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
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
General grouping functions
(Springer, 2020) Santos, Helida; Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Some aggregation functions that are not necessarily associative, namely overlap and grouping functions, have called the attention of many researchers in the recent past. This is probably due to the fact that they are a richer class of operators whenever one compares with other classes of aggregation functions, such as t-norms and t-conorms, respectively. In the present work we introduce a more general proposal for disjunctive n-ary aggregation functions entitled general grouping functions, in order to be used in problems that admit n dimensional inputs in a more flexible manner, allowing their application in different contexts. We present some new interesting results, like the characterization of that operator and also provide different construction methods.
Open Access
Enhancing the efficiency of the interval-valued fuzzy rule-based classifier with tuning and rule selection
(Springer, 2020) Sanz Delgado, José Antonio; Da Cruz Asmus, Tiago; Osa Hernández, Borja de la; Bustince Sola, Humberto; Institute of Smart Cities - ISC; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA1926
Interval-Valued fuzzy rule-based classifier with TUning and Rule Selection, IVTURS, is a state-of-the-art fuzzy classifier. One of the key point of this method is the usage of interval-valued restricted equivalence functions because their parametrization allows one to tune them to each problem, which leads to obtaining accurate results. However, they require the application of the exponentiation several times to obtain a result, which is a time demanding operation implying an extra charge to the computational burden of the method. In this contribution, we propose to reduce the number of exponentiation operations executed by the system, so that the efficiency of the method is enhanced with no alteration of the obtained results. Moreover, the new approach also allows for a reduction on the search space of the evolutionary method carried out in IVTURS. Consequently, we also propose four different approaches to take advantage of this reduction on the search space to study if it can imply an enhancement of the accuracy of the classifier. The experimental results prove: 1) the enhancement of the efficiency of IVTURS and 2) the accuracy of IVTURS is competitive versus that of the approaches using the reduced search space.
Open Access
Improving Michigan-style fuzzy-rule base classification generation using a Choquet-like Copula-based aggregation function
(CEUR Workshop Proceedings (CEUR-WS.org), 2021) Hinojosa-Cardenas, Edward; Sarmiento-Calisaya, Edgar; Camargo, Heloisa A.; Sanz Delgado, José Antonio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA1926
This paper presents a modification of a Michigan-style fuzzy rule based classifier by applying the Choquet-like Copula-based aggregation function, which is based on the minimum t-norm and satisfies all the conditions required for an aggregation function. The proposed new version of the algorithm aims at improving the accuracy in comparison to the original algorithm and involves two main modifications: replacing the fuzzy reasoning method of the winning rule by the one based on Choquet-like Copula-based aggregation function and changing the calculus of the fitness of each fuzzy rule. The modification proposed, as well as the original algorithm, uses a (1+1) evolutionary strategy for learning the fuzzy rulebase and it shows promising results in terms of accuracy, compared to the original algorithm, over ten classification datasets with different sizes and different numbers of variables and clases.
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
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.
Open Access
Interval-valued aggregation functions based on moderate deviations applied to motor-imagery-based brain computer interface
(IEEE, 2021) Fumanal Idocin, Javier; Takáč, Zdenko; Fernández Fernández, Francisco Javier; Sanz Delgado, José Antonio; Goyena Baroja, Harkaitz; Lin, Chin-Teng; Wang, Yu-Kai; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this work we develop moderate deviation functions to measure similarity and dissimilarity among a set of given interval-valued data to construct interval-valued aggregation functions, and we apply these functions in two MotorImagery Brain Computer Interface (MI-BCI) systems to classify electroencephalography signals. To do so, we introduce the notion of interval-valued moderate deviation function and, in particular, we study those interval-valued moderate deviation functions which preserve the width of the input intervals. In order to apply them in a MI-BCI system, we first use fuzzy implication operators to measure the uncertainty linked to the output of each classifier in the ensemble of the system, and then we perform the decision making phase using the new interval-valued aggregation functions. We have tested the goodness of our proposal in two MI-BCI frameworks, obtaining better results than those obtained using other numerical aggregation and interval-valued OWA operators, and obtaining competitive results versus some non aggregation-based frameworks.
Open Access
Motor-imagery-based brain-computer interface using signal derivation and aggregation functions
(IEEE, 2021) Fumanal Idocin, Javier; Wang, Yu-Kai; Lin, Chin-Teng; Fernández Fernández, Francisco Javier; Sanz Delgado, José Antonio; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Brain Computer Interface (BCI) technologies are popular methods of communication between the human brain and external devices. One of the most popular approaches to BCI is Motor Imagery (MI). In BCI applications, the ElectroEncephaloGraphy (EEG) is a very popular measurement for brain dynamics because of its non-invasive nature. Although there is a high interest in the BCI topic, the performance of existing systems is still far from ideal, due to the difficulty of performing pattern recognition tasks in EEG signals. This difficulty lies in the selection of the correct EEG channels, the signal-tonoise ratio of these signals and how to discern the redundant information among them. BCI systems are composed of a wide range of components that perform signal pre-processing, feature extraction and decision making. In this paper, we define a new BCI Framework, named Enhanced Fusion Framework, where we propose three different ideas to improve the existing MI-based BCI frameworks. Firstly, we include an additional pre-processing step of the signal: a differentiation of the EEG signal that makes it time-invariant. Secondly, we add an additional frequency band as feature for the system: the Sensory Motor Rhythm band, and we show its effect on the performance of the system. Finally, we make a profound study of how to make the final decision in the system. We propose the usage of both up to six types of different classifiers and a wide range of aggregation functions (including classical aggregations, Choquet and Sugeno integrals and their extensions and overlap functions) to fuse the information given by the considered classifiers. We have tested this new system on a dataset of 20 volunteers performing motor imagery-based braincomputer interface experiments. On this dataset, the new system achieved a 88.80% of accuracy. We also propose an optimized version of our system that is able to obtain up to 90, 76%. Furthermore, we find that the pair Choquet/Sugeno integrals and overlap functions are the ones providing the best results.
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.