Averaging operators in fuzzy classification systems

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Averaging operators in fuzzy classification systems

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Andrea Mesiarová-Zemánková ∗ , Khurshid Ahmad

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Department of Computer Science, Trinity College, Dublin, Ireland

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Received 8 July 2013; received in revised form 24 February 2014; accepted 16 June 2014

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Abstract

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We examine averaging operators used for aggregation of outputs of fuzzy classification rules. We discuss several methods used in fuzzy rule-based classification systems and show related multi-polar averaging operators. We further define new aggregation methods based on the idea used for definition of OWA operators. We show connection between multi-polar averaging operators and the multi-polar Choquet integral and using this connection we study the conditions under which are the respective averaging operators monotone. We include several examples of special multi-polar OWA operators and their relation to existing bipolar aggregation operators. © 2014 Published by Elsevier B.V.

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The field of automatic classification systems has been a hot topic in recent years. Classification systems based on fuzzy rules (see for example [2,9] and references therein) offer capability to deal with noisy, imprecise or incomplete information while keeping a satisfactory level of approximation and a good interpretability of the system. In classification problems the task is to assign an object to one of the existing categories, while a confidence degree of this classification can also be given: especially in health related classification problems the confidence of the classification is important and there is a big difference between the case when a pattern is classified as negative with a confidence degree 0.99 and the case when this pattern is classified as negative with a confidence degree just 0.1. In this paper we will assume that the confidence (reliability) of the classification is given by an association degree between the pattern and the output class (see [2,9]) which is a well established notion, sometimes called also a soundness degree or a compatibility grade. In classification problems an object is represented as a collection of properties (features) from some feature space and classification techniques are based on them: the properties are grouped in so-called (fuzzy) classification rules,

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1. Introduction

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Keywords: Classification; Aggregation operator; Multi-polarity; OWA operator; Choquet integral

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* Correspondence to: A. Mesiarová-Zemánková, Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia.

E-mail addresses: [email protected] (A. Mesiarová-Zemánková), [email protected] (K. Ahmad). http://dx.doi.org/10.1016/j.fss.2014.06.010 0165-0114/© 2014 Published by Elsevier B.V.

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A. Mesiarová-Zemánková, K. Ahmad / Fuzzy Sets and Systems ••• (••••) •••–•••

where each rule implies a category (or multiple categories) that the object should be assigned to. For an example of a fuzzy classification rule we can assume that we have a collection of fruits that we want to sort. Here the following:

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is an example of a fuzzy classification rule. Classification of several objects is therefore based on the collection of (fuzzy) classification rules. The output of a fuzzy classification rule is either a category along with a confidence degree, or an m-dimensional vector, corresponding to a confidence degree for each or m classes (categories). Some ideas concerning rules with m-dimensional output can be found in [15], however, in this paper we will focus on rules with a pair output, i.e., output of this type of fuzzy classification rules is given by a pair (k, x), where k ∈ {1, . . . , m} denotes the consequent class and x ∈ [0, 1] denotes the absolute value of the output of the rule. The absolute value of the output of the rule can be obtained by a combination of the firing degree of the rule and the rule weight w. This combination is usually based on the product aggregation operator, or more generally it is based on a RET operator [22]. If the absolute value of the output of the rule is 0, i.e., such a rule implies no association between the pattern and the consequent class, then the rule output can be labeled by word ‘unclassified’ and therefore 0 can be taken as a neutral value that belongs to every category k ∈ {1, . . . , m}.

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“IF the color is ‘green’ and the shape is ‘pear-like’ and the pit is ‘big’ THEN the fruit is ‘avocado’ ”

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Definition 1. Let m ∈ N and let K m = {1, . . . , m} be the set of m categories (classes). Assume the input pairs of the form (k, x), with k ∈ K m , x ∈]0, 1] and a neutral input 0 ∈ R (which belongs to each category). Then K m × [0, 1] with the convention 0 = (k, 0) for all k ∈ K m will be called a multi-polar space. Therefore outputs of fuzzy classification rules are points from a multi-polar space. For an input (k, x) ∈ K m ×[0, 1], k will be called a class of the input (k, x) and x will be called an absolute value of the input (k, x). Based on the outputs of all fuzzy classification rules from the rule base of the classification system, every pattern should be assigned to one of the predefined classes from {1, . . . , m}, i.e., the outputs of the fuzzy classification rules should be somehow put together in order to get overall result (output) of the classification. In other words, the outputs of the fuzzy classification rules should be aggregated in order to obtain the final classification for a pattern. Our aim is to investigate these aggregation techniques. Similarly as in the case of rule outputs, also the overall output of the classification system can have several forms: either the output is just a class (category) where the pattern should be assigned to, or it is a class along with a confidence degree, or it is an m-dimensional vector with a confidence degree for every possible class. However, the m-dimensional output is usually only informative and at the end the user is often interested only in one strongest class in which the pattern should be classified. The major part of the literature on fuzzy rule-based classification systems focuses only on the class output of the system (see for example [2] and references therein). Nevertheless, as we mentioned above, the confidence degree of classification is also important and therefore in the discussed aggregation techniques we will try to obtain a confidence degree as well. In such a case the output of the classification is a point from a multi-polar space. Summarizing, our focus is on aggregation techniques that can aggregate input points from a multi-polar space into output point from a multi-polar space. Some authors (see f.e. [2]) aggregate only outputs of the rules that are activated, i.e., their absolute value is bigger than 0. In this paper we will always aggregate all rules, i.e., non-activated rules correspond to inputs 0. Note that in the case when 0 is a neutral point of the aggregation—as it is in the case of maximum rule and maximum vote method—then this distinction plays no role. If we assume monotonicity of the classification then this type of aggregation is based on some multi-polar aggregation operator (see Section 2 and [13]). The aim of this paper is to discuss multi-polar averaging operators used for aggregation of outputs of fuzzy classification rules, propose new ones and investigate their properties. We begin with the definition of multi-polar aggregation operators (Section 2) and define several averaging operators in Section 3. Since the OWA operators (see [19–21]) have been successfully used in a number of fuzzy systems we will use their advantages for improvement of classification methods and define multi-polar OWA operators. In Section 4 we will show connection between multi-polar OWA and averaging operators and multi-polar Choquet integral and we will investigate properties of these operators. We will

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continue by discussing several special cases of multi-polar OWA operators and related capacities (Section 5). We give our conclusions in Section 6

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We begin with the definition of a multi-polar aggregation operator.

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Definition 2. Let m ∈ N. A mapping M: operator if



m n m n∈N (K × [0, 1]) −→ K × [0, 1], is called an m-polar aggregation

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(M1) M is non-decreasing, i.e., if xi ≤ yi then for   M (k1 , x1 ), . . . , (ki−1 , xi−1 ), (ki , xi ), (ki+1 , xi+1 ), . . . , (kn , xn ) = (k, x)   M (k1 , x1 ), . . . , (ki−1 , xi−1 ), (ki , yi ), (ki+1 , xi+1 ), . . . , (kn , xn ) = (p, y) we have one of the following cases: (k = p = ki ) ∧ (x ≤ y), or (k = p = ki ) ∧ (x ≥ y), or (k = p ∧ k = ki ) ∧ (p = ki ). Note that in the case when min(x, y) = 0 we assume the representation where k = p. (M2) M(0, . . . , 0) = 0 and M((k, 1), . . . , (k, 1)) = (k, 1) for all k ∈ K m ;       n-times n-times (M3) for n = 1, M((k, x)) = (k, x) for all (k, x) ∈ K m × [0, 1].

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2. Multi-polar aggregation operators

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The idea hidden in the property (M1) tells us that inputs are ‘dragging’ output of the aggregation in the direction of their respective category and the bigger an input in category ki is, the more belongs the output of the aggregation to this category ki , or the less it belongs to any other category. For more details and motivation for the property (M1) see [15]. In the case when m = 1 (m = 2) a multi-polar aggregation operator reduces to a (bipolar) aggregation operator on [0, 1] ([−1, 1]). Examples of multi-polar aggregation operators include the oriented maximum operator omax (which gives the input with the maximal absolute value, and gives 0 if two inputs with maximal absolute values have different classes), the ordered category projection operator (which gives the standard aggregation of values from the most important class that is present), and the union of the projections to a single coordinate is also an m-polar aggregation operator. Further, for x = ((k1 , x1 ), . . . , (kn , xn )) with ki ∈ K m , xi ∈ [0, 1] let   xi = (i, d1 ), . . . , (i, dn ) for i = 1, . . . , m, where dj = xj if j -th input belongs to i-th category, i.e., kj = i, and dj = 0 otherwise. Then the above is a decomposition of x to m parts according to categories and generalizing the bipolar case [10,12], an ordinal  sum construction for m-polar aggregation operators, the operator M ∗ : n∈N (K m × [0, 1])n −→ K m × [0, 1], is given by   (k, Ak (x)) if ki = k for all i, M ∗ (k1 , x1 ), . . . , (kn , xn ) = 1 1 m m ∗(A (x ), . . . , A (x )) else,  k where A is an aggregation operator for all k ∈ K m and ∗: n∈N (K m × [0, 1])n −→ K m × [0, 1] is an m-polar aggregation operator. The operator M ∗ is an m-polar aggregation operator, which will be called an m-polar ∗-ordinal sum of aggregation operators. As we will see later (Section 3), omax operator corresponds in classification systems to the case when the result is determined by the rule with the biggest absolute value of the output. The aggregation based on projection operators corresponds to the case when classification is based on just one selected rule. Finally, aggregation based on the ordered category projection corresponds to the case when categories are hierarchical. This means, that if the most important category appears among rule outputs then output of classification is in this category independently on absolute values of the rule outputs in different categories. Multi-polar aggregation operators are defined on K m × [0, 1], however, it is just a matter of simple transformation to define a multi-polar aggregation operator working on K m × [0, b] for any b > 0. The following is an example of

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basic multi-polar aggregation operators defined on [0, ∞[ instead of [0, 1]. Here clx(x) denotes the class of the input with the maximal absolute value. If it is not unique we take clx(x) = 1.

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Definition 3. Let A(d) = i=1 di for d = (d1 , . . . , dn ) and for input ((k1 , x1 ), . . . , (kn , xn )) let σ be a permutation such that xσ (1) ≥ · · · ≥ xσ (m) . Further assume 2 ≤ q ≤ m and     U q (k1 , x1 ), . . . , (kn , xn ) = clx(x), max(0, xσ (1) − xσ (2) − · · · − xσ (q) ) .  Then the operation Lq : n∈N (K m × [0, ∞[)n −→ K m × [0, ∞[ given by   (k, A(x)) if ki = k for all i, L (k1 , x1 ), . . . , (kn , xn ) = U q ((1, A(x1 )), . . . , (m, A(xm ))) else,

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will be called a q-summation on K m × [0, ∞[.

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where U 1 = omax. For q > 1 the q-summation extends the bipolar addition and therefore in the multi-polar case, q-summations replace standard addition on [0, ∞[. In [15] we have shown that in real-world example from physics, the multi-polar summation is modelled by Lm operator, i.e., by an m-summation. However, in some problems another operator can be selected as a multi-polar summation depending on the convenience. We conclude this section with the definition of several important properties of multi-polar aggregation operators.

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Definition 4. Let d: (K × [0, ∞[)2 −→ R+ 0 be given by   |x − y| if k1 = k2 d (k1 , x), (k2 , y) = x +y else. Then d will be called an oriented distance on an m-polar aggregation operator. Then

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(i) M is positively homogeneous if for all c > 0 there is M((k1 , c · x1 ), . . . , (kn , c · xn )) = (k, c · x), where M((k1 , x1 ), . . . , (kn , xn )) = (k, x). (i) M is continuous if for all i ∈ {1, . . . , n} and for all ε > 0 there exists a δ > 0 such that if d((ki , xi ), (qi , yi )) < δ then for   x = (k1 , x1 ), . . . , (ki−1 , xi−1 ), (ki , xi ), (ki+1 , xi+1 ) . . . , (kn , xn ) and

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  y = (k1 , x1 ), . . . , (ki−1 , xi−1 ), (qi , yi ), (ki+1 , xi+1 ) . . . , (kn , xn )

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we have   d M(x), M(y) < ε.

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(iii) M is associative if for all i ∈ {1, . . . , n} there is     M (k1 , x1 ), . . . , (kn , xn ) = M (k, y), (q, z) , where (k, y) = M((k1 , x1 ), . . . , (ki , xi )) and (q, z) = M((ki+1 , xi+1 ), . . . , (kn , xn )). (iv) M is commutative if the value of M((k1 , x1 ), . . . , (kn , xn )) does not depend on the order of inputs.

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Additionally, for q = 1 the 1-summation is given by         L1 (k1 , x1 ), . . . , (kn , xn ) = U 1 1, A x1 , . . . , m, A xm ,

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The omax operator is not continuous, neither associative, however, it is positively homogeneous and commutative. L1 summation is non-associative, non-continuous, commutative and positively homogeneous. For q > 1 the q-summation is positively homogeneous, commutative and continuous, however, it is not associative.

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3. Averaging operators used for aggregation of classification rules

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The two main methods for aggregation of fuzzy classification rules are the maximum vote and the maximum rule method (see [9]). The authors of [2] suggest that aggregation should be based on averaging operators, i.e., such operators that are bounded by minimum and maximum, i.e., min ≤ A ≤ max . Indeed, all methods from [2], except the maximum vote method are based on averages. Moreover, as arithmetic mean is just normalized summation, the maximum vote method is also related to an averaging operator. In this section we will discuss averaging operators used in aggregation of fuzzy classification rules and propose new ones. In maximum rule method the output of the classification is determined by the rule with the biggest absolute value of the output. This means that here the aggregation is based on the multi-polar omax operator. Assume that for a given pattern xp the outputs of n fuzzy classification rules in our rule base are (k1 , x1 ), . . . , (kn , xn ), where (ki , xi ) ∈ K m × [0, 1] for i = 1, . . . , n. Then the output of the classification system (k, x) is given by (k, x) = omax(x), where x = (k1 , x1 ), . . . , (kn , xn ). In maximum vote method the output of the classification is determined by the following procedure:

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(k, x) = L (x). 1

The problem with this method is in the fact that L1 is a multi-polar aggregation operator on K m × [0, ∞[ and not on K m × [0, 1]. In unipolar case this corresponds to standard addition which is an aggregation operator on [0, ∞[ but not on [0, 1]. This fact implies that the maximum possible confidence of classification is n for n rules in the rule base instead of 1. However, in this paper we assume that the confidence degree is equal to the association degree between the pattern and the class which cannot exceed 1, i.e., 100% association. Therefore we should modify the maximum vote method accordingly. In unipolar case the summation can be transformed

to an aggregation operator on [0, 1]

x simply by dividing by the number of inputs. In this case xi will change to n i and thus summation is replaced by the arithmetic mean. In the multi-polar case we can do the same.

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Definition 5. Let x = (k1 , x1 ), . . . , (kn , xn ) where (ki , xi ) ∈ K m × [0, 1] for i = 1, . . . , n. Then a mapping  A: n∈N (K m × [0, 1])n −→ K m × [0, 1] given by   L1 (x) x1 xn A(x) = = L1 k1 , , . . . , kn , n n n will be called an m-polar arithmetic mean.

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This means that the aggregation is in this case based on the multi-polar L1 operator, i.e., 1-summation. Therefore the output of the classification system (k, x) is given by

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1. Divide rules into groups with the same consequent classes. 2. Sum absolute values of the rules within each group. 3. Select the class corresponding to the group with the maximum sum of outputs.

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As we mentioned above, 1-summation is the only q-summation that does not extend the bipolar addition. In the case that we want to model the multi-polar addition by another q-summation we define the following.

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Definition 6. Let x = (k1 , x1 ), . . . , (kn , xn ) where (ki , xi ) ∈ K m × [0, 1] for i = 1, . . . , n. Then for q ≤ m a mapping  q A : n∈N (K m × [0, 1])n −→ K m × [0, 1] given by   Lq (x) x1 xn Aq (x) = = Lq k1 , , . . . , kn , n n n

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will be called an m-polar q-arithmetic mean.

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The q-arithmetic mean is a positively homogeneous and commutative multi-polar aggregation operator. For q > 1, q-arithmetic mean is continuous and extends bipolar arithmetic mean. As arithmetic mean is normalized addition we will call a method based on this type of aggregation a normalized maximum vote method. Aggregation based on multi-polar arithmetic mean suppose that every rule has the same weight. In the case that rules have different weights normalization should be done by dividing the output by the sum of weights instead of n. On the other hand, generalizing the arithmetic mean we can introduce a multi-polar weighted mean as follows.

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For q = 1 a multi-polar 1-weighted mean will be called simply a multi-polar weighted mean. The properties of multi-polar weighted means are the same as those of multi-polar arithmetic means. Moreover, is easy to see the following proposition. Proposition 1. Assume a classification system based on fuzzy rules with weights, where for each pattern x the absolute value of the output of each rule is obtained as the product of the firing degree and the rule weight, and aggregation method is based on normalized maximum vote. Then this classification system is equivalent to the classification system based on fuzzy rules without weights, where aggregation method is based on a multi-polar weighted mean, with weights equal to normalized rule weights of the original classification system (i.e., divided by the sum of weights). Similarly as we have introduced weights into maximum vote method we can introduce weights into maximum rule method. First we will define a weighted maximum operator. Definition 8. Let x = (k1 , x1 ), . . . , (kn , xn ) where (ki , xi ) ∈ K m × [0, 1] for i = 1, . . . , n. Further let w = (w1 , . . . , wn ) with wi ≥ 0 and maxni=1 wi = 1 be a weighting vector. Then a mapping omaxw : n∈N (K m × [0, 1])n −→ K m × [0, 1] given by   omaxw (x) = omax(wx) = omax (k1 , w1 x1 ), . . . , (kn , wn xn )

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will be called a weighted oriented maximum.

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The weighted oriented maximum has the same basic properties as oriented maximum operator. Moreover, it is easy to show the following proposition.

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Proposition 2. Assume a classification system based on fuzzy rules with weights, where for each pattern x the absolute value of the output of each rule is obtained as the product of the firing degree and the rule weight, and aggregation method is based on the maximum rule method. Then this classification system is equivalent to the classification system based on fuzzy rules without weights, where aggregation method is based on a multi-polar weighted oriented maximum, with weights equal to normalized rule weights of the original classification system (i.e., divided by the strongest weight).

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will be called an m-polar q-weighted mean.

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× [0, 1] for i =1, . . . , n. Let w = (w1 , . . . , wn ) with Definition 7.

Let x = (k1 , x1 ), . . . , (kn , xn ) where (ki , xi q wi ≥ 0 and ni=1 wi = 1 be a weighting vector. Then for q ≤ m a mapping Ww : n∈N (K m × [0, 1])n −→ K m × [0, 1] given by   W q (x) = Lq (wx) = Lq (k1 , w1 x1 ), . . . , (kn , wn xn )

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Propositions 1 and 2 show the important fact that rule weights can be replaced by introducing weights into aggregation process. This fact solves the problem of interpretability of rule weights. It also shows that the search for the best rule weights can be transformed to the fitting of the proper aggregation operator. Another averaging operator related to the weighted mean on [0, 1] is an OWA (ordered weighted average) operator [19–21]. The OWA operator has been successfully used in a number of fuzzy systems and therefore we would like to introduce the same idea into classification systems. In the case of OWA operator the weights are not assigned to the inputs according to the position, but according to the strength of inputs. If we assume a weighting vector

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w = (w1 , . . . , wn ) with wi ≥ 0 and formula n

OWAw (x) = wi xσ (i) ,

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i=1 wi

= 1, then for an input x = (x1 , . . . , xn ) the OWA operator is given by the

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where σ : {1, . . . , n} −→ {1, . . . , n} is a permutation such that xσ (1) ≤ · · · ≤ xσ (n) . Note that in original paper [19] an equivalent non-increasing order was used, however, as we will later investigate the connection between OWA and the Choquet integral a non-decreasing order is preferable. Several extensions of the OWA operator to the bipolar case are described in [11]. Similar procedure can be adopted also for the case of multi-polar inputs. Following the idea that weights should be assigned according to the strength of inputs we see that in multi-polar case the permutation σ should give the absolute values of the inputs into nondecreasing order. The problem with this procedure is in the fact that inputs from different classes can have the same absolute value. While in the unipolar case for inputs 0.3, 0.3 the OWA operator gives the same output independent on the permutation σ , already in bipolar case inputs (1, 0.3), (2, 0.3) do not give the same result. For example for w = (0.2, 0.8) we get in the first case, when we assume x1 ≤ x2 , that multi-polar OWA of these two inputs is L2 ((1, 0.06), (2, 0.24)) = (2, 0.18), but for x2 ≤ x1 we get L2 ((1, 0.24), (2, 0.06)) = (1, 0.18). Therefore, for multipolar OWA operators we should solve the problem of ties. In [11] several possibilities how to solve such a tie in the bipolar case were shown. We will define multi-polar OWA operators in a similar way. The simplest choice how to define a multi-polar OWA operator is to divide inputs into separate classes, aggregate this classes by means of the unipolar (standard) OWA operator and then put the result together by means of a selected addition. Then

the multipolar q-symmetric OWA operator with respect to a weighting vector w = (w1 , . . . , wn ) with wi ≥ 0 and ni=1 wi = 1 is given by       q MSOWAw (x) = Lq 1, OWAw x1 , . . . , m, OWAw xm . However, this approach aggregate all classes separately and thus does not allow possible synergy between classes. A multi-polar fusion OWA operator allows for this synergy. Let

x = (k1 , x1 ), . . . , (kn , xn ) where (ki , xi ) ∈ K m × [0, 1] n for i = 1, . . . , n. Further, let w = (w 1 , . . . , wmn ) with win≥ 0 and m i=1 wi = 1 be a weighting vector. Then a multi-polar fusion OWA operator MFOWAw : n∈N (K × [0, 1]) −→ K × [0, 1] is given by the following procedure. 1. Define value classes of the input x, i.e., classes C0 , C1 , . . . , Cp , where C0 = {j ∈ {1, . . . , n} | xj = 0} and sets Ci ⊆ {1, . . . , n}, Ci =  ∅ for i = 1, . . . , p are such that 0 < xk = xj for all k, j ∈ Ci and xk < xj for all k ∈ Ci , p j ∈ Cr with i < r and i=0 Ci = {1, . . . , n}. 2. Divide the weighting vector into groups of weights, each corresponding to one value class, i.e., if Card(Ci ) = si for i = 0, 1, . . . , p then to C0 corresponds the group of weights W0 = (w1 , . . . , ws0 ), the group of weights W1 = (ws0 +1 , . . . , ws0 +s1 ) corresponds to C1 and so on. 0 = ( 3. Average the weights within the weight groups, i.e., we obtain a new weight groups W w1 , . . . , w  s0 ), where

s0

w

i w i = i=1 for i = 1, . . . , s0 and similarly for all weight groups. s0 4. Use the procedure for OWA operator and compute multi-polar OWA operator with respect to the weighting vector  n ). w = ( w1 , . . . , w

40 41 42 43 44 45 46 47 48 49 50 51 52

1 2

i=1

5 6

n

7

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Therefore the formula for the multi-polar q-fusion OWA operator is the following:   q MFOWAw (x) = Lq (k1 , v1 · x1 ), . . . , (kn , vn · xn ) ,

nj

i=nj −1 +1

wi

41 42 43

j

where for i ∈ Cj we have vi = , sj = Card(Cj ) for j = 0, 1, . . . , p, nj = u=1 su . sj Thus the multi-polar fusion OWA operator solves the problem of ties by averaging the weights that belong to the inputs with the same absolute value. However, this procedure does not take into account how many inputs with the same absolute value belong to the same class. Therefore the modification of the weight averaging procedure yield a multi-polar group OWA (MGOWA) operator. The computation of MGOWA operator follows from the following philosophy: assume that the categories are ordered, i.e., there exists a permutation τ : {1, . . . , m} −→ {1, . . . , m} such that for categories 1, . . . , m we have τ (1) ≤ · · · ≤ τ (m).

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1 2 3 4 5 6 7

Then the ties in the computation of the multi-polar OWA operator can be solved by further ordering of inputs with the same absolute values into a non-decreasing order according to the categories. Therefore the formula for the ordered q MGOWA operator is given by MGOWAw (x) = Lq ((kσ (1) , w1 ·xσ (1) ), . . . , (kσ (n) , wn ·xσ (n) )), where σ : {1, . . . , n} −→ {1, . . . , n} is a permutation such that xσ (1) ≤ · · · ≤ xσ (n) and for all i < j if xσ (i−1) < xσ (i) = · · · = xσ (j ) < xσ (j +1) then kτ (σ (i)) ≤ · · · ≤ kτ (σ (j )) . In the case when there is no order defined on the set of categories we take the average through all possible orderings of categories. The procedure for computation of the (unordered) MGOWA is therefore the following:

8 9 10 11 12 13 14 15

18 19 20

1. Define value classes of the input x. 2. Divide the weighting vector into groups of weights, each corresponding to one value class. 3. Divide each value class Ci into classes Cik for k = 1, . . . , m such that j ∈ Cik if and only if kj = k. Denote rik = Card(Cik ). Then si = ri1 + · · · + rim . 4. Define a function F : P(K m ) −→ R by

rik F (S) = for S ∈ P(K m ) and F (∅) = 0. Further denote Q(S, k) := F (S) + k. Assume that the group of weights corresponding to the value class Ci is wj1 , . . . , wjsi and denote vt = wjt for t = 1, . . . , si . Then we define a new weight group { v1 , . . . ,  vsi }, with  v1 = · · · =  vr 1 ,  vr 1 +1 = · · · =  vr 1 +r 2 , and so on . . .; by i

 v1 =

25

28 29 30 31 32 33 34 35 36 37 38

ri1 k=1 vQ(S,k) S∈P (K m \{1}) r 1  m−1  i Card(S)

m

26 27

i

i

i

41 42 43 44

47 48 49 50 51 52

6 7

 vr 1 +1 = i

9 10 11 12 13 14 15

S∈P (K m \{2})

m

17 18 19 20

23

,

ri2 vQ(S,k) k=1  m−1  ri2 Card(S)

24 25 26

,

and so on . . . . The set S in the summation index simply shows how many classes are smaller in current ordering than the class for which we the new weights. As the number of the subsets of the cardinality s of the set  compute  with cardinality m − 1 is m−1 and there are m possible cardinalities: 0, 1, . . . , m − 1, we have to divide the new s weights accordingly. Then we again denote w jt =  vt for t = 1, . . . , si . w = ( w1 , . . . , w n ) and a 5. Compute multi-polar ordered MGOWA operator with respect to the weighting vector  permutation τ (k) = k for k ∈ K m . Therefore the formula for the multi-polar (unordered) q-group OWA operator is the following:   q 1 · xσ (1) ), . . . , (kσ (n) , w n · xσ (n) ) , MGOWAw (x) = Lq (kσ (1) , w

27 28 29 30 31 32 33 34 35 36 37 38 39

w is described in the above procedure and σ is a permutation such that xσ (1) ≤ · · · ≤ xσ (n) where the weighting vector  and for all i < j if xσ (i−1) < xσ (i) = · · · = xσ (j ) < xσ (j +1) then kσ (i) ≤ · · · ≤ kσ (j ) . We conclude this section with a combined multi-polar OWA operator. The combination of the approach using value classes and the basic multi-polar symmetric OWA operator yields a multi-polar balancing OWA operator. The procedure for computation of this multi-polar OWA operator is the following:

45 46

5

22

39 40

4

21



22

24

3

16

k∈S

21

23

2

8

16 17

1

40 41 42 43 44 45

1. Define value classes of the input x. 2. Divide the weighting vector into groups of weights, each corresponding to one value class. 3. Divide each value class Ci into classes Cik for k = 1, . . . , m such that j ∈ Cik if and only if kj = k. Denote rik = Card(Cik ). Then si = ri1 + · · · + rim . 4. Aggregate inputs in every value class by means of the multi-polar symmetric OWA operator with respect to the corresponding weight group (without the final summation). 5. Sum the outputs by the selected summation.

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Therefore the formula for the multi-polar q-balancing OWA operator is the following:   q 1 · x1 ), . . . , (kn , w n · xn ) , MBOWAw (x) = Lq (k1 , w where the weighting vector  w is given by

nj w u=nj −rjk +1 ju w i = , rjk where i ∈ and wj1 , . . . , wjsj are weights corresponding to the value class Cj in an original order and nj = s0 + · · · + sj . All multi-polar OWA operators that we defined, as well as weighted multi-polar average and weighted oriented maximum are positively homogeneous and moreover, they are also positively homogeneous with respect to weights. This means that if  w is not normalized and  w = c · w then A w (x) = c · Aw (x). For the fuzzy rule-based classification systems this means that if we multiply all rule weights by the same constant c > 0 the category output of the classification will not change and the absolute value of the output will be multiplied by c. Cjk

16 17 18 19 20 21 22 23

Remark 1. For all four multi-polar OWA operators, for m n, where m is the number of classes and n is the number of inputs, the most computationally expensive operation is sorting of inputs according to absolute values. All other operations, i.e., weights grouping, weights averaging, computation of OWA (of sorted inputs), even permutations on classes are of the computational complexity O(n) (assuming the fixed-precision floating-point arithmetic). Thus computational complexity of these multi-polar OWA operators is the same as the computational complexity of the sorting algorithm. On the web page http://www.sorting-algorithms.com we can see a summary of sorting algorithms, were the best algorithms have the computational complexity of O(n · log(n)).

28 29 30 31 32

4. Multi-polar averages and multi-polar Choquet integral In this section we will discuss relation between multi-polar averaging and OWA operators and a multi-polar Choquet integral. The Choquet integral (see [1,3,17,18]) was used in a number of applications and it was shown that it works as an interpolator between the grid points given by the respective capacity [5]. In unipolar case, the OWA operators were shown to be in one-to-one correspondence with symmetric capacities via the Choquet integral (see [4]). We would like to obtain a similar result also for the multi-polar case. We begin with the definitions of several multipolar Choquet integrals with respect to a (unipolar) capacity.

37 38 39 40

43 44 45 46 47 48 49 50 51 52

6 7 8 9 10 11 12 13 14 15

17 18 19 20 21 22 23

25

27 28 29 30 31 32

34 35

(i) A capacity m: 2X −→ [0, 1] is a set function such that μ(∅) = 0, μ(X) = 1 and μ(A) ≤ μ(B) whenever A ⊆ B ⊆ X. (ii) A capacity m: 2X −→ [0, 1] is called additive if for all A ⊆ X there is

  μ(A) = μ {i} . i∈A

41 42

5

33

Definition 9.

35 36

4

26

33 34

3

24

26 27

2

16

24 25

1

(iii) A capacity

m: 2X

μ(A) = μ(B)

38 39 40

42 43 44

for all A, B ⊆ X with Card(A) = Card(B). (iv) The (discrete) Choquet integral C(μ, x) for x ∈ [0, 1]n is defined by C(μ, x) =

37

41

−→ [0, 1] is called symmetric if

n

36

xσ (i)

 μ(Aσ,i ) − μ(Aσ,i+1 ) ,



i=1

where σ is a permutation of the vector (1, . . . , n) such that xσ (1) ≤ · · · ≤ xσ (n) and Aσ,i = {σ (i), . . . , σ (n)}, with convention Aσ,n+1 = ∅.

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10

1

Extension of the Choquet integral into multi-polar scale yield several multi-polar Choquet integrals (see [14]).

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

2

−→ [0, 1] be a capacity on X for i = 1, . . . , m, and denote Definition 10. Let X = {1, . . . , n}, m ∈ N, let μi Ψ = {μ1 , . . . , μm }. Then for an m-polar aggregation operator O, such that O(0, . . . , 0, (k, x), 0, . . . , 0) = (k, x) for k ∈ K, x ∈ [0, 1], the m-polar O-Choquet integral with respect to Ψ , OMC(Ψ, ·): (K m × [0, 1])n −→ K m × [0, 1], is given by       OMC(Ψ, x) = O 1, C μ1 , x 1 , . . . , m, C μm , x m ,

3

= Lq

9

: 2X

= L1 )

(O the m-polar O-Choquet integral is called the where x = ((k1 , x1 ), . . . , (kn , xn )). In the case when O m-polar q-Choquet integral (Choquet integral) MCq (MC). When μ1 = · · · = μm we say that the m-polar O-Choquet integral is symmetric.

46

49

(i) For an input vector x = ((k1 , x1 ), . . . , (kn , xn )) consider a set C0 = {j ∈ {1, . . . , n} | xj = 0} and sets Ci ⊆ {1, . . . , n}, Ci = ∅ with i = 1, . . . , p, such that 0 < xk = xj for all k, j ∈ Ci and xk < xj for all k ∈ Ci , j ∈ Cr with p q p i < r and i=0 Ci = {1, . . . , n}. Denote Ci = {j ∈ Ci | kj = q} for q = 1, . . . , m. Further denote Di = k=i Ck and |Ci | = xj for j ∈ Ci . The sets Ci will be called (m-polar) value classes of input x. (ii) Let μ: 2X −→ [0, 1] be a capacity on X. Then for an m-polar aggregation operator O, such that O(0, . . . , 0, (k, x), 0, . . . , 0) = (k, x) for k ∈ K, x ∈ [0, 1], the m-polar balancing O-Choquet integral OMBC(μ, ·): (K m × [0, 1])n −→ K m × [0, 1] of input x with respect to the capacity μ is given by OMBC(μ, x)     p p

    1    m  |Ci | μ Ci ∪ Di+1 − μ(Di+1 ) , . . . , m, |Ci | μ Ci ∪ Di+1 − μ(Di+1 ) , = O 1, i=1

i=1

where Dp+1 = ∅. In the case when O = Lq (O = L1 ) the m-polar balancing O-Choquet integral is called the m-polar q-balancing Choquet integral (balancing Choquet integral) MBCq (MBC).

52

6 7 8

10 11 12

16 17 18 19 20 21 22 23 24 25 26 27 28 29

31 32

Γ = {σ | σ is a permutation, xσ (1) ≤ · · · ≤ xσ (n) }

33

with Card(Γ ) = S for an input x = ((k1 , x1 ), . . . , (kn , xn )). Then for an m-polar aggregation operator O, such that O(0, . . . , 0, (k, x), 0, . . . , 0) = (k, x) for k ∈ K, x ∈ [0, 1], the m-polar fusion O-Choquet integral OMFC(μ, ·): (K m × [0, 1])n −→ K m × [0, 1] of input x is given by    μ(Aσ,i ) − μ(Aσ,i+1 ) n OMFC(μ, x) = O CSσ ∈Γ CSi=1 kσ (i) , xσ (i) , S where CS: (K m × [0, 1])n −→ [0, 1]m is the category summation given by     n n

xj1 , . . . , m, xjm . CS(x) = 1, j =1

15

30

Definition 12. Let μ: 2X −→ [0, 1] be a capacity on X and let

34 35 36 37 38 39 40 41 42 43

j =1

44

In the case when O = Lq (O = L1 ) the m-polar fusion O-Choquet integral is called the m-polar q-fusion Choquet integral (fusion Choquet integral) MFCq (MFC).

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In the following proposition we show relation between multi-polar symmetric OWA operator and the symmetric m-polar Choquet integral.

50 51

5

14

47 48

4

13

Definition 11.

44 45

1

48 49 50

Proposition 3. The symmetric m-polar q-Choquet integral with respect to a symmetric capacity multi-polar q-symmetric OWA operator.

μ: 2X

−→ [0, 1] is a

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11

The proof of the previous proposition follows from the fact that the (unipolar) Choquet integral with respect to a symmetric capacity is just an OWA operator with respect to a weighting vector given by wi = μ(n − i + 1) − μ(n − i), where for a symmetric capacity μ the μ(i) denotes the measure of any set A with cardinality Card(A) = i. Thus we have shown that m-polar q-symmetric OWA operators and symmetric capacities are in one-to-one correspondence via the symmetric m-polar q-Choquet integral. Similar result can be shown for the multi-polar fusion OWA operator.

6 7 8

11 12 13 14 15

Theorem 1. The m-polar q-fusion Choquet integral with respect to a symmetric capacity m-polar q-fusion OWA operator.

−→ [0, 1] is an

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

4 5

7

9

−→ [0, 1] and q ≤ m. Then the q-fusion Choquet integral with respect Proof. Assume a symmetric capacity to μ is given by    μ(Aσ,i ) − μ(Aσ,i+1 ) q q n MFC (μ, x) = L CSσ ∈Γ CSi=1 kσ (i) , xσ (i) , S

10

where

15

μ: 2X

11 12 13 14

16

Γ = {σ | σ is a permutation, xσ (1) ≤ · · · ≤ xσ (n) }

17

19

3

8

16

18

2

6

μ: 2X

9 10

1

17

with Card(Γ ) = S for an input x = ((k1 , x1 ), . . . , (kn , xn )). If we denote wi = μ(n − i + 1) − μ(n − i) we get    wi MFCq (μ, x) = Lq CSσ ∈Γ CSni=1 kσ (i) , xσ (i) . S Dividing the input x into value classes C0 , C1 , . . . , Cp we can see that the weights wi corresponding to all inputs from the same value class are for every σ ∈ Γ always the same. For a value class Ci we denote Card(Ci ) = si . Then there are si ! possible permutations within the value class Ci and Card(Γ ) = S = s0 ! · · · si ! · · · sp !. Therefore the q-fusion Choquet integral with respect to a symmetric capacity can be computed separately for each value class, i.e.,  p   MFCq (μ, x) = Lq CSj =0 MFCq wj , xj , where wi = μ(n − i + 1) − μ(n − i), w = (w1 , . . . , wn ) and wj is the group of weights corresponding to a value class j , i.e.,  j j  wj = w1 , . . . , wsj = (ws0 +s1 +···+sj −1 +1 , . . . , ws0 +s1 +···+sj −1 +sj ), xj

j j j j = ((k1 , x1 ), . . . , (ksj , xsj ))

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

is a vector composed of the inputs from the class j and

34

 j   w sj j MFCq wj , xj = CSτ ∈Ω CSi=1 uj · kτ (i) , i , sj !

35 36 37 38

40

where uj is the absolute value of the inputs belonging to the class j and   Ω = τ | τ : {1, . . . , sj } −→ {1, . . . , sj } is a permutation .

41

Further,

41

38 39

42 43

q



j

MFC w , x

j



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sj = uj · CSi=1 CSτ ∈Ω

51 52

40

j

 j kτ (i) ,

42

wi , sj !

43 44 45

and thus

46

q



j

MFC w , x

j



j sj wi = uj · CSi=1 sj

sj  j  kt , 1 . CSt=1

49 50

39

We conclude that MFC (wj , xj ) = uj ·

sj j sj wi p j Lq (CSj =0 CSt=1 (kt , uj · i=1 )) and sj q

sj i=1 sj

thus

j

wi

sj sj j j CSt=1 (kt , 1) = CSt=1 (kt , uj

47 48

·

sj i=1 sj

49 j

wi

). Summarizing MFC (μ, x) = q

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12

1

  MFCq (μ, x) = Lq (k1 , x1 · v1 ) . . . , (kn , xn · vn ) ,

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

nj

i=nj −1 +1

where for i ∈ j there is vi =

sj

wi

, with nj =

1

j

i=1 si ,

2

thus finishing the proof.

3

2

4

We have shown that m-polar q-fusion OWA operators and symmetric capacities are in one-to-one correspondence via the m-polar q-fusion Choquet integral. Remark 2. For m = 2 the m-polar balancing Choquet integral with respect to a symmetric capacity μ: 2X −→ [0, 1] is an m-polar (unordered) group OWA operator (see [11]). However, for m > 2 this is no longer true. However, similarly as in the case of the multi-polar fusion Choquet integral a multi-polar group Choquet integral can be defined as an average of the Choquet integrals through all permutations of categories.

26

29 30 31

34

and let Γ be the set of all permutations σ : {1, . . . , n} −→ {1, . . . , n} such that xσ (1) ≤ · · · ≤ xσ (n) and for each i < j with xσ (i−1) < xσ (i) = · · · = xσ (j ) < xσ (j +1) there exists a τ ∈ Ω such that kτ (σ (i)) ≤ · · · kτ (σ (j )) , where Card(Γ ) = S for input x = ((k1 , x1 ), . . . , (kn , xn )). Then for an m-polar aggregation operator O, such that O(0, . . . , 0, (k, x), 0, . . . , 0) = (k, x) for k ∈ K, x ∈ [0, 1], the m-polar group O-Choquet integral OMGC(μ, ·): (K m × [0, 1])n −→ K m × [0, 1] of input x is given by    μ(Aσ,i ) − μ(Aσ,i+1 ) OMGC(μ, x) = O CSσ ∈Γ CSni=1 kσ (i) , xσ (i) , S

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

9 10 11 12

16 17 18 19 20 21 22 23 24

where in the case when O = Lq (O = L1 ) the m-polar group O-Choquet integral is called the m-polar q-group Choquet integral (group Choquet integral) MGCq (MGC).

25 26 27

Similarly as in the case of m-polar q-fusion OWA operators we can show that m-polar q-group OWA operators and symmetric capacities are in one-to-one correspondence via the m-polar q-group Choquet integral. The combined multi-polar balanced OWA operator has the similar structure as other multi-polar OWA operators. Namely, in the following proposition we show its connection to the balancing Choquet integral.

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Proposition 4. The m-polar q-balancing Choquet integral with respect to a symmetric capacity μ: 2X −→ [0, 1] is an m-polar q-balancing OWA operator.

35 36

8

15

32 33

7

14

27 28

6

13

Definition 13. Let μ: 2X −→ [0, 1] be a capacity on X and   Ω = τ | τ is a permutation, τ : {1, . . . , m} −→ {1, . . . , m}

24 25

5

33 34 35

j

The proof of the above proposition follows easily from the fact that if wi = μ(n − i + 1) − μ(n − i) then μ(Ci ∪ Di+1 ) − μ(Di+1 ) = ws +···+s −r j +1 + · · · + ws0 +···+si . 0

i

i

Remark 3. Beside the Choquet integral, in unipolar case the Sugeno integral plays an important role. The (discrete) Sugeno integral S(μ, x) for x ∈ [0, 1]n is defined by S(μ, x) =

n 

xσ (i) ∧ μ(Aσ,i ),

i=1

where σ is a permutation of the vector (1, . . . , n) such that xσ (1) ≤ · · · ≤ xσ (n) and Aσ,i = {σ (i), . . . , σ (n)}, with convention Aσ,n+1 = ∅. The Sugeno integral with respect to a maxitive capacity, i.e., such capacity μ that    μ {i} μ(A) = i∈A

is just a weighted maximum. The Sugeno integral can be generalized to a multi-polar scale similarly as the Choquet integral. The weighted oriented maximum for aggregation of fuzzy rules by weighted maximum rule method can be then obtained as the multi-polar Sugeno integral with respect to a maxitive capacity.

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4.1. Monotonicity of multi-polar OWA operators

1

2 3 4 5

2

In the following we will examine when are the multi-polar OWA operators also multi-polar aggregation operators, i.e., we want to see under which conditions are respective multi-polar OWA operators monotone. The first proposition is very easy to see.

6 7

10 11

14 15 16

8

Indeed, if an input xi is increased then OWA applied on the xki is increased and OWA’s in all other classes are unchanged. Therefore if the original output was from the class ki the output will increase, otherwise it will either decrease or stay unchanged.

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

9 10 11 12

Proposition

6. If a multi-polar q-fusion OWA operator with respect to a weighting vector w = (w1 , . . . , wn ) with wi ≥ 0 and ni=1 wi = 1, is monotone then weights are increasing, i.e., w1 ≤ w2 ≤ · · · ≤ wn . For q = 1, m also the opposite holds, i.e., if w1 ≤ w2 ≤ · · · ≤ wn then a multi-polar q-fusion OWA operator with respect to a weighting vector w is monotone.

17 18

5

7

12 13

4

6

Proposition 5. A multi-polar SOWA operator is monotone for all q ≤ m.

8 9

3

13 14 15 16 17

Proof. The necessity follows from the fact that in bipolar case the multi-polar fusion integral is monotone if and only if the capacity μ is submodular (see [11]). From Theorem 1 we have wi = μ(n − i + 1) − μ(n − i) and wi ≤ wi+1 means that μ(n − i + 1) − μ(n − i) ≤ μ(n − i) − μ(n − i − 1)

18 19 20 21

which holds because of the submodularity of μ. For the sufficiency assume that w1 ≤ w2 ≤ · · · ≤ wn and that xi change to xi + c for some c = 0 such that xi + c ∈ [0, 1]. Now there are two cases, either c > 0 or c < 0. We will assume c > 0, the other case can be shown analogically. Assume i ∈ Cj . If j = p we have increased an input from the strongest value class and by this we have created a new value class. If j < p and we take any s ∈ Cj +1 we have two possibilities: either xi + c < xs or xi + c ≥ xs . However, in the case that xi + c ≥ xs we can divide the interval [xi , xi + c] into several subintervals by points xi = xu0 , xi + c0 , xu1 , xu1 + c1 , . . . , xi + c where uk ∈ Cj +k , ck > 0 and xuk + ck < xuk+1 for k = 0, 1, . . . , q and either xi + c = xv for v ∈ Cj +q+1 or xi + c = xuq + cq . Thus this possibility can be shown by composition of several cases when xi + c < xs in combination with the cases when c < 0 and xi + c > xu for u ∈ Cj −1 . Summarizing, it is enough to show that if we increase an input by a small amount such that the new input will be smaller than absolute value of the inputs in the subsequent value class, or the increased input belongs to the strongest value class, the output will increase (or stay unchanged) with respect to the corresponding category. From the procedure for computing of MFOWA we can see that the MFOWA can be computed in each value class separately and then summed together. Therefore we will examine the change in the value class Cj . Denoting si = Card(Ci ) we see that the corresponding weight group for the value class Cj is ws0 +···+sj −1 +1 , . . . , ws0 +···+sj . For simplicity we will denote ws0 +···+sj −1 +d = vd for d = 1, . . . , sj . Denote further r d = Card(Cjd ) for d = 1, . . . , m. Then the output of the MFOWA in the value class Cj is   W + vs j W + vs j 1, xi · r 1 , . . . , m, xi · r m , sj sj

sj −1 where W = i=1 vi . If we increase xi with i ∈ Cj by a small value we will create a new value class. Sum of the outputs of MFOWA in these two value classes is then  k W k, xi · r sj − 1 for all k = ki and for ki we get    W + (xi + c)vsj . ki , xi r ki − 1 sj − 1

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

51

We have to show that the output of the aggregation will not decrease with respect to the category ki . The difference

52

of the output in each category is the following: for k = ki the output will decrease by

W +vs xi (r k sj j

−

r k sjW−1 )

=

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14

vsj − s W−1

vs (sj −1)−W

xi · r k jsj (sj −1) = xi · r k ki the output will increase by 

j

sj

≥ 0. The last inequality follows from the fact that v1 ≤ · · · ≤ vsj . In the category

19 20 21 22 23 24 25 26 27 28 29 30 31 32

W + vs j  W = xi + (xi + c)vsj − xi · r ki sj − 1 sj

xi · r ki − 1

(sj − r ki )(vsj − sj

W sj −1 )

4

+ c · vs j ≥ 0

s=1,s=k,ki

vsj − s W−1

m

35 36 37 38 39 40 41 42 43 44 45 46 47

50 51 52

8 9 10 11 12 13 14 15 16

18

rs

Since from the above computations we know that B ≥ 0 we get A ≥ 0 thus finishing the proof.

19 20

2

Remark 4. In the bipolar case q ≤ m admits only q = 1, 2 and therefore if m = 2 for all q ≤ m we have that a multi-polar q-fusion OWA operator with respect to a weighting vector w is monotone if and only if w1 ≤ · · · ≤ wn . The same holds for m-polar case for m ∈ N in the case of two inputs. Obviously, two inputs cannot belong to more than two classes. As we have seen from the proof of the previous proposition, the problem with the sufficiency part for q = 1, m is in the case when classes different than ki are used for final summation in computation of q-fusion OWA operator. Indeed, if ki is the winning class of the original aggregation then the monotonicity holds without problem. Also in the case that the winning class of the original aggregation is k = ki , but ki is used for final summation in computation of q-fusion OWA operator. In this case the absolute value of the output will decrease by       xi · B r k − R + xi R + r k B + c · vsj = 2 · xi · B · r k + xi · B r σ (m) + · · · + r σ (q+1) − r ki + c · vsj ,

21 22 23 24 25 26 27 28 29 30 31 32 33

where σ is the permutation corresponding to Lq summation. Note that r ki is subtracted because it is equal to one of r σ (m), . . . , r σ (q + 1). Thus the decrease in evidently non-negative. However, in the case when ki is not among classes used for final summation the difference in output values will be   x i · B r k − R σ + c · vsj ,

q where R σ = s=2 r σ (s) and σ (q) < σ (ki ), which is not necessarily non-negative. The problem arises here when r k < R σ and c is small enough. Summarizing, monotonicity is violated in the case when increased input is from the class not involved in the final summation and r k < R σ . If we have only three inputs, if these belong only to two classes, by the bipolar case MFOWA is monotone. Suppose that from the three inputs each is in a separate class. If the original output is 0 there is no problem with monotonicity and thus assume that output is not zero and the winning class is k. This means that the corresponding input has a unique maximal absolute value. In order not to include ki into computation necessarily the corresponding input has a unique minimal absolute value. However, then r k = 0 and R σ = 0 thus there is no problem with monotonicity. Summarizing, for three inputs MFOWA is monotone if and only if w1 ≤ w2 ≤ w3 . The case with four inputs can be seen in the following example.

48 49

7

17

We have to show that A ≥ 0. If we denote B = and s=1,s=k,ki = R we have sj     A = xi · B r k − R + xi R + r k B + c · vsj = 2 · xi · B · r k + c · vsj . j

5 6

where again the last inequality follows from the fact that v1 ≤ · · · ≤ vsj . Thus we see that if q = 1 we have the result since if the category output of MFOWA was the category ki then output will increase, if it was k = ki the output will either decrease or change category to ki . In the case when q = m we can distinguish two cases. When the original output of MFOWA was ki then the new output will remain in category ki and will increase. In the case that the result was k = ki we get the following: If the category of the new output will change then the new output category will be ki since it is the only category where the output was increased. If the category of the new output will remain the same k = ki then the absolute value will decrease by the following:   m vs − sjW−1 (sj − r ki )(vsj − sjW−1 ) vsj − sjW−1

k j s A = xi · r + xi + c · v s j − xi r . sj sj sj

33 34

2 3

17 18

1

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Example 1. Assume an input x = ((1, 0.25), (2, 0.25), (2, 0.5), (3, 1)) and let w = (0.1, 0.2, 0.3, 0.4) be a weighting vector. Then MFOWA1w (x) = (3, 0.4), MFOWA2w (x) = (3, 0.2125), and MFOWA3w (x) = 0.175. If we increase x1 by a small enough c > 0 we get y = ((1, 0.25 + c), (2, 0.25), (2, 0.5), (3, 1)) and then MFOWA1w (x) = (3, 0.4), MFOWA2w (x) = (3, 0.225), and MFOWA3w (x) = 0.175 − 0.2 · c. Thus we see that by increasing an input in class 1 we

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15

have decreased MFOWA3w , however, we have increased MFOWA2w , what violates the monotonicity of MFOWA operator. By including additional inputs and weights equal to zero we can extend this example to any number of inputs. Similar example can be found for any number of classes for m > 3.

4 5 6 7

10 11 12 13 14 15 16

− Rσ ) + c · v

Remark 4 shows that the problem is hidden in the difference xi sj which can be negative. Since by a proper selection of inputs we can have r k < R σ , to overcome this problem there are two possibilities: either B = 0 and then the difference is always non-negative, or avoid this situation, i.e., the operator will be monotone in the case that the situation that ki is not involved in the final summation cannot occur. First assume that B = 0. As B =

vsj − s W−1 j sj

in such a case vsj = which together with w1 ≤ w2 ≤ · · · ≤ wn gives v1 = · · · = vsj . By proper selection of inputs for MFOWA we can derive that for q = 1, m the MFOWA is monotone only if w1 = · · · = wn−1 = wn . In such a case all inputs are paired with the same weight, which means that w1 = · · · = wn = n1 and in such a case we retrieve just the multi-polar arithmetic mean. Next we should examine the case when ki is always contained in the computation of the final sum of the new input. This we can, however, ensure for an arbitrary combination of inputs only in the case when w1 = · · · = wn−1 = 0, wn = 1. In this case we obtain a monotone averaging operator, i.e., a multi-polar aggregation operator (see Section 5). W sj −1

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

39 40 41 42 43

Remark 5. The above results indicate the hypothesis that the multi-polar q-fusion Choquet integral is for q = 1, m monotone if and only if the capacity μ is submodular.

n

Proposition 7. Assume a weighting vector w = (w1 , . . . , wn ) with wi ≥ 0 and i=1 wi = 1. If a multi-polar q-balancing OWA operator with respect to w is monotone then weights are increasing, i.e., w1 ≤ w2 ≤ · · · ≤ wn . For q = 1 also the opposite holds, i.e., if w1 ≤ w2 ≤ · · · ≤ wn then a multi-polar q-balancing OWA operator with respect to a weighting vector w is monotone. Proof. The connection with the balancing Choquet integral and the bipolar case implies the necessity (see [11]). Sufficiency: suppose that w1 ≤ w2 ≤ · · · ≤ wn . First let us observe that similarly as for multi-polar FOWA, also multi-polar BOWA can be computed separately within value classes. Further, using the notation of the previous proposition assume i ∈ Cj . Then for a class k the output of MBOWA within the value class Cj is equal to (k, xi (wnj + · · · + wnj −r k +1 )). Similarly as in the proof for the MFOWA let us increase xi by a small amount. Then for a class ki the output of MBOWA within the two corresponding value classes will be xi (wnj + · · · + wnj −r k +1 ) + c · wnj , i.e., it will increase by c · wnj . The output in the class k = ki will decrease to xi (wnj −1 + · · · + wnj −r k ), i.e., the difference is xi (wnj − wnj −r k ) which is non-negative since wnj ≥ wnj −r k . Therefore the result for 1-summation L1 holds. 2 The counter example for q = 1 can be seen in the following example.

50

8 9 10 11 12 13 14 15 16

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

38 39 40 41 42 43

45

47 48

The proof of this proposition follows easily from the fact that for m = 2 the m-polar q-group OWA operator coincides with the m-polar q-balancing OWA operator.

49 50 51

51 52

7

46

Proposition 8. If for q ≤ m the m-polar q-group OWA operator with respect to w is monotone then w1 ≤ · · · ≤ wn .

48 49

6

44

We conclude this section by examining the monotonicity of the m-polar q-group OWA operator.

46 47

5

37

Example 2. Assume an input x = ((1, 0.25), (2, 0.25), (2, 0.5), (3, 1)) and let w = (0.1, 0.2, 0.3, 0.4) be a weighting vector. Then we obtain MBOWA1w (x) = (3, 0.4), MBOWA2w (x) = (3, 0.2), and MBOWA3w (x) = (3, 0.15). If we increase x1 by a small enough c > 0 we get y = ((1, 0.25 + c), (2, 0.25), (2, 0.5), (3, 1)) and for this input we see that MBOWA1w (y) = (3, 0.4), MBOWA2w (y) = (3, 0.225), and MBOWA3w (y) = (3, 0.175 − 0.2c). Thus we see that by increasing an input in class 1 we have increased MBOWA2w , as well as MBOWA3w , what violates the monotonicity of MBOWA operator.

44 45

3

17

37 38

2

4

· B(r k

8 9

1

Proposition 9. For q = 1 the m-polar q-group OWA operator is monotone if and only if w1 ≤ · · · ≤ wn .

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16

Proof. Necessity follows from the previous proposition. For the sufficiency assume w1 ≤ · · · ≤ wn and similarly as in the previous proofs, for an input x = ((k1 , x1 ), . . . , (kn , xn )) we will increase xi with i ∈ Cj by a small amount. For the original input, the weights in the value class Cj for all categories are given by

4

6

 vs =

rjk

u=1 vQ(S,u) S∈P (K m \{k}) r k  m−1  j Card(S)

6 7 8

for all  vs corresponding to an input from Cjk . For k = ki we have

9

10

10

13

rjk

u=1 vQ(S,u) S∈P (K m \{k}) r k  m−1  j Card(S)

14

m

11 12

15

=

rjk

u=1 vQ(S,u) S∈P (K m \{ki ,k}) r k  m−1  j Card(S)

+

rjk

u=1 vQ(S∪{ki },u)  m−1  rjk Card(S∪{k i })

m

11 12 13

.

14 15

Therefore the sum for class k within the value class Cj is

16

16

rjk

vQ(S,u) u=1  m−1  S∈P (K m \{ki ,k})

17 18



24

 vs =

rjk

u=1 vQ(S,u) S∈P (K m \{ki ,k}) r k  m−1  j Card(S)

17

.

18 19 20

+

21 22

rjk

23

u=1 vQ(S∪{ki },u−1)  m−1  rjk Card(S∪{k i })

24 25

m

26 27

u=1 vQ(S∪{ki },u)  m−1  Card(S∪{ki })

When we increase xi by a small c > 0 the new weights are for the class k = ki given by

23

25

rjk

m

20

22

+

Card(S)

k, xi

19

21

26 27

and therefore the sum for class k within the value class Cj is

28

28

rjk

vQ(S,u) u=1  m−1  S∈P (K m \{ki ,k})

29 30 31



34

+

Card(S)

k, xi

rjk

u=1 vQ(S∪{ki },u−1)  m−1  Card(S∪{ki })

m

32 33

29

.

30 31 32

To show that the sum in each category k = ki decreased we have to show that the difference between the original sum and the new sum is non-negative, i.e., that

35

rjk



37

x i · D = xi

S∈P (K m \{ki ,k})

45 46 47 48

m

u=1 vQ(S∪{ki },u−1)  m−1  Card(S∪{ki })

34

36 37

≥ 0.

D=

Card(S∪{ki })

m

and since vQ(S∪{ki },r k ) ≥ vQ(S∪{ki },0) for all k ∈ K m and all S ∈ P(K m \ {ki , k}) it is evident that D ≥ 0. Further we j would like to show that in class ki the new sum is bigger than the original sum and then the monotonicity for q = 1 will be ensured. In class ki the new weights are given by k

rj i −1



50 51

 vs =

u=1 vQ(S,u)  m−1  S∈P (K m \{ki }) ki (rj −1) Card(S)

m

38 39 40

vQ(S∪{k },r k ) −vQ(S∪{ki },0) i j  m−1  m S∈P (K \{ki ,k})

49

52

− xi

S∈P (K m \{ki ,k})



42

44



rjk

Easy computation gives us

41

43

u=1 vQ(S∪{ki },u)  m−1  Card(S∪{ki })

m

39 40

33

35

36

38

3

5

m

8 9

2

4

5

7

1

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for all s = i, s ∈ Cjki and  vi = vr 1 +···+r m = vsj . Therefore the new sum for class ki within the value class Cj and a j j newly created value class is

3 4 5 6 7 8



Card(S)

ki , x i

13

D = xi

Card(S)

m

16 17 18 19 20 21 22 23 24

27 28 29

32 33 34 35 36 37 38

and since vsj ≥ v

41 42

49

k

52

v

11 12

15

+ (xi + c)vsj

16

2

17 18 19 20



k Q(S,rj i )

S∈P (K m \{ki }) 

D = −xi

m−1  Card(S)

vQ(S∪ki ,0) S∈P (K m \{ki })  m−1 

+ (xi + c)vsj = xi

Card(S)

m and for class k = ki a decrease given by

Dk = x i

m

B =− −

29

vQ(S∪{ki ,k},0) −vQ(S∪{ki },0)  m−1  S∈P (K m \{ki ,k})

30

Card(S∪{ki })

Card(S)

k∈K m ,k=p,ki S∈P (K m \{ki ,k})

≥

S∈P (K m \{ki })

vQ(S∪{ki },0)  m−1  + Card(S)

32 33 34 35 36 37 38

S∈P (K m \{ki ,p})

40

vQ(S∪{ki ,p},0) − vQ(S∪{ki },0)  m−1 

41 42

Card(S∪{ki })

43

vQ(S∪{ki ,k},0) − vQ(S∪{ki },0)  m−1 

44 45

Card(S∪{ki })

46

B we see that A = xi m + c · vsj and thus if we show that B ≥ 0 we will get A ≥ c · vsj . Thus we want to proof that

vQ(S∪{ki ,p},0) − vQ(S∪{ki },0) m · vs j +  m−1 

31

39

vQ(S∪ki ,0)  m−1  + mvsj +

S∈P (K m \{ki ,p})

24

27



Card(S∪{ki })

S∈P (K m \{ki })

23

28

vQ(S∪{k },r k ) −vQ(S∪{ki },0) i j  m−1  S∈P (K m \{ki ,k})

22

26

+ (xi + c)vsj

k∈K m ,k=p,ki

If we denote

21

25

= xi . m m In the case that the class of the original output was ki then the result clearly holds. In the case that the original and new output classes are not equal, the only possibility is that the new output class is ki since that is the only class where the sum has increased. Therefore we have to investigate only the case when the original and the new output class is the same p = ki . In this case for q = m the difference in absolute values of outputs is given by

Dk . A = D + Dp −

50 51

≥ 0.

14

m for any S ∈ P(K m \ {ki }) we see that D ≥ c · vsj ≥ 0.



45

48

m

10

Proof. Necessity follows from the Proposition 8. For the sufficiency we will use results from the proof of the previous proposition, i.e., we know that if we increase xi by a small c > 0 then the change in sums for the two value classes, Cj and the newly created value class, will be for class ki an increase given by

44

47

+ (xi + c)vsj − xi

Card(S)

9

Proposition 10. For q = m the m-polar q-group OWA operator is monotone if and only if w1 ≤ · · · ≤ wn .

43

46

m−1  Card(S)

Q(S,rj i )

39 40

8

k

rj i

vQ(S,u) u=1  m−1  S∈P (K m \{ki })

k Q(S,rj i )

S∈P (K m \{ki }) 

D = −xi

30 31

7

13

v



25 26

6

We have

14 15

5

k

rj i −1

vQ(S,u) u=1  m−1  S∈P (K m \{ki })

12

2

4

+ (xi + c)vsj .

m We have to show that the difference between the new sum and the original sum is non-negative, i.e., that

10

1

3

k

rj i −1 vQ(S,u) u=1  m−1  S∈P (K m \{ki })



9

11

17

Card(S∪{ki })



k∈K m ,k=p,ki S∈P (K m \{ki ,k})

vQ(S∪{ki ,k},0) − vQ(S∪{ki },0) .  m−1  Card(S∪{ki })

47 48 49 50

(1)

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18

1 2

Since

1

3 4

S∈P (K m \{ki })

=

8

+

9

13 14

17 18 19 20

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

vQ(S∪{ki },0) vQ(S∪{ki ,p},0)  m−1  +  m−1  Card(S)



Card(S∪{ki })



m · vQ(K m ,0) +

≥ +

vQ(S∪{ki },0)  m−1  +

4

6

vQ(S∪{ki ,k},0) − vQ(S∪{ki },0)  m−1  Card(S∪{ki })



vQ(S∪{ki ,k,p},0) − vQ(S∪{ki ,p},0) + ,   m−1 Card(S∪{ki ,p})

vQ(S∪{ki ,p},0) vQ(S∪{ki },0)  m−1  +   m−1 Card(S∪{ki })

S∈P (K m \{ki ,p})





k∈K m ,k=p,ki

S∈P (K m \{ki ,k,p})

concluding the proof.

2

51 52

9 10

14

Card(S∪{ki ,p})

15

vQ(S∪{ki },0)  m−1 

16 17

Card(S)



18

vQ(S∪{ki ,k},0) vQ(S∪{ki ,k,p},0)  m−1  +   . m−1 Card(S∪{ki })

19 20

Card(S∪{ki ,p})

21

Card(S)

Card(S)

Card(S)+1

Card(S)+1

The counter example for q = 1, m can be seen in the following example. Example 3. Assume an input x = ((1, 0.25), (2, 0.25), (2, 0.5), (3, 1)) and let w = (0.1, 0.2, 0.3, 0.4) be a weighting vector. Since in one value class there are maximally two inputs the aggregation of this input vector will be the same as in the case of the multi-polar fusion OWA operator, i.e., MGOWA1w (x) = (3, 0.4), MGOWA2w (x) = (3, 0.2125), and MGOWA3w (x) = 0.175. If we increase x1 by a small enough c > 0 we get y = ((1, 0.25 + c), (2, 0.25), (2, 0.5), (3, 1)) and then MGOWA1w (y) = (3, 0.4), MGOWA2w (y) = (3, 0.225), and MGOWA3w (y) = 0.175 − c · 0.2. Thus we see that by increasing an input in class 1 we have decreased MGOWA1w and MGOWA3w , however, we have increased MGOWA2w , what violates the monotonicity of MGOWA operator.

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

5. Special cases of multi-polar OWA operators

49 50

8

13

47 48

7

12



We will now compute coefficients corresponding to weights v1, . . . , vsj that appear in our inequality. First observe, that in our inequality there appears, with a non-zero coefficient, only weights of the form vQ(S∪{ki },0) for S ∈ P(K m \ {ki }). Further we can divide the weights into two groups: ones of the form vQ(S∪{ki ,p},0) and ones of the form vQ(S∪{ki },0) for S ∈ P(K m \ {ki , p}), i.e., those that contain p in the set S and those that do not contain p. We will put weights of the form vQ(S∪{ki ,p},0) to the left hand-side of the inequality and weights of the form vQ(S∪{ki },0) to the right-hand side of the inequality. Then since vQ(S∪{ki ,p},0) ≥ vQ(S∪{ki },0) for all S ∈ P(K m \ {ki , p}), if we can show that the coefficient dS+ of the weight vQ(S∪{ki ,p},0) on the left-hand side is bigger or equal than the coefficient dS− of the weight vQ(S∪{ki },0) on the right-hand side for all S ∈ P(K m \ {ki , p}), we will have the result. First let us compute coefficient + + − 1 m−2 for vQ(K m ,0) : dK m \{p} = m − (m − 2) and thus dK m \{p} = 2. For vQ(K m \{p},0) : dK m \{p} = 1 + m−1 + m−1 = 2. + − Thus we have dK m \{p} = dK m \{p} . Now assume an arbitrary set S ∈ P(K m \ {ki , p}), S = K m \ {ki , p}. By similar



2 1 1  − computation as before we get dS+ = k∈S,k=ki ,p  m−1 k ∈S,k / =ki ,p  m−1  =  m−1  . On the other hand Card(S)+1 Card(S)+2 Card(S)+1



+ − 1 1  2 1  1  +  m−1 + k∈S,k=ki ,p  m−1 − k ∈S,k we get dS− =  m−1 / =ki ,p  m−1  =  m−1  . We get dS = dS thus Card(S)+1

3

5

Card(S∪p)

k∈K m ,k=p,ki S∈P (K m \{ki ,k,p})

S∈P (K m \{ki ,p}) Card(S∪{ki })

2

11



21 22

k∈K m ,k=p,ki S∈P (K m \{ki ,k})

vQ(S∪{ki ,k},0) − vQ(S∪{ki },0)  m−1 

the inequality (1) is transformed to

15 16



k∈K m ,k=p,ki S∈P (K m \{ki ,k,p})

10

12



S∈P (K m \{ki ,p})

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11

Card(S)

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vQ(S∪{ki },0)  m−1  +

48 49

Using the relations from the previous section for symmetric capacity, we can investigate what operators we will obtain when using other special capacities. We will investigate multi-polar Choquet integrals with respect to three special capacities: an additive capacity (which is in fact a class of capacities), the minimal and the maximal capacity.

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5.1. Multi-polar symmetric Choquet integral

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1 2

Let μ be additive. Since the standard OWA operator with respect to an additive capacity is just the weighted mean with respect to weights wi = μ({i}), the multi-polar symmetric Choquet integral with respect to an additive capacity is just the multi-polar weighted mean. As the only symmetric additive capacity is such that μ(A) = Card(A) n the multi-polar symmetric Choquet integral with respect to an additive symmetric capacity is just the multi-polar arithmetic mean, i.e., weights in this case are w1 = · · · = wn = n1 . For the strongest capacity μ∗ : 2X −→ [0, 1] given by 0 if A = ∅, μ∗ (A) = 1 else, the symmetric m-polar q-Choquet integral is given by     MSC μ∗ , x = Lq (1, y1 ), . . . , (m, ym ) , where yi = max(xi ). For m = 1, 2 the above equation reduces to known results for the unipolar scale where MSC(μ∗ , x) = max(x1 , . . . , xn ); and for the bipolar scale where MSC(μ∗ , x) = max(x1 , . . . , xn , 0) − max(−x1 , . . . , −xn , 0) (see [11]). Note that the strongest capacity μ∗ corresponds to weights w1 = · · · = wn−1 = 0, wn = 1. Similarly, for the weakest capacity μ∗ : 2X −→ [0, 1] given by 1 if A = X, μ∗ (A) = 0 else, the symmetric m-polar Choquet integral is given by   MSC(μ∗ , x) = L (1, y1 ), . . . , (m, ym ) , where yi = scale there is

min(xi ).

For m = 1, 2 we get MSC(μ∗ , x) = min(x1 , . . . , xn ) on the unipolar scale and on the bipolar 

min(x1 , . . . , xn ) if min(x1 , . . . , xn ) > 0, MSC(μ∗ , x) = max(x1 , . . . , xn ) if max(x1 , . . . , xn ) < 0, 0 otherwise. Here the weakest capacity μ∗ corresponds to weights w1 = 1, w2 = · · · = wn = 0. 5.2. Multi-polar fusion Choquet integral Let μ be additive. Then the multi-polar q-fusion Choquet integral is given by    xσ (i) μ({σ (i)}) MFCq (μ, x) = Lq CSσ ∈Γ CSni=1 kσ (i) , , S and thus MFCq (μ, x) = L

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

 q





   k1 , x1 μ {1} , . . . , kn , xn μ {n} ,

which means that MFCq (μ, x) is just a multi-polar weighted mean with respect to the weights wi = μ({i}). The m-polar fusion q-Choquet integral with respect to the strongest capacity μ∗ is given by   nm n1 q q MFC (μ, x) = L , . . . , m, c · , 1, c · n n

where c = max(x1 , . . . , xn ) and nj = Card{i | xi = c ∧ ki = j } for j ∈ K m and n = m i=1 ni . Note that for q > 1 this is nothing else than m-polar R ∗ aggregation of the inputs {(ki , xi ) | xi = c}, see [14,23]. As we have mentioned in the previous section, the m-polar q-fusion Choquet integral with respect to μ is not monotone when μ is not submodular. The weakest capacity μ∗ is not submodular and therefore the m-polar q-fusion Choquet integral with respect to the weakest capacity is not an m-polar aggregation operator. However, for completeness we introduce this case as well. The m-polar q-fusion Choquet integral with respect to μ∗ is given by   qm q1 q q MFC (μ, x) = L , . . . , m, p · , 1, p · q q

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where p = min(x1 , . . . , xn ) and qj = Card{i | xi = p ∧ ki = j } for j ∈ K m and q = m-polar R ∗ aggregation of the inputs {(ki , xi ) | xi = p}.

m

i=1 qi .

If q > 1 we get the

3 4

7 8

4 5

Let μ be additive. Then the multi-polar q-balancing Choquet integral is given by     p p

 1  m q q , |Ci |μ Ci , . . . , m, |Ci |μ Ci 1, MBC (μ, x) = L

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i=1

i=1

and therefore we get again       MBCq (μ, x) = Lq k1 , x1 μ {1} , . . . , kn , xn μ {n} , which means that MBCq (μ, x) is just a multi-polar weighted mean with respect to the weights wi = μ({i}). The m-polar q-balancing Choquet integral with respect to the strongest capacity μ∗ is equal to the oriented maximum. Similarly as in the case of the m-polar q-fusion Choquet integral also the m-polar q-balancing Choquet integral is not monotone when the capacity is not submodular. The m-polar q-balancing Choquet integral with respect to the weakest capacity μ∗ is equal to the oriented minimum given for x = ((k1 , x1 ), . . . , (kn , xn )) with ki ∈ K m , xi ∈ [0, 1] by     omin (k1 , x1 ), . . . , (kn , xn ) = clm(k1 , . . . , kn ), min(x1 , . . . , xn ) if {ki | xi = min(x1 , . . . , xn )} = {k} for some k ∈ K m (which means that all minimal inputs are from the same category) with clm(k1 , . . . , kn ) = ki if xi = min(x1 , . . . , xn ), in all other cases omin((k1 , x1 ), . . . , (kn , xn )) = 0. However, since μ∗ is not submodular, the oriented minimum is not non-decreasing and therefore it is not an m-polar aggregation operator. 5.4. Multi-polar group Choquet integral

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6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Similarly as in the case of the m-polar q-fusion Choquet integral we can show that the m-polar q-group Choquet integral with respect to an additive capacity is just a multi-polar weighted mean with respect to weights wi = μ({i}). The m-polar q-group Choquet integral with respect to the strongest capacity μ∗ and the weakest capacity μ∗ behaves similarly as the m-polar q-balancing Choquet integral, however, the result is divided by the number of categories containing the maximal input or the minimal input, respectively. This means that the m-polar q-group Choquet integral with respect to the strongest capacity μ∗ is just the oriented maximum divided by the number of categories containing the maximal input. However, if there is more than 1 category containing the maximal input then the oriented maximum is equal to 0 and thus for m-polar q-group Choquet integral we retrieve just the oriented maximum again. Similarly, the m-polar q-group Choquet integral with respect to the weakest capacity μ∗ is just the oriented minimum.

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6. Conclusions

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5.3. Multi-polar balancing Choquet integral

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38

We have discussed averaging operators for aggregation of fuzzy classification rules. We have shown that the rule weights can be translated into aggregation process thus solving a problem with interpretability of the rule weights. This also shows that the tuning of the weights can be obtained by fitting of the best multi-polar aggregation operator. We have examined multi-polar averages, weighted averages and oriented maximum and weighted maximum. We have introduced several multi-polar OWA operators and showed connection between multi-polar averaging operators and the multi-polar Choquet integral. Connection with the Sugeno integral was also mentioned. We have investigated monotonicity of the newly defined multi-polar OWA operators and showed that this can be achieved by selecting L1 (Lm ) as a multi-polar aggregation operator for the (final) interclass summation. We have shown that all mentioned multi-polar Choquet integrals with respect to an additive capacity yield the same weighted mean which shows that all multi-polar OWA operators with respect to the weighting vector containing equal weights coincide with the multi-polar arithmetic mean. The oriented maximum, the oriented minimum and R ∗ aggregation defined by Yager and Rybalov were shown to be multi-polar Choquet integrals with respect to the strongest/weakest capacity. We expect applications of our results in fuzzy rule-based classification systems and in other related domains of science where inputs from multiple multi-polar categories occur. For a fuzzy rule-based classifier that employ the MFOWA aggregation see [16].

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Acknowledgements

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This work was supported by AXA Research Fund and grants APVV-0073-10 and VEGA 2/0049/14.

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References

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[1] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953–54) 131–295. [2] O. Cordón, M.J. Del Jesus, F. Herrera, A proposal on reasoning methods in fuzzy rule-based classification systems, Int. J. Approx. Reason. 20 (1) (1999) 21–45. [3] D. Denneberg, Non-additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, 1994. [4] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets Syst. 69 (1995) 279–298. [5] M. Grabisch, The Choquet integral as a linear interpolator, in: Proc. of Tenth Internat. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004), 2004, pp. 373–378. [6] M. Grabisch, The symmetric Sugeno integral, Fuzzy Sets Syst. 139 (2003) 473–490. [7] M. Grabisch, The M˝obius function on symmetric ordered structures and its application to capacities on finite sets, Discrete Math. 287 (13) (2004) 17–34. [8] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge Univ. Press, Cambridge, U.K., 2009. [9] H. Ishibuchi, T. Morisawa, T. Nakashima, Voting schemes for fuzzy-rule-based classification systems, in: Proceedings of the Fifth IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’96), 1996, pp. 614–620. [10] R. Mesiar, B. De Baets, New construction methods for aggregation operators, in: Proc. IPMU’2000, Madrid, 2000, pp. 701–707. [11] R. Mesiar, A. Mesiarová-Zemánková, K. Ahmad, Discrete Choquet integral and some of its symmetric extensions, Fuzzy Sets Syst. 184 (2011) 148–155. [12] A. Mesiarová, J. Lazaro, Bipolar aggregation operators, in: T. Calvo, et al. (Eds.), Proc. AGOP’2003, Alcalá de Henares, Spain, 2003, pp. 119–123. [13] A. Mesiarová-Zemánková, K. Ahmad, Multi-polar aggregation, in: Proc. IPMU’2012, Catania, Italy, 2012, pp. 379–387. [14] A. Mesiarová-Zemánková, K. Ahmad, Multi-polar Choquet integral, Fuzzy Sets Syst. 220 (2013) 1–20. [15] A. Mesiarová-Zemánková, K. Ahmad, Extended multi-polarity and multi-polar-valued fuzzy sets, Fuzzy Sets Syst. 234 (2014) 61–78. [16] A. Mesiarová-Zemánková, Multi-polar aggregation operators in reasoning methods for fuzzy rule-based classification systems, IEEE Trans. Fuzzy Syst. (2014), http://dx.doi.org/10.1109/TFUZZ.2014.2298878. [17] E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995. [18] J. Šipoš, Integral with respect to a premeasure, Math. Slovaca 29 (1979) 141–145. [19] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1) (1988) 183–190. [20] R.R. Yager, D.P. Filev, Induced ordered weighted averaging operators, IEEE Trans. Syst. Man Cybern. 29 (2) (1999) 141–150. [21] R.R. Yager, J. Kacprzyk, The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer, Boston, MA, 1997. [22] R.R. Yager, Uninorms in fuzzy systems modeling, Fuzzy Sets Syst. 122 (2001) 167–175. [23] R.R. Yager, A. Rybalov, Bipolar aggregation using the Uninorms, Fuzzy Optim. Decis. Mak. 10 (1) (2011) 59–70.

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