New classes of the moderate deviation functions

At present, in the field of aggregation of various input values, attention is focused on the construction of aggregation functions using other functions that can affect the resulting aggregated value. This resulting value should characterize the properties of the individual input values as accurately as possible. Attention is also paid to aggregation using the so-called moderate deviation function. Using this function in aggregation ensures that all properties of aggregation functions are preserved. This work offers constructions of the moderate deviation functions using negations and automorphisms on the symmetric interval [−1, 1] and a general closed interval [a, b] ⊂ [−∞, ∞].The work of Jana Špirková has been supported by the Slovak Scientific Grant Agency VEGA no. 1/0150/21. The work of Humberto Bustince, Javier Fernandez and Mikel Sesma-Sara has been supported by grand PID2019-108392GB-I00 (AEI/10.13039/501100011033)

Local properties of strengthened ordered directional and other forms of monotonicity

In this study we discuss some of the recent generalized forms of monotonicity, introduced in the attempt of relaxing the monotonicity condition of aggregation functions. Specifically, we deal with weak, directional, ordered directional and strengthened ordered directional monotonicity. We present some of the most relevant properties of the functions that satisfy each of these monotonicity conditions and, using the concept of pointwise directional monotonicity, we carry out a local study of the discussed relaxations of monotonicity. This local study enables to highlight the differences between each notion of monotonicity. We illustrate such differences with an example of a restricted equivalence function.This work is supported by the project TIN2016-77356-P (AEI/FEDER, UE), by the Public University of Navarra under the project PJUPNA13 and by Slovak grant APVV-14-0013

Generalized forms of monotonicity in the data aggregation framework

El proceso de agregación trata el problema de combinar una colección de valores numéricos en un único valor que los represente y las funciones encargadas de esta operación se denominan funciones de agregación. A las funciones de agregación se les exige que cumplan dos condiciones de contorno y, además, han de ser monótonas con respecto a todos sus argumentos. Una de las tendencias en el área de investigación de las funciones de agregación es la relajación de la condición de monotonía. En este respecto, se han introducido varias formas de monotonía relajada. Tal es el caso de la monotonía débil, la monotonía direccional y la monotonía respecto a un cono. Sin embargo, todas estas relajaciones de monotonía están basadas en la idea de crecer, o decrecer, a lo largo de un rayo definido por un vector real. No existe noción de monotonía que permita que la dirección de crecimiento dependa de los valores a fusionar, ni tampoco existe noción de monotonía que considere el crecimiento a lo largo de caminos más generales, como son las curvas. Además, otra de las tendencias en la teoría de la agregación es la extensión a escalas más generales que la de los números reales y no existe relajación de monotonía disponible para este contexto general. En esta tesis, proponemos una colección de nuevas formas de monotonía relajada para las cuales las direcciones de monotonía pueden variar dependiendo del punto del dominio. En concreto, introducimos los conceptos de monotonía direccional ordenada, monotonía direccional ordenada reforzada y monotonía direccional punto a punto. Basándonos en funciones que cumplan las propiedades de monotonía direccional ordenada, proponemos un algoritmo de detección de bordes que justifica la aplicabilidad de estos conceptos. Por otro lado, generalizamos el concepto de monotonía direccional tomando, en lugar de direcciones en Rn, caminos más generales: definimos el concepto de monotonía basado en curvas. Por último, combinando ambas tendencias en la teoría de la agregación, generalizamos el concepto de monotonía direccional a funciones definidas en escalas más generales que la de los números reales.The process of aggregation addresses the problem of combining a collection of numerical values into a single representative number and the functions that perform this operation are called aggregation functions. Aggregation functions are required to satisfy two boundary conditions and to be monotone with respect to all their arguments. One of the trends in the research area of aggregation functions is the relaxation of the condition of monotonicity. In that attempt, various relaxed forms of monotonicity have been introduced. This is the case of weak, directional and cone monotonicity. However, all these relaxed forms of monotonicity are based on the idea of increasing, or decreasing, along a fixed ray defined by a real vector. There is no notion of monotonicity allowing the direction of increasingness to depend on the specific values to aggregate, nor there exists any other notion that considers increasingness along more general paths, such as curves. Additionally, another trend in the theory of aggregation is the extension to handle more general scales than real numbers and there is no relaxation of monotonicity available in that general context. In this dissertation, we propose a collection of new relaxed forms of monotonicity for which the directions of monotonicity may vary from one point of the domain to another. Specifically, we introduce the concepts of ordered directional monotonicity, strengthened ordered directional monotonicity and pointwise directional monotonicity. Based on the concept of ordered directionally monotone functions, we propose an edge detection algorithm that justifies the applicability of these concepts. Furthermore, we generalize the concept of directional monotonicity so that, instead of directions in Rn, more general paths are considered: we define curve-based monotonicity. Finally, combining both trends in the theory of aggregation, we generalize the concept of directional monotonicity to functions that are defined on more general scales than real numbers.Programa de Doctorado en Ciencias y Tecnologías Industriales (RD 99/2011)Industria Zientzietako eta Teknologietako Doktoretza Programa (ED 99/2011