2. Geographic classification problem

Geographic classification problem aims to find the region for an unassigned olive oil sample. This problem comes to exist to support the traceability of denominated protected origin policy for olive oil samples. Especially, the definition of a methodology is an important issue for A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators http://dx.doi.org/10.5772/intechopen.79962 89

Figure 1. Geographical classification problem scheme for olive oil.

Turkey. In literature, it is seen that the scholars generally prefer to study on the classification of olive oils [9, 10]. Principal component analysis, linear discriminant, probabilistic neural networks, and classification binary tree were preferred techniques to evaluate the parameters [9, 10]. Back propagation artificial neural networks (BP-ANN) is also used to solve [11] this kind of problem. In [12], the adulteration in olive oil was defined by near-infrared spectroscopy and using chemometric techniques such as principal component analysis, partial least squares regression (PLS), and applied methods for data pretreatments such as signal detection correction. Principal component analysis and SIMCA classification model [13] are other methods to support the geographic classification problem given in Figure 1.

#### 3. Preliminaries

manner among these agricultural food products. It is necessary to observe the properties of olive oil produced from different kinds of regions or different types of olive varieties. Geographical classification problem investigates the relationship among the chemical and senso-

Nowadays, machine learning discipline and chemical data structures come together with the information age. Machine learning is interested in the design and development of algorithms for computers. It aims to observe the relationships among the data structure and to make knowledge mining without assumptions. There are several machine learning algorithms to

Decision trees as machine learning tasks, are most commonly used in machine learning discipline. There are several types of decision tree algorithms such as ID3, C4.5, CART, etc. Nowadays, fuzzy logic is adapted into decision tree algorithms to handle the uncertainty. The decision trees adapted with fuzzy logic are called as fuzzy decision tree [1–3]. It consists of nodes for testing attributes, edges for branching by test values of fuzzy sets, and leaves for

The chemical measurements have also uncertainty [4–8]. In this study, geographical classification problem uses chemical measurements. This study aims to propose an improved methodological approach for the classification of olive oil samples based on fuzzy ID3 classification approach. This novel proposed system constructs the rules by using fuzzy decision tree algorithm. Its reasoning procedure is based on weighted rule-based system adapted into the fuzzy reasoning handled with different T-operators. The model is examined by using different decision tree approaches (C4.5 and standard version fuzzy ID3 algorithm) and FID3 reasoning method with eight different T-operators. This study is examined on 101 virgin olive oil samples collected from four different regions (North Aegean, South Aegean, Mediterranean, and South East) by using measurements of chemical parameters. Min-max normalization was applied into the dataset. The nonparametric methods were preferred for the statistical analysis because of the data structure. Leave-one-out procedure was performed in order to measure the performances of the algorithms. The Friedman aligned rank test and pairwise comparisons were performed to evaluate fuzzy reasoning method based on different T-operators. And, the comparison between unweighted and weighted fuzzy reasoning approaches was done. The rest of the paper is organized as follows: Section 2 presents the geographical classification problem definition and related works. The preliminaries such as fuzzification, fuzzy ID3 algorithm, and fuzzy rulebased classification system are given in Section 3. Experimental study on unweighted and weighted fuzzy rule-based approach to Geographic Classification of Virgin Olive Oil Using T-

Operators is given in Section 4, and finally, the conclusion is represented in Section 5.

Geographic classification problem aims to find the region for an unassigned olive oil sample. This problem comes to exist to support the traceability of denominated protected origin policy for olive oil samples. Especially, the definition of a methodology is an important issue for

rial parameters for each region.

deciding class according to class membership.

2. Geographic classification problem

search the knowledge.

88 Potential of Essential Oils

We briefly explain fuzzy logic and fuzzy c-means algorithm as fuzzification tool. Also, we review briefly fuzzy ID3 builder combined with fuzzy rule-based classification and its reasoning method. We give information about T-operators and we suggest fuzzy ID3 weighted reasoning method approach via different types of T-operators in subsections.

#### 3.1. Fuzzy logic and fuzzy c-means algorithm as fuzzification tool

In 1965, fuzzy set theory was first proposed in [14]. A fuzzy subset of the universe of discourse U is described by a membership function μvð Þ V : U ! ½ � 0; 1 , which represents the degree to which uEU belongs to the set v. Each value defines by a membership degree. The transformation process into membership degrees for each term of fuzzy variables is called as fuzzification. In literature, there are many types of membership functions, triangular membership functions, trapezoidal membership functions, Gaussian membership functions, etc. [15]. In general, triangular membership functions are preferred. Otherwise, fuzzy c-means (FCM) algorithm, which was suggested in [16] and it was improved in [17], can be used for the transformation of membership degrees for each term of fuzzy variables. This algorithm is a kind of clustering algorithm. This clustering algorithm aims to reach a fuzzy C partition matrix U. The objective function Jm is minimized as follows for fuzzy partition (Eq. (1)):

$$J\_m(\mathcal{U}, \boldsymbol{\upsilon}) = \sum\_{k=1}^{n} \sum\_{i=1}^{c} \left(\mu\_{i\mathbf{k}}\right)^m (d\_{i\mathbf{k}})^2 \tag{1}$$

There is a class Cj from a preassigned class set C ¼ f g C1;C2;…;CM to an object, which is a part of a certain feature space x∈ SN and a classifier is to realize an assignment for an appropriate

A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators

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In general, the classifier includes a set of fuzzy rules. It can be a neural network, a decision tree, fuzzy decision tree etc. If the classifier produces a set of fuzzy rules, the system is called a

The antecedents of fuzzy rules defined by fuzzy variables provide computational flexibility. Using a set of training samples and a classifier solves a classification problem. The model provides the class of a new sample. The scheme of classification problem with fuzzy ID3 algorithm combined with fuzzy rule-based classification system is summarized in Figure 2 as follows.

In this study, it is seen that fuzzy interactive dichotomizer 3 (fuzzy ID3) algorithm is preferred as a classifier. This algorithm generate rules, fuzzy ID3 algorithm constructs a tree in learning process. Fuzzy entropy is applied to find the attributes, which has the maximum information whereas minimum uncertainty. Each path of the tree shows the rules. Each leaf node has rule weight (RW) for each class. RWj represents jth rule's weight handled from fuzzy confidence value CFj which equals to RWj. After the rules induction, fuzzy rule-based reasoning is

In literature, there are three definitions for fuzzy rules [23]. In this study, the following type of

<sup>N</sup> then Y is Cj with rk

<sup>1</sup> and…:and xN is A<sup>k</sup>

rules is used for the experiments constructed from the fuzzy decision trees.

Fuzzy rules with a class and a certainty degree in the consequent [24].

Figure 2. A classification problem with fuzzy ID3 algorithm combined with FRBC.

Rk : If x<sup>1</sup> is Ak

fuzzy rule-based classification system (its acronym is FRBCS).

performed to handle the classification task.

class, (<sup>D</sup> <sup>¼</sup> SN ! <sup>C</sup><sup>Þ</sup> [23].

where

$$d\_{ik} = d(\mathbf{x}\_k, \mathbf{v}\_i) = \left[ \sum\_{j=1}^p \left( \mathbf{x}\_{kj} - \mathbf{v}\_{ij} \right)^2 \right]^{1/2}, k = 1, \dots, n; \mathbf{i} = 1, \dots, \mathbf{c} \tag{2}$$

and, μik is explained as the membership degree of the kth data point in ith class. Dimensionality of the data space is indicated by 'p'. The parameter mEð Þ 1; ∞ demonstrates sharpness of the fuzzification process. In Eq. (2), dik indicates any distance measure (usually the Euclidean distance) between kth data point and i th cluster center in p dimensional space. Then, vi displays i th cluster center. Eq. (3) calculates each of the clusters centers for each class:

$$\upsilon\_{i\dot{\jmath}} = \frac{\sum\_{k=1}^{n} \mu\_{i\dot{k}}^{m} \mathbf{x}\_{k\dot{\jmath}}}{\sum\_{k=1}^{n} \mu\_{i\dot{k}}^{m}}, \dot{\imath} = 1, 2..., c; \dot{\jmath} = 1, 2..., p. \tag{3}$$

Membership degrees are calculated according to the Eq. (4):

$$\mu\_{ik} = \frac{1}{\sum\_{z=1}^{c} \left(\frac{\|\mathbf{x}\_k - \mathbf{v}\_i\|}{\|\mathbf{x}\_k - \mathbf{v}\_z\|}\right)^{\frac{2}{m-1}}}, i = 1, 2, \dots, c; k = 1, \dots, n \tag{4}$$

Validity indicators are used in order to determine the number of clusters (c) [18–20]. One of them is partition coefficient formulized as below (Eq. (5)):

$$V\_{\rm PC} = \frac{1}{n} \sum\_{i=1}^{c} \sum\_{j=1}^{n} \mu\_{ij}^{2} \tag{5}$$

whereas optimal cluster number is determined by the calculation of max Vð Þ PC; U; c . Each cluster number represents the number of fuzzy linguistic term for each fuzzy variable.

#### 3.2. Fuzzy rule-based classification system (FRBCS)

Fuzzy rule-based classification system (FRBCS) is very useful for the solution of classification problems. In real life, they have been applied into the different kinds of problems, such as image processing [21], medical problems [22], etc.

There is a class Cj from a preassigned class set C ¼ f g C1;C2;…;CM to an object, which is a part of a certain feature space x∈ SN and a classifier is to realize an assignment for an appropriate class, (<sup>D</sup> <sup>¼</sup> SN ! <sup>C</sup><sup>Þ</sup> [23].

In general, the classifier includes a set of fuzzy rules. It can be a neural network, a decision tree, fuzzy decision tree etc. If the classifier produces a set of fuzzy rules, the system is called a fuzzy rule-based classification system (its acronym is FRBCS).

The antecedents of fuzzy rules defined by fuzzy variables provide computational flexibility. Using a set of training samples and a classifier solves a classification problem. The model provides the class of a new sample. The scheme of classification problem with fuzzy ID3 algorithm combined with fuzzy rule-based classification system is summarized in Figure 2 as follows.

In this study, it is seen that fuzzy interactive dichotomizer 3 (fuzzy ID3) algorithm is preferred as a classifier. This algorithm generate rules, fuzzy ID3 algorithm constructs a tree in learning process. Fuzzy entropy is applied to find the attributes, which has the maximum information whereas minimum uncertainty. Each path of the tree shows the rules. Each leaf node has rule weight (RW) for each class. RWj represents jth rule's weight handled from fuzzy confidence value CFj which equals to RWj. After the rules induction, fuzzy rule-based reasoning is performed to handle the classification task.

In literature, there are three definitions for fuzzy rules [23]. In this study, the following type of rules is used for the experiments constructed from the fuzzy decision trees.

Fuzzy rules with a class and a certainty degree in the consequent [24].

In literature, there are many types of membership functions, triangular membership functions, trapezoidal membership functions, Gaussian membership functions, etc. [15]. In general, triangular membership functions are preferred. Otherwise, fuzzy c-means (FCM) algorithm, which was suggested in [16] and it was improved in [17], can be used for the transformation of membership degrees for each term of fuzzy variables. This algorithm is a kind of clustering algorithm. This clustering algorithm aims to reach a fuzzy C partition matrix U. The objective

function Jm is minimized as follows for fuzzy partition (Eq. (1)):

dik ¼ d xk ð Þ¼ ; vi

vij ¼

Membership degrees are calculated according to the Eq. (4):

<sup>μ</sup>ik <sup>¼</sup> <sup>1</sup> P<sup>c</sup> z¼1

them is partition coefficient formulized as below (Eq. (5)):

3.2. Fuzzy rule-based classification system (FRBCS)

image processing [21], medical problems [22], etc.

P<sup>n</sup> <sup>k</sup>¼<sup>1</sup> <sup>μ</sup><sup>m</sup> ikxkj

P<sup>n</sup> <sup>k</sup>¼<sup>1</sup> <sup>μ</sup><sup>m</sup> ik

distance) between kth data point and i

where

90 Potential of Essential Oils

i

Jmð Þ¼ <sup>U</sup>; <sup>v</sup> <sup>X</sup><sup>n</sup>

X p

2 4

j¼1

th cluster center. Eq. (3) calculates each of the clusters centers for each class:

k k xk�vi k k xk�vz � � <sup>2</sup> m�1

> VPC <sup>¼</sup> <sup>1</sup> n Xc i¼1

k¼1

xkj � vij � �<sup>2</sup>

and, μik is explained as the membership degree of the kth data point in ith class. Dimensionality of the data space is indicated by 'p'. The parameter mEð Þ 1; ∞ demonstrates sharpness of the fuzzification process. In Eq. (2), dik indicates any distance measure (usually the Euclidean

Validity indicators are used in order to determine the number of clusters (c) [18–20]. One of

whereas optimal cluster number is determined by the calculation of max Vð Þ PC; U; c . Each cluster number represents the number of fuzzy linguistic term for each fuzzy variable.

Fuzzy rule-based classification system (FRBCS) is very useful for the solution of classification problems. In real life, they have been applied into the different kinds of problems, such as

Xn j¼1 μ2

Xc i¼1

> 3 5

1=2

μik � �<sup>m</sup>ð Þ dik

<sup>2</sup> (1)

, k ¼ 1, …, n; i ¼ 1, …, c (2)

th cluster center in p dimensional space. Then, vi displays

, i ¼ 1, 2…:, c; j ¼ 1, 2…:, p: (3)

, i ¼ 1, 2…::, c; k ¼ 1, …:, n (4)

ij (5)

Rk : If x<sup>1</sup> is Ak <sup>1</sup> and…:and xN is A<sup>k</sup> <sup>N</sup> then Y is Cj with rk

Figure 2. A classification problem with fuzzy ID3 algorithm combined with FRBC.

where rk is the certainty degree of the classification in the class Cj for a pattern belonging to the fuzzy substance restricted by the fuzzy antecedent.

Step 2d: Select the attribute (Attr) that maximizes the gain information (Gk) [27].

Step 2e: Assign the selected attribute as the root node and the linguistic labels as candidate

Step 3: Pick out one branch to analyze. Remove the branch if it is containing nothing. If the branch is nonentity, calculate the relative frequencies via (Eq. (6)) of all objects within the branch into each class. If the relative frequency of each class is above the given threshold θ<sup>r</sup> or all the attributes have been expanded for this branch, stop the branch as a leaf. Otherwise, select the attribute from among those, which have not been extended yet in this branch with the smallest average fuzzy classification entropy (Eq. (9)) as a new decision node for the branch and add its linguistic labels as candidates branches to analyze. At each leaf, each class will have its relative frequency [27].

Step 4: Repeat Step 3 while there are branches to analyze. If there are no candidate branches

After the fuzzy decision tree induction, the rules are generated from each branch. Each branch

Rule Rj: If x<sup>1</sup> is Aj<sup>1</sup> and … and xn is Ajn then Class ¼ Cj with RWj, where Rj is the label of the jth rule. x ¼ ð Þ x1; x2;…; xn is an n-dimensional pattern vector. This vector is used to represent the example. Aji is a fuzzy set. CjE C is the class label, and RWj is the rule weight. In fuzzy decision tree, at each leaf node has rule weights. These rule weights are founded via the relative

Fuzzy reasoning method (FRM) is defined as an inference procedure. This inference procedure aims to achieve an assignment from a set of fuzzy if then rules. It makes the combination between the information of the rules fires and the pattern to be classified. This ability of FRM supports the generalization capability of the classification system [25]. We will analyze this idea in this section according to the following structure. In this section, the adaptation of the general model of fuzzy reasoning is represented with the classical FRM. After that, we talk about a general model of reasoning that involves different possibilities as reasoning methods, we suggest eight alternative FRMs as some particular new proposals, which are adapted with the general reasoning model. Finally, in the last section, we present the experiments carried out, displaying the advantageous behavior of the alternative proposed reasoning methods.

Let xp <sup>¼</sup> xp1;…; xpn be the pth example of the training set, which is composed of <sup>P</sup> examples, where xpi is the value of the ith attribute ð Þ i ¼ 1; 2; …; n of the pth sample. Each example

ð Þ Gk , where Gk ¼ Ek � Entrki (9)

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A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators

Attr ¼ max 1 ≤ k ≤ n

The rule structure generated from each branch of the fuzzy decision tree.

behaves as path. The rule Rj is given as follows [27]:

frequency for each class (as given in Step 3) [27].

3.4.1. General model of fuzzy reasoning

3.4. Fuzzy reasoning method based on T-operators

branches of the tree.

then the decision tree is totaled [27].

#### 3.3. Fuzzy interactive dichotomizer 3

Fuzzy decision tree is the adaptation of decision tree structure with fuzzy logic. There are many types of decision tree algorithms, which are adapted with fuzzy logic to construct a fuzzy decision tree. A tree is generated and the decision rules are achieved by using each path from the root to the leaves of the tree. Fuzzy interactive dichotomizer 3 (Fuzzy ID3) defined in [2] is widely used as a classification tree builder algorithm. It is the adaptation of ID3 algorithm proposed by Quinlan in [25] with fuzzy logic. One of the important advantages is to deal with crisp and fuzzy variables defined by the user. This algorithm separates the data set according to a data attribute, which is selected by using a measure called as information gain based on fuzzy entropy. It seeks the attributes, which has the information with the highest degree of resolution.

Let a training set consists of <sup>p</sup> samples, xp <sup>¼</sup> xp1;…; xpn � � be the pth sample of the training set where xpi is the value of the ith attribute ð Þ i ¼ 1; 2;…; n of the pth training sample. Each sample belongs to a class shown as ypEC ¼ f g C1;C2;…;Cm , where m is the number of classes of the problem [26]. Assume there are N labeled fuzzified patterns and n attributes A ¼ f g A1; A2;…; An . For each k assume that 1ð Þ ≤ k ≤ n . The attribute Ak takes mk values of fuzzy subsets Ak1; Ak2;…; Akmk ð Þ. C denotes the classification target attribute, taking m values C1, C2, …, Cm. The symbol Mð Þ: is used to denote the cardinality of a given fuzzy set, that is, the sum of the membership values of the fuzzy set [2, 26].

The induction process of fuzzy ID3 is given as follows:

Step 1: Produce a root node, which contains a set of all data. Each data is fuzzified, and each membership degree equals to 1 for all data for the initialization.

Step 2: The attribute for each internal node is selected by using the following steps:

Step 2a: Compute its relative frequencies with respect to class Cj ð Þ j ¼ 1; 2; …; m for each linguistic label Aki ð Þ i ¼ 1; 2;…; mk ,

$$p\_{ki}(j) = \frac{M(A\_{ki} \cap \mathbb{C}\_j)}{M(A\_{ki})} \tag{6}$$

Step 2b: Compute its fuzzy classification entropy for each linguistic label Aki ð Þ i ¼ 1; 2;…; mk :

$$Entr\_{ki} = -\sum\_{j=1}^{m} p\_{ki}(j) \log \left( p\_{ki}(j) \right) \tag{7}$$

Step 2c: Compute the average fuzzy classification entropy (Ek) of each attribute.

$$E\_k = \sum\_{i=1}^{m\_k} \frac{M(A\_{ki})}{\sum\_{j=1}^{m\_k} M(A\_{kj})} Entropy \tag{8}$$

Step 2d: Select the attribute (Attr) that maximizes the gain information (Gk) [27].

where rk is the certainty degree of the classification in the class Cj for a pattern belonging to the

Fuzzy decision tree is the adaptation of decision tree structure with fuzzy logic. There are many types of decision tree algorithms, which are adapted with fuzzy logic to construct a fuzzy decision tree. A tree is generated and the decision rules are achieved by using each path from the root to the leaves of the tree. Fuzzy interactive dichotomizer 3 (Fuzzy ID3) defined in [2] is widely used as a classification tree builder algorithm. It is the adaptation of ID3 algorithm proposed by Quinlan in [25] with fuzzy logic. One of the important advantages is to deal with crisp and fuzzy variables defined by the user. This algorithm separates the data set according to a data attribute, which is selected by using a measure called as information gain based on fuzzy entropy. It seeks the attributes, which has the information with the highest degree of resolution.

where xpi is the value of the ith attribute ð Þ i ¼ 1; 2;…; n of the pth training sample. Each sample belongs to a class shown as ypEC ¼ f g C1;C2;…;Cm , where m is the number of classes of the problem [26]. Assume there are N labeled fuzzified patterns and n attributes A ¼ f g A1; A2;…; An . For each k assume that 1ð Þ ≤ k ≤ n . The attribute Ak takes mk values of fuzzy subsets Ak1; Ak2;…; Akmk ð Þ. C denotes the classification target attribute, taking m values C1, C2, …, Cm. The symbol Mð Þ: is used to denote the cardinality of a given fuzzy set, that is,

Step 1: Produce a root node, which contains a set of all data. Each data is fuzzified, and each

Step 2a: Compute its relative frequencies with respect to class Cj ð Þ j ¼ 1; 2; …; m for each

Step 2b: Compute its fuzzy classification entropy for each linguistic label Aki ð Þ i ¼ 1; 2;…; mk :

j¼1

M Að Þ ki mk <sup>j</sup>¼<sup>1</sup> M Akj

M Aki ∩ Cj � � M Að Þ ki

Step 2: The attribute for each internal node is selected by using the following steps:

pkiðÞ¼ j

Entrki ¼ �X<sup>m</sup>

Step 2c: Compute the average fuzzy classification entropy (Ek) of each attribute.

Ek <sup>¼</sup> <sup>X</sup>mk i¼1 P � � be the pth sample of the training set

pkið Þ<sup>j</sup> log pkið Þ<sup>j</sup> � � (7)

� � Entrki (8)

(6)

fuzzy substance restricted by the fuzzy antecedent.

Let a training set consists of p samples, xp ¼ xp1;…; xpn

the sum of the membership values of the fuzzy set [2, 26].

membership degree equals to 1 for all data for the initialization.

The induction process of fuzzy ID3 is given as follows:

linguistic label Aki ð Þ i ¼ 1; 2;…; mk ,

3.3. Fuzzy interactive dichotomizer 3

92 Potential of Essential Oils

$$Attr = \max\_{1 \le k \le n} (\mathcal{G}\_k)\_\text{\textquotedblleft} \mathcal{G}\_k = E\_k - Entr\_{ki} \tag{9}$$

Step 2e: Assign the selected attribute as the root node and the linguistic labels as candidate branches of the tree.

Step 3: Pick out one branch to analyze. Remove the branch if it is containing nothing. If the branch is nonentity, calculate the relative frequencies via (Eq. (6)) of all objects within the branch into each class. If the relative frequency of each class is above the given threshold θ<sup>r</sup> or all the attributes have been expanded for this branch, stop the branch as a leaf. Otherwise, select the attribute from among those, which have not been extended yet in this branch with the smallest average fuzzy classification entropy (Eq. (9)) as a new decision node for the branch and add its linguistic labels as candidates branches to analyze. At each leaf, each class will have its relative frequency [27].

Step 4: Repeat Step 3 while there are branches to analyze. If there are no candidate branches then the decision tree is totaled [27].

The rule structure generated from each branch of the fuzzy decision tree.

After the fuzzy decision tree induction, the rules are generated from each branch. Each branch behaves as path. The rule Rj is given as follows [27]:

Rule Rj: If x<sup>1</sup> is Aj<sup>1</sup> and … and xn is Ajn then Class ¼ Cj with RWj, where Rj is the label of the jth rule. x ¼ ð Þ x1; x2;…; xn is an n-dimensional pattern vector. This vector is used to represent the example. Aji is a fuzzy set. CjE C is the class label, and RWj is the rule weight. In fuzzy decision tree, at each leaf node has rule weights. These rule weights are founded via the relative frequency for each class (as given in Step 3) [27].

#### 3.4. Fuzzy reasoning method based on T-operators

Fuzzy reasoning method (FRM) is defined as an inference procedure. This inference procedure aims to achieve an assignment from a set of fuzzy if then rules. It makes the combination between the information of the rules fires and the pattern to be classified. This ability of FRM supports the generalization capability of the classification system [25]. We will analyze this idea in this section according to the following structure. In this section, the adaptation of the general model of fuzzy reasoning is represented with the classical FRM. After that, we talk about a general model of reasoning that involves different possibilities as reasoning methods, we suggest eight alternative FRMs as some particular new proposals, which are adapted with the general reasoning model. Finally, in the last section, we present the experiments carried out, displaying the advantageous behavior of the alternative proposed reasoning methods.

#### 3.4.1. General model of fuzzy reasoning

Let xp <sup>¼</sup> xp1;…; xpn be the pth example of the training set, which is composed of <sup>P</sup> examples, where xpi is the value of the ith attribute ð Þ i ¼ 1; 2; …; n of the pth sample. Each example belongs to class ypEC ¼ f g C1;C2; …;Cm , where m is the number of classes of the problem. It is assumed that xp is a novel example to be classified FID3 reasoning procedure given in [2]. Fuzzy reasoning method for FARC-HD in [28] is summarized in four steps. In our approach, fuzzy ID3 reasoning method is combined with T-operators. T-operators were developed from the triangular inequalities [29, 30]. The combination of fuzzy set theory and T-operators are used to intersect and reunite two fuzzy sets [31, 32]. There are different types of T-operators, which are also called T-norms and T-conorms in literature [33]. These operators are used in different types of problems [33]. T-operators are two placed functions from 0½ �� ; 1 ½ � 0; 1 to 0½ � ; 1 that are monotonic, commutative, and associative [33].

T-norm is used to find the intersection of two fuzzy sets A and B. The intersection of two fuzzy sets A and B is a fuzzy set C, written as C ¼ A and B, whose MF is related to those of A and B by

$$
\mu\_{\mathbb{C}}(\mathbf{x}) = \left(\mu\_{A}(\mathbf{x}) \bigwedge \mu\_{\mathbb{B}}(\mathbf{x})\right) \tag{10}
$$

3.4.2. Fuzzy rule evaluation measures in data mining

membership function of the antecedent fuzzy set Aqi.

the support of Aq¼)Cq is written as follows [39–46]:

3.4.3. Heuristic methods for rule weight specification

Let us assume that m labeled patterns,

There are two measures called as confidence and support in the field data mining to evaluate rules. Assume that fuzzy rule Rj is defined as Aq¼)Cq where Aq <sup>¼</sup> Aq1;…; Aqn � �. In [34–37],

A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators

In literature [38–40], the compatibility grade of each training pattern xp with the antecedent Aq is defined by the product operation as <sup>μ</sup>Aqð Þ xp <sup>¼</sup> <sup>μ</sup>Aq1ð Þ xp<sup>1</sup> � … � <sup>μ</sup>Aqnð Þ xpn , where <sup>μ</sup>Aqið Þ: is the

> P xpEClass Cq

The confidence is a numerical approximation of the conditional probability. On the other hand,

P xpEClass Cq

While the determination of the consequent class, there are many ways to give weights to the rules [38–40]. In general, the consequent Cq of the fuzzy rule Aq¼)Cq in [38] is settled with the

While a set of antecedent fuzzy sets is given for each attribute, the antecedent part of each fuzzy rule (i.e. Aq) is defined with the combination of antecedent fuzzy sets for n attributes. In [36], it is seen that the confidence is directly used for each class for the fuzzy rule with multiple

P<sup>m</sup>

xp <sup>¼</sup> xp1;…; xpn � �, p <sup>¼</sup> <sup>1</sup>, …:, m (12)

μAq xp � �

μAq xp � �

� � <sup>¼</sup> max c Aq¼)Class h � �j<sup>h</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M � � (15)

RWqh <sup>¼</sup> c Aq¼)Class h � �, h <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …, M: (16)

� � can be used as the rule weight RWq of the fuzzy rule Aq¼)Cq.

� � (13)

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95

<sup>m</sup> (14)

<sup>p</sup>¼<sup>1</sup> <sup>μ</sup>Aq xp

fuzzy versions of two rule evaluation measures were explained as below:

are given from M classes for an n-dimensional pattern classification problem.

The confidence of the fuzzy rule Aq¼)Cq is written as follows [39, 40]:

c Aq¼)Cq � � <sup>¼</sup>

s Aq¼)Cq � � <sup>¼</sup>

The support measures the coverage of the training patterns by Aq¼)Cq.

class who has the maximum confidence for the antecedent Aq.

c Aq¼)Cq

The confidence c Aq¼)Cq

consequent classes [23].

On the other hand, T-conorm is performed to achieve the union of two fuzzy sets A and B is a fuzzy set C, written as C ¼ A or B, whose membership function (MF) is related to those of A and B by

$$
\mu\_{\mathbb{C}}(\mathbf{x}) = \left(\mu\_A(\mathbf{x}) \bigvee \mu\_B(\mathbf{x})\right) \tag{11}
$$

T-Operators used in fuzzy reasoning method are given in Table 1 [27].


Table 1. T-Operators used in fuzzy reasoning method.

#### 3.4.2. Fuzzy rule evaluation measures in data mining

There are two measures called as confidence and support in the field data mining to evaluate rules. Assume that fuzzy rule Rj is defined as Aq¼)Cq where Aq <sup>¼</sup> Aq1;…; Aqn � �. In [34–37], fuzzy versions of two rule evaluation measures were explained as below:

Let us assume that m labeled patterns,

belongs to class ypEC ¼ f g C1;C2; …;Cm , where m is the number of classes of the problem. It is assumed that xp is a novel example to be classified FID3 reasoning procedure given in [2]. Fuzzy reasoning method for FARC-HD in [28] is summarized in four steps. In our approach, fuzzy ID3 reasoning method is combined with T-operators. T-operators were developed from the triangular inequalities [29, 30]. The combination of fuzzy set theory and T-operators are used to intersect and reunite two fuzzy sets [31, 32]. There are different types of T-operators, which are also called T-norms and T-conorms in literature [33]. These operators are used in different types of problems [33]. T-operators are two placed functions from 0½ �� ; 1 ½ � 0; 1 to 0½ � ; 1

T-norm is used to find the intersection of two fuzzy sets A and B. The intersection of two fuzzy sets A and B is a fuzzy set C, written as C ¼ A and B, whose MF is related to those of A and B by

<sup>μ</sup>Cð Þ¼ <sup>x</sup> <sup>μ</sup>Að Þ<sup>x</sup> ^μBð Þ<sup>x</sup>

On the other hand, T-conorm is performed to achieve the union of two fuzzy sets A and B is a fuzzy set C, written as C ¼ A or B, whose membership function (MF) is related to those of

T-Operators used in fuzzy reasoning method are given in Table 1 [27].

Ref T-norm operators T-conorm operators

ð Þ <sup>x</sup>þy�x:<sup>y</sup> <sup>T</sup><sup>∗</sup>

<sup>λ</sup>þð Þ <sup>1</sup>�<sup>λ</sup> ð Þ <sup>x</sup>þy�x:<sup>y</sup> <sup>T</sup><sup>∗</sup>

� �

� �<sup>λ</sup> � �<sup>1</sup>=<sup>λ</sup> <sup>T</sup><sup>∗</sup>

maxð Þ <sup>x</sup>;y;<sup>λ</sup> T<sup>∗</sup>

� � T<sup>∗</sup>

<sup>1</sup>þ<sup>λ</sup> ; <sup>0</sup>

Ref T-norm operators T-conorm operators Parametric

; 0

T∗

Zadeh [14] <sup>T</sup>1ð Þ¼ <sup>x</sup>; <sup>y</sup> minð Þ <sup>x</sup>; <sup>y</sup> <sup>T</sup><sup>∗</sup>

Product Sum [41, 42] <sup>T</sup>2ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>:<sup>y</sup> <sup>T</sup><sup>∗</sup>

<sup>T</sup>3ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>:<sup>y</sup>

Yager [44] <sup>T</sup>5ð Þ¼ <sup>x</sup>; <sup>y</sup> max 1 � ð Þ <sup>1</sup> � <sup>x</sup> <sup>p</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>y</sup> <sup>p</sup> ð Þ<sup>1</sup>=<sup>p</sup>

<sup>T</sup>7ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>:<sup>y</sup>

<sup>1</sup><sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup>�<sup>1</sup> <sup>λ</sup> <sup>þ</sup> <sup>1</sup> <sup>y</sup>�1 � �

<sup>μ</sup>Cð Þ¼ <sup>x</sup> <sup>μ</sup>Að Þ<sup>x</sup> <sup>⋁</sup>μBð Þ<sup>x</sup> � � (11)

<sup>1</sup> ¼ maxð Þ x; y

<sup>2</sup> ¼ x þ y � x:y

<sup>3</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>þy�2:x:<sup>y</sup> 1�x:y

<sup>4</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>þy�ð Þ <sup>2</sup>�<sup>λ</sup> :x:<sup>y</sup>

<sup>6</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>1</sup> <sup>1</sup><sup>þ</sup> <sup>1</sup> ð Þ <sup>x</sup>�<sup>1</sup> �<sup>λ</sup>

<sup>7</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> ð Þ <sup>1</sup>�<sup>x</sup> :ð Þ <sup>1</sup>�<sup>y</sup> max 1ð Þ �x;1�y;λ

<sup>5</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> min <sup>x</sup><sup>p</sup> <sup>þ</sup> <sup>y</sup><sup>p</sup> ð Þ<sup>1</sup>=<sup>p</sup>

<sup>λ</sup>þð Þ <sup>1</sup>�<sup>λ</sup> ð Þ <sup>1</sup>�x:<sup>y</sup> <sup>λ</sup> <sup>≥</sup> <sup>0</sup>

<sup>þ</sup> <sup>1</sup> <sup>y</sup>�1 � ��<sup>λ</sup> � ��1=<sup>λ</sup> <sup>λ</sup> <sup>&</sup>gt; <sup>0</sup>

<sup>8</sup>ð Þ¼ x; y minð Þ x þ y þ λ:x:y; 1 λ > �1

; 1 � � p > 0 (10)

Range

λ ¼ ½ � 0; 1

that are monotonic, commutative, and associative [33].

A and B by

94 Potential of Essential Oils

Nonparametric Hamacher [43] ð Þ λ ¼ 0

Dubois and Prade [46]

Parametric operators [27]

Hamacher [43] <sup>T</sup>4ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>x</sup>:<sup>y</sup>

Dombi [45] <sup>T</sup>6ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>1</sup>

Weber [41] <sup>T</sup>8ð Þ¼ <sup>x</sup>; <sup>y</sup> max <sup>x</sup>þy�1þλ:x:<sup>y</sup>

Table 1. T-Operators used in fuzzy reasoning method.

Nonparametric operators [27]

$$\mathbf{x}\_p = (\mathbf{x}\_{p1}, \dots, \mathbf{x}\_{pn}), p = 1, \dots, m \tag{12}$$

are given from M classes for an n-dimensional pattern classification problem.

In literature [38–40], the compatibility grade of each training pattern xp with the antecedent Aq is defined by the product operation as <sup>μ</sup>Aqð Þ xp <sup>¼</sup> <sup>μ</sup>Aq1ð Þ xp<sup>1</sup> � … � <sup>μ</sup>Aqnð Þ xpn , where <sup>μ</sup>Aqið Þ: is the membership function of the antecedent fuzzy set Aqi.

The confidence of the fuzzy rule Aq¼)Cq is written as follows [39, 40]:

$$\mathcal{L}\{A\_{\boldsymbol{\eta}} \Longrightarrow \mathsf{C}\_{\boldsymbol{\eta}}\} = \frac{\sum\_{\mathsf{x}\_{\boldsymbol{p} \in \mathsf{Long}} \mathsf{C}\_{\boldsymbol{q}}} \mu\_{A\_{\boldsymbol{q}}}(\mathbf{x}\_{\boldsymbol{p}})}{\sum\_{\boldsymbol{p}=1}^{m} \mu\_{A\_{\boldsymbol{q}}}(\mathbf{x}\_{\boldsymbol{p}})} \tag{13}$$

The confidence is a numerical approximation of the conditional probability. On the other hand, the support of Aq¼)Cq is written as follows [39–46]:

$$\text{res}\{A\_{q}\Longrightarrow \mathbb{C}\_{q}\} = \frac{\sum\_{\text{x}\_{\text{pr}\text{Class }\text{C}\_{q}}\mu\_{A\_{q}}(\mathbf{x}\_{p})}{m} \tag{14}$$

The support measures the coverage of the training patterns by Aq¼)Cq.

#### 3.4.3. Heuristic methods for rule weight specification

While the determination of the consequent class, there are many ways to give weights to the rules [38–40]. In general, the consequent Cq of the fuzzy rule Aq¼)Cq in [38] is settled with the class who has the maximum confidence for the antecedent Aq.

$$c\left(A\_q \Longrightarrow \mathbb{C}\_q\right) = \max\left\{c\left(A\_q \Longrightarrow \text{Class } h\right) \middle| h = 1, 2, \ldots, M\right\} \tag{15}$$

The confidence c Aq¼)Cq � � can be used as the rule weight RWq of the fuzzy rule Aq¼)Cq.

While a set of antecedent fuzzy sets is given for each attribute, the antecedent part of each fuzzy rule (i.e. Aq) is defined with the combination of antecedent fuzzy sets for n attributes. In [36], it is seen that the confidence is directly used for each class for the fuzzy rule with multiple consequent classes [23].

$$RW\_{\eta h} = c\big(A\_{\eta} \Longrightarrow \text{Class } h\big), h = 1, 2, 3, \dots, M. \tag{16}$$

The adaptation of generalized model with weighted fuzzy reasoning based on T-operators.

The steps are given below combined with FID3 reasoning based on T-operators:

Step 1: Antecedent degree of a rule: In this step, the strength of activation of the if-part for all rules handled from each path of the fuzzy decision tree in the RB with the pattern xp is computed

$$\mu\_{A\_{\!\!\!\!=}}(\mathbf{x}\_{\mathcal{P}}) = T\left(\mu\_{A\_{\!\!\!=}}(\mathbf{x}\_{\mathcal{P}1}), \ldots, \ldots, \mu\_{A\_{\!\!=}}(\mathbf{x}\_{\mathcal{P}n\_{\!\!\!=}})\right) \tag{17}$$

T-operators is observed. Then, the weighted and unweighted fuzzy reasoning methods based

A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators

http://dx.doi.org/10.5772/intechopen.79962

97

Olives were collected from certain trees of the cultivars, which were determined subject matter of this work: Ayvalik, Memecik, Kilis Yaglik, and Nizip Yaglik. The samples collected in 2002– 2003, 2004–2005, and 2005–2006 harvest seasons. About 101 olive oil samples [47] were used for the experimental study. These samples were collected from different regions [North Aegean (33), South Aegean (53), Mediterranean (4), and South East (11)]. The detail information about the chemical analysis of the samples was given in pioneer studies [27, 47, 48]. PCA was applied in SPSS 20.0, partition coefficients and fuzzy c-means algorithm were handled in MATLAB 2015. The software is designed named as OliveDeSoft in the Visual C# for the experimental study (intel i7, 2.4 GHz, 4 Gb RAM) [48]. The data fuzzification process was applied by using fuzzy c-means (FCM). Partition coefficient determined the number of clusters [19, 20]. The

calculated partition coefficient value for each cluster is given in former study [27].

In former study [27], principal component analysis is performed on this data set in order to explore the data structure. It is seen that the geographic origin of virgin olive oils on the results handled from the chemical analyses are explained clearly. Yet one region (Mediterranean) has less data than the other regions, so it is not explained. The data implementation is done in IBM SPSS 20. The chemical measurements have fuzziness. So, we prefer to use fuzzy ID3 algorithm based on fuzzy logic for the classification in our study. In classical case, ID3 algorithm works with categorical variables. It is an advantage of fuzzy ID3 algorithm. This algorithm carries out numerical variables via fuzzy variables. Each numeric variable is converted to fuzzy variable. Fuzzy c-means algorithm is performed for the fuzzification. This proposed approach displays eight different T-operators in the reasoning procedure. The performances of standard fuzzy ID3 represented in [2, 27] and C4.5 [49] algorithms are examined in the experimental study. Leave one out validation procedure was performed for the performances measurement of the algorithms. Accuracy rate is preferred to test different methods [13]. In experimental study, threshold value for fuzzy decision tree is set to θ<sup>r</sup> ¼ 0:75. Parameters of parametric operators are fixed as Yager p = 2, Hamacher p = 0.25, Dombi = 1, Dubois = 0.25, and Weber = 15 for fuzzy reasoning procedure. The comparison of the performances of unweighted and weighted fuzzy

Studying fuzzy reasoning method with nonparametric operators: C4.5 algorithm also uses entropy as splitting criteria. It is the improved version of ID3 algorithm. It was presented by Quinlan in 1994 to work on the numerical data [27]. The performance of it is 86.14%. Then, it is observed that the performance of fuzzy ID3 algorithm with reasoning method in [2] is 86.14% too [27]. The performance results of nonparametric approaches given in Table 2 shows that the result handled from three nonparametric operators have the same performance value with handled from C4.5 algorithm. Yet, the accuracy handled with Zadeh T-operators is smaller value with

on different T-operators are compared.

4.2. Performance measure and statistical tests

reasoning approaches is performed.

82.18%.

4.1. Olive oil samples

where μAj xpi is the matching degree of the example with ith antecedent of the rule Rj, which is handled from a leaf node at the end of each path. T is a T-norm (listed in Table 1) and nj is the number of antecedents of the rule.

Step 2: Consequent degree for a class: The consequent degree favor of class l by the rule Rj for the pattern xp is computed as follows where RWjl the weight is computed according to the multiple consequent classes (Eq. (16))

$$\left(\boldsymbol{b}\_{j}^{l}\right) = T\left(\mu\_{A\_{j}}\left(\mathbf{x}\_{p}\right), \boldsymbol{R}\boldsymbol{W}\_{jl}\right) \tag{18}$$

Step 3: Confidence degree for a class: In this stage, the confidence degree for the class l according to all rules in RB is computed. To obtain the confidence degree of a class, the association degrees of the rules of that class are aggregated by using conjunction operators, where T\* is a T-conorm (listed in Table 1) [2, 27].

$$\text{conf}\_l(\mathbf{x}\_p) = T^\*\left(b\_1^l(\mathbf{x}\_p), b\_2^l(\mathbf{x}\_p), \dots, b\_R^l(\mathbf{x}\_p)\right) \tag{19}$$

where bl <sup>j</sup> xp , <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, R, is the association degree of the pattern xp, to the class <sup>l</sup>, according to the j.th rule.

Step 4: Classification: The class is obtained with the highest confidence degree assign as the predicted one [2, 27].

$$\text{Class} = \arg\max\_{l=1,2...,m} \left( \text{conf}\_l(\mathbf{x}\_p) \right) \tag{20}$$
