Preaggregation Functions: Construction and an Application Lucca, Giancarlo; Sanz, Jose Antonio; Dimuro, Gracaliz Pereira ...
IEEE transactions on fuzzy systems,
2016-April, 2016-4-00, 20160401, Volume:
24, Issue:
2
Journal Article
Peer reviewed
Open access
In this paper, we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only ...monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build preaggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators such as the maximum and the Choquet integral.
Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary ...conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties.
Overlap functions are aggregation operators specially introduced to be used in applications involving the overlap problem and/or when the associativity property is not strongly required for the ...aggregation operator, as in classification problems and decision making based on fuzzy preference relations. This paper considers the existent results on residual implication induced by fuzzy conjunctions to introduce the concept of residual implication derived from overlap functions O, denoted by RO-implication, preserving the residuation property. RO-implications are weaker than R-implications constructed from positive and continuous t-norms, in the sense that RO-implications do not necessarily satisfy certain properties satisfied by such R-implications, as the exchange principle, but only weaker versions of these properties. However, in general, such properties are not demanded for many applications. The objectives of this paper are: (a) to analyse the main properties satisfied by RO-implications, establishing under which conditions of an overlap function O the derived RO-implication satisfies the properties of fuzzy implications and (b) to provide two particular characterization of RO-implications derived from (i) the sub-class of overlap functions O that have 1 as neutral element and (ii) the more general sub-class of overlap functions O satisfying the condition O(x,1)⩽x.
Overlap functions and grouping functions are special kinds of aggregation operators that have been recently proposed for applications in classification problems, like, e.g., imaging processing. ...Overlap and grouping functions can also be applied in decision making based on fuzzy preference relations, where the associativity property is not strongly required and the use of t-norms or t-conorms as the combination/separation operators is not necessary. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow the application in several different contexts. This paper introduces new interesting results related to overlap and grouping functions, investigating important properties, such as migrativity, homogeneity, idempotency and the existence of generators. De Morgan triples are introduced in order to study the relationship between those dual concepts. In particular, we introduce important results related to the action of automorphisms on overlap and grouping functions, analyzing the preservation of those properties and also the Lipschitzianity condition.
•We make a revision of recent generalizations of the Choquet integral that appear in the literature.•We show some of the most relevant theoretical features of these extensions.•We also discuss some ...applications where these extensions have provided good results.
In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely CT-integrals (by t-norm T), CF-integrals (by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), CF1F2-integrals (by a pair of fusion functions (F1, F2) under some specific constraints) and their generalization gCF1F2-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.
Overlap functions are aggregation functions that express the overlapping degree between two values. They have been used both as a conjunction in several practical problems (e.g., image processing and ...decision making), and to generate overlap indices between two fuzzy sets, which can be used to construct fuzzy confidence values to be applied in fuzzy rule based classification systems. Some generalizations of overlap functions were recently proposed, such as n-dimensional and general overlap functions, which allowed their application in n-dimensional problems. More recently, the concept of interval-valued overlap functions was presented, mainly to deal with uncertainty in providing membership functions. In this paper, we introduce: (i) the concept of n-dimensional interval-valued overlap functions, studying their representability, (ii) the definition of general interval-valued overlap functions, providing their characterization and some construction methods. Moreover, we also define the concept of interval-valued overlap index, as well as some constructing methods. In addition, we show an illustrative example where those new concepts are applied.
Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem ...and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.
Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. ...The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications RO-implications, the strong implications (G,N)-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N. Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x,I(x,y))≤y, for any fuzzy implication I and overlap function O, and, in particular, for RO-implications, (G,N)-implications, QL-implications and D-implications derived from tuples (O,G,N), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T-conditionality and U-conditionality, for t-norms T and uninorms U, respectively.
Traditionally, Convolutional Neural Networks make use of the maximum or arithmetic mean in order to reduce the features extracted by convolutional layers in a downsampling process known as pooling. ...However, there is no strong argument to settle upon one of the two functions and, in practice, this selection turns to be problem dependent. Further, both of these options ignore possible dependencies among the data. We believe that a combination of both of these functions, as well as of additional ones which may retain different information, can benefit the feature extraction process. In this work, we replace traditional pooling by several alternative functions. In particular, we consider linear combinations of order statistics and generalizations of the Sugeno integral, extending the latter’s domain to the whole real line and setting the theoretical base for their application. We present an alternative pooling layer based on this strategy which we name “CombPool” layer. We replace the pooling layers of three different architectures of increasing complexity by CombPool layers, and empirically prove over multiple datasets that linear combinations outperform traditional pooling functions in most cases. Further, combinations with either the Sugeno integral or one of its generalizations usually yield the best results, proving a strong candidate to apply in most architectures.
Choquet integral is a widely used aggregation operator on one-dimensional and interval-valued information, since it is able to take into account the possible interaction among data. However, there ...are many cases where the information taken into account is vectorial, such as Long Short-Term Memories (LSTM). LSTM units are a kind of Recurrent Neural Networks that have become one of the most powerful tools to deal with sequential information since they have the power of controlling the information flow. In this paper, we first generalize the standard Choquet integral to admit an input composed by <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>-dimensional vectors, which produces an <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>-dimensional vector output. We study several properties and construction methods of vector Choquet integrals. Then, we use this integral in the place of the summation operator, introducing in this way the new VCI-LSTM architecture. Finally, we use the proposed VCI-LSTM to deal with two problems: sequential image classification and text classification.
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