, saving us a significant amount of work. The proposed methodology improves and automates the clustering and finds the impact of various fuzzy implications.

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.9	1	.89	.89	.89	.89	.89	1	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.9
.77	.77	.77	.77	.77	.77	.77	.77	1	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.85	.77	.77
.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77	.77	.91	.91	.98	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.8	.97	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.8	1	.8	.77	.77	.77	.77	.8	.8	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.97	.8	1	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	1	.95	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	.95	1	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.98	.77	.77	.77	.91	.91	1	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	1	.77	.77	.95	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.95	.8	.95	.77	.77	.77	.77	1	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.92	.8	.92	.77	.77	.77	.77	.92	1	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	.95	.77	.77	1	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	1	.61	.61	.61
.77	.77	.77	.77	.77	.77	.77	.77	.85	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77
.9	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.9
.91	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	1

Automatic fuzzy rules production based on clustering and implication selection

Abstract

Keywords

1 Introduction

2 Terminology

2.1 Fundamentals of fuzzy set theory

2.2 Fuzzy relations on fuzzy numbers

2.3 Fuzzy equivalence relation

2.5 Fuzzy rule based inference systems

3.1 Genuine type fuzzy implications

5.1 Creation of output subsets for each partition derived from clustering

5.2 Generation of fuzzy numbers on the Y j partitions

5.3 Fuzzy comparison between the created fuzzy numbers Y ˜ j and the fuzzy terms L ˜ Y ( i )

7.1 Fuzzy inference choices

7.3 Performance evaluation

8 Case study

10 Conclusions

References

5.2 Generation of fuzzy numbers on the Y_j partitions

5.3 Fuzzy comparison between the created fuzzy numbers ${\tilde{Y}}_{j}$ and the fuzzy terms ${\tilde{L}}_{Y}^{(i)}$

1	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.91
.9	1	.89	.89	.89	.89	.89	1	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.9
.89	.89	1	.96	.96	.96	.96	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	1	.98	.98	.99	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.98	1	.99	.98	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.98	.99	1	.98	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.89	.89	.96	.99	.98	.98	1	.89	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.89	.89
.9	1	.89	.89	.89	.89	.89	1	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	.9
.77	.77	.77	.77	.77	.77	.77	.77	1	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.85	.77	.77
.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77	.77	.91	.91	.98	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.8	.97	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.8	1	.8	.77	.77	.77	.77	.8	.8	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.97	.8	1	.77	.77	.77	.77	.95	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	1	.95	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.91	.77	.77	.77	.95	1	.91	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.98	.77	.77	.77	.91	.91	1	.81	.77	.77	.81	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	1	.77	.77	.95	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.95	.8	.95	.77	.77	.77	.77	1	.92	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.92	.8	.92	.77	.77	.77	.77	.92	1	.77	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.81	.77	.77	.77	.81	.81	.81	.95	.77	.77	1	.74	.61	.61	.61
.61	.61	.61	.61	.61	.61	.61	.61	.61	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	.74	1	.61	.61	.61
.77	.77	.77	.77	.77	.77	.77	.77	.85	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	1	.77	.77
.9	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	1	.9
.91	.9	.89	.89	.89	.89	.89	.9	.77	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.61	.77	.9	1