Aggregation theory is devoted to the fusing of several values into a unique output that summarizes the given information. Typically, the aggregation process is formalized in terms of an increasing mathematical function that maps the input values to the result, fulfilling some boundary conditions. However, this formalization can be too restrictive for some scenarios. In some cases, the inputs can be seen as observations of random variables, the aggregation result being also a random variable. In others, the aggregation process can be identified as a program that performs the aggregation rather than a mathematical function. In this direction, the concepts of aggregation of random variables and computable aggregation have been defined in the literature. This paper is devoted to the definition of computable aggregation of random variables, which are computer programs, not functions, that aggregate random variables, not numbers. Special attention is given to different possible alternatives to modelize random variables and monotonicity. The implementation of some examples is also provided.

The concept of hierarchical structure is central when considering complex systems. On one hand, many complex systems exhibit a hierarchical structure, on the other hand, the idea of defining a hierarchical structure to cope with the complexity of the system is widely present in literature related to many different fields. Hierarchies are also present in the field of aggregation with the definition of hierarchical aggregation processes. Broadly speaking, a hierarchical system is a system formed by several components (subsystems) structured at different levels, implying a sort of ranking or ordering relation among them. Positioning in the levels could be related to many different aspects or properties of the components: priority, abstraction, granularity, specificity, precision, etc.Computable aggregations have recently been introduced as a natural approach to classical aggregation functions, in which the emphasis is placed on the program (the implementation) that makes possible the aggregation instead of simply considering the function or the algorithm that permits the aggregation process. Considered as a program that implements an aggregation process, a computable aggregation is also suitable for being interpreted in terms of a hierarchical process. In this paper, the idea of hierarchical computable aggregation is considered, exploring those situations where the aggregation process involves some intrinsic structure that can be interpreted in hierarchical terms (like priorities and veto), as well as those other situations where the hierarchical approach is mostly related to computational considerations (like recursion and parallelization). These and other types of hierarchical computable aggregation will be presented and analyzed.

The idea of computable aggregation operators was introduced as a generalization of aggregation operators, allowing the replacement of the mathematical function usually considered for aggregation, by a program that performs the aggregation process. There are different reasons to justify this extension. One of them is the interest in exploring some computational properties not directly related to the aggregation itself but to its implementation (complexity, recursivity, parallelisation, etc). Another reason, the one driving to the present paper, is the need to define a framework where the quite common process of first sampling (over a large data set) and then aggregating the sample, could be analysed as a formal aggregation process. This process does not match with the idea of an aggregation function, due to its non-deterministic nature, but could easily be adapted to that of a (non-deterministic) computable aggregation. The idea of non-deterministic aggregation requires the extension of the concept of monotonicity (a key aspect of aggregation operators) to this new framework. The present paper will explore this kind of non-deterministic aggregation processes, first from an empirical point of view and then in terms of populations, adapting the idea of monotonicity to both of them and finally defining a common framework for its analysis.

Computable aggregation operators can be seen as a generalization of aggregation operators where the mathematical function is replaced by a program that performs the aggregation process. This extension allows the introduction of new aggregation processes not feasible under the classical framework. Particularly interesting are some non-deterministic processes widely considered to merge information. However, especially in non-deterministic processes, the extension of some of the well-known concepts for aggregation operators such as monotony, is needed. In this work, a new concept of monotonicity is proposed, from a probabilistic perspective, for non-deterministic computable aggregation operators. To be consistent, the concept coincides with the classical definition in the deterministic case. In addition, some cases of interest are analysed.

The concept of pre-aggregation function defined in [0,1]nhas been recently extended to that of generalized pre-aggregation function in the framework of a totally ordered set T with maximum and minimum value. To do so, the concept of monotonicity is transformed in that of conditioned monotonicity based on the chains in Tn, generalizing the idea of directional monotonicity. In the present paper we explore the concept of conditioned monotonicity considering some specific conditioning structures (covers, partitions and projections). On this basis we consider some situations where conditioned monotonicity ensures monotonicity. Finally we use these definitions and properties to define some pre-aggregation and aggregation functions that are applied to image preprocessing problems.

In this paper the relation between aggregation functions, algorithms and computer programs is revisited, extending the concept of recursive aggregation operator by means of the recursive computable aggregation. In particular, two different recursive computable aggregation are distinguished: on the one hand, the hard recursive computable aggregation, which appears when there is a unique recursive function generating the aggregation, and are fully related to associativity; and on the other hand, the soft recursive computable aggregation, which appears when the number of elements to be aggregated is needed. Some illustrative examples are provided.

Complexity is a typical criteria to analyze interpretability of fuzzy systems: number of variables, terms, rules, etc. Hierarchical fuzzy systems have shown a great potential to reduce fuzzy systems complexity. On the other hand, the counterpart to complexity reduction is the presence of synthetic variables generated at intermediate levels of the hierarchy. Those synthetic variables are generally affected by an absolute absence of semantics, minimizing their interpretability. As a consequence, when analyzing interpretability in hierarchical fuzzy systems, complexity should only be a part of the equation, since semantics should also be considered. And particularly, semantics of the intermediate variables added to reduce complexity. The use of hierarchical fuzzy systems will only produce an effective interpretability improvement when the design of the hierarchical structure was driven by the semantics of the intermediate variables. In other words, intermediate variables should be interpretable in terms of the problem under analysis. This consideration, made in the framework of hierarchical fuzzy systems, could be extended to any kind of hierarchical system defined under the umbrella of explainable artificial intelligence. The present paper will consider different aspects of interpretability concerning hierarchical fuzzy systems, and will explore the role of intermediate variables in accordance with the previous comments. The appropriate selection of those intermediate variables will drive to a process where subsystems will be decoupled for design and interpretation. Under this assumption we will finally consider the measurement of interpretability in hierarchical fuzzy systems as a process where the interpretability of each component (fuzzy system) of the hierarchy is first evaluated and then aggregated to achieve an overall interpretability evaluation.

Indoor localization is increasingly required for applications like deployment of rescue teams in emergency situations, proactive care for the elders, and so on. The quick growing of coverage of WiFi networks makes WiFi technology a very promising choice for indoor localization. But, this localization should be linked to a map to be useful. This work presents an open-access software designed for that purpose. It is composed of two different applications, a desktop software for research purposes and an Android application for user friendly localization. We address the localization task as a high dimensional classification problem. So far, we have developed classifiers based on the classic Nearest Neighbour, Support Vector Machines (SVM) and fuzzy rule-based classifiers. This work is made in the context of the ABSYNTHE project which is aimed at creating human-robot teams. We show a use case of the new software in one of the scenarios of the ABSYNTHE project.

This work presents the full integration of fuzzy inference-grams (Fingrams) in KEEL to visual analysis of fuzzy association rules. Fingrams graphically represent fuzzy rule-based systems (FRBSs) in 2D graphs that illustrate the interaction among fuzzy rules in terms of rule cofiring, i.e., paying attention to rule pairs simultaneous fired by a given input. The new module allows to generate Fingrams for fuzzy association rules created in the suite KEEL, that can be afterwards analyzed to comprehend the system behavior and improve it. We sketch the use and potentials in an illustrative example built in KEEL over a real-world dataset including qualitative assessments of a set of design chairs.

Since Zadeh’s proposal and Mamdani’s seminal ideas, interpretability is acknowledged as one of the most appreciated and valuable characteristics of fuzzy system identification methodologies. It represents the ability of fuzzy systems to formalize the behavior of a real system in a human understandable way, by means of a set of linguistic variables and rules with a high semantic expressivity close to natural language. Interpretability analysis involves two main points of view: readability of the knowledge base description (regarding complexity of fuzzy partitions and rules) and comprehensibility of the fuzzy system (regarding implicit and explicit semantics embedded in fuzzy partitions and rules, as well as the fuzzy reasoning method). Readability has been thoroughly treated by many authors who have proposed several criteria and metrics. Unfortunately, comprehensibility has usually been neglected because it involves some cognitive aspects related to human reasoning, which are very hard to formalize and to deal with. This paper proposes the creation of a new paradigm for fuzzy system comprehensibility analysis based on fuzzy systems’ inference maps, so-called fuzzy inference-grams (fingrams), by analogy with scientograms used for visualizing the structure of science. Fingrams show graphically the interaction between rules at the inference level in terms of co-fired rules, i.e., rules fired at the same time by a given input. The analysis of fingrams offers many possibilities: measuring the comprehensibility of fuzzy systems, detecting redundancies and/or inconsistencies among fuzzy rules, identifying the most significant rules, etc. Some of these capabilities are explored in this study for the case of fuzzy models and classifiers.

The so-called fuzzy inference-grams (fingrams) constitute a novel and powerful tool for expert analysis of fuzzy rule bases at inference level in terms of co-fired rules, i.e., rules fired at the same time by a given input vector. A fuzzy rule base can be seen as a population made up of a set of individual rules which are competing and collaborating among them with the aim of yielding both good generality-specificity and interpretability-accuracy trade-offs. Thus, system behavior turns up from the existing relations among rules which can be easily analyzed by looking at the rule base as a fingram, i.e., as a social network made of nodes representing fuzzy rules. This paper introduces a new software module for fingram generation and analysis which is provided with the last version of GUAJE, a free software tool devoted to the generation of understandable and accurate fuzzy systems. Notice that, fingram analysis consists of studying the interaction among nodes in the network for the purpose of understanding the structure and behavior of the fuzzy rule base under consideration. It is based on the basic principles of social network analysis which have been properly incorporated into GUAJE and adapted to the design of fuzzy systems.

Assessing interpretability of fuzzy systems still remains an open and challenging problem. Defining a good index is extremely difficult mainly due to the inherent subjective nature of interpretability. It strongly depends on the background of the person who makes the assessment according to its own knowledge, but also taking into account its previous experience and preferences. Since looking for fuzzy systems with a good accuracy-interpretability trade-off is required for many applications, guiding the whole design process by a good quality index would be extremely appreciated. Such index must be aware of both accuracy and interpretability. This paper introduces a framework that makes possible defining an index easily adaptable to the context of each problem by means of incorporating user's preferences and quality criteria. To do so, all aspects related to interpretability are first identified and then combined into a decision hierarchy framework. It is derived from a previous experimental study based on a web poll. The top of the hierarchy represents the quality index while the bottom includes all fuzzy systems to be evaluated. It consists of k decision levels structured as suggested by the classical analytic hierarchy process (AHP) defined by Saaty. In addition, the aggregation process is made using the ordered weighted averaging (OWA) operators defined by Yager. Such AHP+OWA combination was already proposed by Yager for solving multi-criteria decision problems. A simple example shows how the proposed method becomes effective but also efficient when assessing several fuzzy systems in an automatic process. The index is easily adaptable for providing those rankings expected by different users.

The goal of this paper is to study some of the most important WiFi signal variations, large and small scale variations and how they affect to WiFi localization systems. Moreover, the paper shows how to use soft computing techniques to deal with these uncertainties in WiFi localization systems. This work describes how to reduce uncertainty produced by small scale variations in indoor environments using fuzzy techniques. Some experimental results and conclusions are presented.

We have designed and built a new open hardware/software board that lets miniaturized robots communicate and at the same time obtain the range and bearing of the source of emission. The open E-puck Range & Bearing board improves an existing infrared relative localization/communication software library (libIrcom) developed for the e-puck robot and based on its on-board infrared sensors. The board allows the robots to have an embodied, decentralized and scalable communication system. Its use and capabilities are demonstrated via an alignment experiment.

This work presents a new process for building comprehensible fuzzy systems for classification problems. Firstly, a feature selection procedure based on crisp decision trees is carried out. Secondly, strong fuzzy partitions are generated for all the selected inputs. Thirdly, a set of linguistic rules are defined combining the previously generated linguistic variables. Then, a linguistic simplification procedure guided by a novel interpretability index is applied to get a more compact and general set of rules without losing accuracy. Finally, an efficient and simple local search strategy increases the system accuracy while preserving the high interpretability. Results obtained in several benchmark classification problems are encouraging because they show the ability of the new methodology for generating highly interpretable fuzzy rule-based classifiers while yielding accuracy comparable to that achieved by other methods like neural networks and C4.5.

This work describes a new methodology for making easier the design process of interpretable knowledge bases. It considers both expert knowledge and knowledge extracted from data. The combination of both kinds of knowledge is likely to yield robust compact systems with a good trade-off between accuracy and interpretability. Fuzzy logic offers an integration framework where both types of knowledge are represented using the same formalism. However, as two knowledge bases may convey contradictions and/or redundancies, the integration process must be made carefully. Results obtained, in four well-known benchmark classification problems, show that our methodology leads to highly interpretable knowledge bases with a good accuracy, comparable to that achieved by other methodologies. © 2008 Wiley Periodicals, Inc.

This work shows how to achieve a good interpretability-accuracy trade-off through keeping the strong fuzzy partition property along the whole fuzzy modeling process. First, a small compact knowledge base is built. It is highly interpretable and reasonably accurate. Second, an optimization procedure, which only affects the fuzzy partitions defining the system variables, is carried out. It improves the system accuracy while preserving the system interpretability. Two optimization strategies are compared: Solis-Wetts, a local search based strategy; and Genetic Tuning, a global search based strategy. Results obtained in a well-known benchmark medical classification problem, related to breast cancer diagnosis, show that our methodology is able to achieve knowledge bases with high interpretability and accuracy comparable to that obtained by other methodologies.

This paper presents a user-friendly portable tool designed and developed in order to make easier knowledge extraction and representation for fuzzy logic based systems. KBCT is an open source software that could be executed under Linux or Windows operating systems. Main goal of KBCT is the generation or refinement of fuzzy knowledge bases with a particular interest of obtaining interpretable partitions and rules. The use of fuzzy logic simplifies the knowledge extraction process and increase interpretability of rules because of the fuzzy rule expression is closed to expert natural language. KBCT lets the user define expert variables and rules, but also provide induction capabilities for partitions and rules. Both types of knowledge, expert and induced, are integrated under the expert control. In addition to this, the user can check consistency and quality of rule base at any moment. A simplify option is implemented in order to allow the user to reduce the size of rule base. The main objective consists of ensuring interpretability, non-redundancy and consistency of the knowledge base along the whole process.

This work presents an intelligent transportation system (ITS) that was implemented on an autonomous vehicle designed to perform global navigation missions on a network of unmarked roads. This is the first step toward the complete implementation of ITS in urban environments, which is the long-term goal of this work. Using a global positioning system, global navigation is achieved by means of a global planner and a task manager that recurrently coordinate the execution of vision-based perception tasks for the road tracking of nonstructured roads and the navigation of intersections. In addition, a vision-based vehicle-detection task has been developed, which endows the global navigation system with a reactive capacity. The complete system has been tested on the BABIECA prototype vehicle, which was autonomously driven for hundreds of kilometers around a private circuit, designed to emulate an urban quarter, at speeds of up to 50 km/h, successfully carrying out different navigation missions. During the tests, the vehicle drove itself across crossroads and performed the appropriate turning maneuvers at intersections. It also demonstrated its robustness with regard to shadows, road texture, weather conditions, and changing illumination.

This article analyzes the role of context in hierarchical fuzzy controllers based on the decomposition of the input space. The usual consideration in most hierarchical fuzzy systems is the reduction of dimensionality problems. This article will analyze how to profit from the qualities of context as a key question in the definition of a fuzzy controller, to reduce the design efforts by making it easier to introduce the expert knowledge in that process. The idea is to use the output of a level of the hierarchy as the method to define (or adjust) the normalization functions (considered as contextual information) applied to the variables of the following level of that hierarchy. Two different situations will be analyzed, including an application example for each case. In the first case the decomposition will affect variables placed at the same level of description (abstraction) regarding the problem to be solved. In the second case, the decomposition process works on variables placed at different levels of description of the problem (descriptions with a different level of abstraction). © 2002 Wiley Periodicals, Inc.

In recent years the use of genetic or evolutionary techniques has produced interesting results in the automatic generation of knowledge bases for fuzzy logic controllers. Three different representations of the rule base have been considered: lists of rules, relational matrices, and decision tables. The use of lists of rules reduces the size of the rule base, but presents some handicaps in crossover since it usually requires some kind of list ordering before applying the operator. A new crossover operator, working with lists (sets) of rules, is designed in such a way that maintaining the advantage of working with a reduced set of rules incorporates the characteristic of easy crossover by using the virtual structure of decision table. © 1998 John Wiley & Sons, Inc.