Open Access
Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions
(Elsevier, 2021) Da Cruz Asmus, Tiago; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Sanz Delgado, José Antonio; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. They have been successfully applied as a fuzzy conjunction operation in several problems in which associativity is not required, such as image processing and classification. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees, as in interval-valued fuzzy rule-based classification systems. In this context, the choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of (a,b)-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval and a new notion of increasingness with respect to a pair of partial orders (≤1,≤2); (iv) present and compare three construction methods for these width-limited interval-valued overlap functions, considering a pair of orders (≤1,≤2), which may be admissible or not, showcasing the adaptability of our developments.
Open Access
On the normalization of interval data
(MDPI, 2020) Santiago, Regivan; Bergamaschi, Flaulles; Bustince Sola, Humberto; Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The impreciseness of numeric input data can be expressed by intervals. On the other hand, the normalization of numeric data is a usual process in many applications. How do we match the normalization with impreciseness on numeric data? A straightforward answer is that it is enough to apply a correct interval arithmetic, since the normalized exact value will be enclosed in the resulting 'normalized' interval. This paper shows that this approach is not enough since the resulting 'normalized' interval can be even wider than the input intervals. So, we propose a pair of axioms that must be satisfied by an interval arithmetic in order to be applied in the normalization of intervals. We show how some known interval arithmetics behave with respect to these axioms. The paper ends with a discussion about the current paradigm of interval computations.
Open Access
IVTURS: A linguistic fuzzy rule-based classification system based on a new interval-valued fuzzy reasoning method with tuning and rule selection
(IEEE, 2013) Sanz Delgado, José Antonio; Fernández, Alberto; Bustince Sola, Humberto; Herrera, Francisco; Automática y Computación; Automatika eta Konputazioa
Interval-valued fuzzy sets have been shown to be a useful tool for dealing with the ignorance related to the definition of the linguistic labels. Specifically, they have been successfully applied to solve classification problems, performing simple modifications on the fuzzy reasoning method to work with this representation and making the classification based on a single number. In this paper we present IVTURS, a new linguistic fuzzy rule-based classification method based on a new completely interval-valued fuzzy reasoning method. This inference process uses interval-valued restricted equivalence functions to increase the relevance of the rules in which the equivalence of the interval membership degrees of the patterns and the ideal membership degrees is greater, which is a desirable behaviour. Furthermore, their parametrized construction allows the computation of the optimal function for each variable to be performed, which could involve a potential improvement in the system’s behaviour. Additionally, we combine this tuning of the equivalence with rule selection in order to decrease the complexity of the system. In this paper we name our method IVTURS-FARC, since we use the FARC-HD method to accomplish the fuzzy rule learning process. The experimental study is developed in three steps in order to ascertain the quality of our new proposal. First, we determine both the essential role that interval-valued fuzzy sets play in the method and the need for the rule selection process. Next, we show the improvements achieved by IVTURS-FARC with respect to the tuning of the degree of ignorance when it is applied in both an isolated way and when combined with the tuning of the equivalence. Finally, the significance of IVTURS-FARC is further depicted by means of a comparison by which it is proved to outperform the results of FARC-HD and FURIA, which are two high performing fuzzy classification algorithms.
Open Access
Pre-aggregation functions: construction and an application
(IEEE, 2015) Lucca, Giancarlo; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Mesiar, Radko; Kolesárová, Anna; Bustince Sola, Humberto; Automática y Computación; Automatika eta Konputazioa
In this work 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 pre-aggregation 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 like the maximum and the Choquet integral.
Open Access
A first study on the use of interval-valued fuzzy sets with genetic tuning for classification with imbalanced data sets
(Springer, 2009) Sanz Delgado, José Antonio; Fernández, Alberto; Bustince Sola, Humberto; Herrera, Francisco; Automática y Computación; Automatika eta Konputazioa
Classification with imbalanced data-sets is one of the recent challenging problems in Data Mining. In this framework, the class dis- tribution is not uniform and the separability between the classes is often difficult. From the available techniques in the Machine Learning field, we focus on the use of Fuzzy Rule Based Classification Systems, as they provide an interpretable model for the end user by means of linguistic variables. The aim of this work is to increase the performance of fuzzy modeling by adding a higher degree of knowledge by means of the use of Interval- valued Fuzzy Sets. Furthermore, we will contextualize the Interval-valued Fuzzy Sets with a post-processing genetic tuning of the amplitude of their upper bounds in order to enhance the global behaviour of this methodology.
Open Access
Evolution in time of L-fuzzy context sequences
(Elsevier, 2016) Alcalde, Cristina; Burusco Juandeaburre, Ana; Bustince Sola, Humberto; Jurío Munárriz, Aránzazu; Sanz Delgado, José Antonio; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this work, we consider a complete lattice L and we study L-fuzzy context sequences which represent the evolution in time of an L-fuzzy context. To carry out this study, in the first part of the paper, we consider n-ary OWA operators in complete lattices, which enable us to make a general analysis and a temporal analysis at any moment in time of L-fuzzy context sequences. After that, evolution in time of the relationship between the objects and the attributes is considered. In particular, we analyze the concepts of Trend and Persistent formal contexts. Finally, we illustrate our results with an example where we consider the particular lattice L = J ([0, 1]).
Open Access
Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning
(Elsevier, 2010) Sanz Delgado, José Antonio; Fernández, Alberto; Bustince Sola, Humberto; Herrera, Francisco; Automática y Computación; Automatika eta Konputazioa
Among the computational intelligence techniques employed to solve classification problems, Fuzzy Rule-Based Classification Systems (FRBCSs) are a popular tool because of their interpretable models based on linguistic variables, which are easier to understand for the experts or end-users. The aim of this paper is to enhance the performance of FRBCSs by extending the Knowledge Base with the application of the concept of Interval-Valued Fuzzy Sets (IVFSs). We consider a post-processing genetic tuning step that adjusts the amplitude of the upper bound of the IVFS to contextualize the fuzzy partitions and to obtain a most accurate solution to the problem. We analyze the goodness of this approach using two basic and well-known fuzzy rule learning algorithms, the Chi et al.’s method and the fuzzy hybrid genetics-based machine learning algorithm. We show the improvement achieved by this model through an extensive empirical study with a large collection of data-sets.
Open Access
N-dimensional admissibly ordered interval-valued overlap functions and its influence in interval-valued fuzzy rule-based classification systems
(IEEE, 2021) Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Overlap functions are a type of aggregation functions that are not required to be associative, generally used to indicate the overlapping degree between two values. They have been successfully used as a conjunction operator in several practical problems, such as fuzzy rulebased classification systems (FRBCSs) and image processing. Some extensions of overlap functions were recently proposed, such as general overlap functions and, in the interval-valued context, n-dimensional interval-valued overlap functions. The latter allow them to be applied in n-dimensional problems with interval-valued inputs, like interval-valued classification problems, where one can apply interval-valued FRBCSs (IV-FRBCSs). In this case, the choice of an appropriate total order for intervals, like an admissible order, can play an important role. However, neither the relationship between the interval order and the n-dimensional interval-valued overlap function (which may or may not be increasing for that order) nor the impact of this relationship in the classification process have been studied in the literature. Moreover, there is not a clear preferred n-dimensional interval-valued overlap function to be applied in an IV-FRBCS. Hence, in this paper we: (i) present some new results on admissible orders, which allow us to introduce the concept of n-dimensional admissibly ordered interval-valued overlap functions, that is, n-dimensional interval-valued overlap functions that are increasing with respect to an admissible order; (ii) develop a width-preserving construction method for this kind of function, derived from an admissible order and an n-dimensional overlap function, discussing some of its features; (iii) analyze the behaviour of several combinations of admissible orders and n-dimensional (admissibly ordered) interval-valued overlap functions when applied in IV-FRBCSs. All in all, the contribution of this paper resides in pointing out the effect of admissible orders and n-dimensional admissibly ordered interval-valued overlap functions, both from a theoretical and applied points of view, the latter when considering classification problems.
Open Access
CFM-BD: a distributed rule induction algorithm for building compact fuzzy models in Big Data classification problems
(IEEE, 2020) Elkano Ilintxeta, Mikel; Sanz Delgado, José Antonio; Barrenechea Tartas, Edurne; Bustince Sola, Humberto; Galar Idoate, Mikel; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Interpretability has always been a major concern for fuzzy rule-based classifiers. The usage of human-readable models allows them to explain the reasoning behind their predictions and decisions. However, when it comes to Big Data classification problems, fuzzy rule based classifiers have not been able to maintain the good tradeoff between accuracy and interpretability that has characterized these techniques in non-Big-Data environments. The most accurate methods build models composed of a large number of rules and fuzzy sets that are too complex, while those approaches focusing on interpretability do not provide state-of-the-art discrimination capabilities. In this paper, we propose a new distributed learning algorithm named CFM-BD to construct accurate and compact fuzzy rule-based classification systems for Big Data. This method has been specifically designed from scratch for Big Data problems and does not adapt or extend any existing algorithm. The proposed learning process consists of three stages: Preprocessing based on the probability integral transform theorem; rule induction inspired by CHI-BD and Apriori algorithms; and rule selection by means of a global evolutionary optimization. We conducted a complete empirical study to test the performance of our approach in terms of accuracy, complexity, and runtime. The results obtained were compared and contrasted with four state-of-the-art fuzzy classifiers for Big Data (FBDT, FMDT, Chi-Spark-RS, and CHI-BD). According to this study, CFM-BD is able to provide competitive discrimination capabilities using significantly simpler models composed of a few rules of less than three antecedents, employing five linguistic labels for all variables.
Open Access
A fuzzy association rule-based classifier for imbalanced classification problems
(Elsevier, 2021) Sanz Delgado, José Antonio; Sesma Sara, Mikel; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Imbalanced classification problems are attracting the attention of the research community because they are prevalent in real-world problems and they impose extra difficulties for learning methods. Fuzzy rule-based classification systems have been applied to cope with these problems, mostly together with sampling techniques. In this paper, we define a new fuzzy association rule-based classifier, named FARCI, to tackle directly imbalanced classification problems. Our new proposal belongs to the algorithm modification category, since it is constructed on the basis of the state-of-the-art fuzzy classifier FARC–HD. Specifically, we modify its three learning stages, aiming at boosting the number of fuzzy rules of the minority class as well as simplifying them and, for the sake of handling unequal fuzzy rule lengths, we also change the matching degree computation, which is a key step of the inference process and it is also involved in the learning process. In the experimental study, we analyze the effectiveness of each one of the new components in terms of performance, F-score, and rule base size. Moreover, we also show the superiority of the new method when compared versus FARC–HD alongside sampling techniques, another algorithm modification approach, two cost-sensitive methods and an ensemble.