Pre-aggregation Functions Research Articles

Probabilistic study of Induced Ordered Linear Fusion Operators for time series forecasting

The aggregation of several predictors in time series forecasting has been used intensely in the last decade in order to construct a better resulting model. Some of the most used alternatives are the ones related to the Induced Ordered Weighted Averaging (IOWA), in which the prediction values are ordered using a secondary vector, often related to the accuracy of the prediction model in the last prediction. Although the time series study has been historically a subject related to statistics and stochastic processes, the random behaviour of the aggregation process is typically not considered. In addition, extensions of aggregation functions with a weaker notion of monotonicity, pre-aggregation functions, are appearing as better alternative for some topics such us classification. In this paper, a pre-aggregation extension of the IOWA operator, the Induced Ordered Linear Fusion (IOLF), is defined as a way to aggregate time series model predictions and its behaviour is studied from a probabilistic point of view. The IOLF operator over random vectors is defined, its properties studied and the relation between some averaging aggregation functions established. The expressions of the optimal weights according to statistical criteria are derived. The advantages and consequences of the use of the IOLF operator are studied, and its behaviour is compared to the usual procedures. Numerical results illustrate its performance on a practical example.

Systematic Review of Aggregation Functions Applied to Image Edge Detection

Edge detection is a crucial process in numerous stages of computer vision. This field of study has recently gained momentum due to its importance in various applications. The uncertainty, among other characteristics of images, makes it difficult to accurately determine the edge of objects. Furthermore, even the definition of an edge is vague as an edge can be considered as the maximum boundary between two regions with different properties. Given the advancement of research in image discontinuity detection, especially using aggregation and pre-aggregation functions, and the lack of systematic literature reviews on this topic, this paper aims to gather and synthesize the current state of the art of this topic. To achieve this, this paper presents a systematic review of the literature, which selected 24 papers filtered from 428 articles found in computer databases in the last seven years. It was possible to synthesize important related information, which was grouped into three approaches: (i) based on both multiple descriptor extraction and data aggregation, (ii) based on both the aggregation of distance functions and fuzzy C-means, and (iii) based on fuzzy theory, namely type-2 fuzzy and neutrosophic sets. As a conclusion, this review provides interesting gaps that can be explored in future work.

From pre-(quasi-)grouping functions to directional monotonic fuzzy implications

Introducing directional monotonicity into aggregation functions is an important research content in the research of aggregation functions. This paper aims to study the properties, constructions and applications of pre-(quasi-)grouping functions. Firstly, the concept of pre-(quasi-)grouping functions is proposed by relaxing the monotonicity of (quasi-)grouping functions to directional monotonicity, and some basic properties are discussed. Subsequently, the relationships between pre-(quasi-)grouping functions and some other pre-aggregation functions are exhibited. Moreover, several construction methods of pre-(quasi-)grouping functions are presented. Finally, based on pre-(quasi-)grouping functions G, fuzzy negations N and overlap functions O, the constructions of (G,N)-directional monotonic fuzzy implications and QL-directional monotonic operations are given, and some important properties and characterizations are investigated.

Cardinality-limiting extended pre-aggregation functions

Aggregation functions, which are at the heart of a number of information fusion processes, allow summarization of multiple inputs into a single representative value. Extended aggregation functions are defined such that the input data can be of varying cardinality, with the implication that there is some consistency across the methods of calculation. This article formalizes an approach to extended aggregation such that contributions of repeated inputs or regions of high density are limited in their ability to influence the final value. We establish important definitions and properties, in particular around whether such functions will be monotone or directionally monotone. We then propose a powerful construction method for extended pre-aggregation functions. Illustrative examples are provided throughout.

Directional Shift-Stable Functions

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [0,1]n, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of n=2 of directionally shift-stable aggregation and pre-aggregation functions are also given.

On some classes of directionally monotone functions

In this work we consider some classes of functions with relaxed monotonicity conditions generalizing some other given classes of fusion functions. In particular, directionally increasing aggregation functions (called also pre-aggregation functions), directionally increasing conjunctors, or directionally increasing implications, etc., generalize the standard classes of aggregation functions, conjunctors, or implication functions, respectively. We analyze different properties of these classes of functions and we discuss a construction method in terms of linear combinations of t-norms.

Generalized [formula omitted]-integrals: From Choquet-like aggregation to ordered directionally monotone functions

This paper introduces the theoretical framework for a generalization of CF1F2-integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by gCF1F2-integrals, is based on the so-called pseudo pre-aggregation function pairs (F1,F2), which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the gCF1F2-integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of gCF1F2-integrals. We study several properties of gCF1F2-integrals, considering different constraints for the functions F1 and F2, and state under which conditions gCF1F2-integrals present or not averaging behaviors. Several examples of gCF1F2-integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation functions.

Cognitive Integrals With Its Generalized and Adapted Forms

Motivated by some cognition styles, this study first proposes a new type of preaggregation functions called cognitive integrals with its generalized forms. Rather than being affected only by selected capacity, generalized cognitive integrals consider other two parameters, the cognitive strength and the involved semicopula. Our main focus is on cognitive strength, by which the integrals values will be monotonically decreasing with respect to our cognitive strength. After some appropriated adaptations, we later propose the concept of adapted cognitive integrals with two equivalent forms of itself. Not only being a type of the preaggregation functions, we prove that adapted cognitive integrals are indeed one new type of aggregation functions, still equipped with the two new parameters like in cognitive integrals.

CF-integrals: A new family of pre-aggregation functions with application to fuzzy rule-based classification systems

This paper introduces the family of CF-integrals, which are pre-aggregations functions that generalizes the Choquet integral considering a bivariate function F that is left 0-absorbent. We show that CF-integrals are 1→-pre-aggregation functions, studying in which conditions they are idempotent and/or averaging functions. This characterization is an important issue of our approach, since we apply these functions in the Fuzzy Reasoning Method (FRM) of a fuzzy rule-based classification system and, in the literature, it is possible to observe that non-averaging aggregation functions usually provide better results. We carry out a study with several subfamilies of CF-integrals having averaging or non-averaging characteristics. As expected, the proposed non-averaging CF-integrals obtain more accurate results than the averaging ones, thus, offering new possibilities for aggregating accurately the information in the FRM. Furthermore, it allows us to enhance the results of classical FRMs like the winning rule and the additive combination.

Preaggregation Functions: Construction and an Application

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.