Gil González-Rodríguez

Fuzzy data treated as functional data: A one-way ANOVA test approach

The use of the fuzzy scale of measurement to describe an important number of observations from real-life attributes or variables is first explored. In contrast to other well-known scales (like nominal or ordinal), a wide class of statistical measures and techniques can be properly applied to analyze fuzzy data. This fact is connected with the possibility of identifying the scale with a special subset of a functional Hilbert space. The identification can be used to develop methods for the statistical analysis of fuzzy data by considering techniques in functional data analysis and vice versa. In this respect, an approach to the FANOVA test is presented and analyzed, and it is later particularized to deal with fuzzy data. The proposed approaches are illustrated by means of a real-life case study.

Looking for a good fuzzy system interpretability index: An experimental approach

Interpretability is acknowledged as the main advantage of fuzzy systems and it should be given a main role in fuzzy modeling. Classical systems are viewed as black boxes because mathematical formulas set the mapping between inputs and outputs. On the contrary, fuzzy systems (if they are built regarding some constraints) can be seen as gray boxes in the sense that every element of the whole system can be checked and understood by a human being. Interpretability is essential for those applications with high human interaction, for instance decision support systems in fields like medicine, economics, etc. Since interpretability is not guaranteed by definition, a huge effort has been done to find out the basic constraints to be superimposed during the fuzzy modeling process. People talk a lot about interpretability but the real meaning is not clear. Understanding of fuzzy systems is a subjective task which strongly depends on the background (experience, preferences, and knowledge) of the person who makes the assessment. As a consequence, although there have been a few attempts to define interpretability indices, there is still not a universal index widely accepted. As part of this work, with the aim of evaluating the most used indices, an experimental analysis (in the form of a web poll) was carried out yielding some useful clues to keep in mind regarding interpretability assessment. Results extracted from the poll show the inherent subjectivity of the measure because we collected a huge diversity of answers completely different at first glance. However, it was possible to find out some interesting user profiles after comparing carefully all the answers. It can be concluded that defining a numerical index is not enough to get a widely accepted index. Moreover, it is necessary to define a fuzzy index easily adaptable to the context of each problem as well as to the user quality criteria.

A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view.

Pyrogenic organic matter production from wildfires: a missing sink in the global carbon cycle.

Wildfires release substantial quantities of carbon (C) into the atmosphere but they also convert part of the burnt biomass into pyrogenic organic matter (PyOM). This is richer in C and, overall, more resistant to environmental degradation than the original biomass, and, therefore, PyOM production is an efficient mechanism for C sequestration. The magnitude of this C sink, however, remains poorly quantified, and current production estimates, which suggest that ∽1-5% of the C affected by fire is converted to PyOM, are based on incomplete inventories. Here, we quantify, for the first time, the complete range of PyOM components found in-situ immediately after a typical boreal forest fire. We utilized an experimental high-intensity crown fire in a jack pine forest (Pinus banksiana) and carried out a detailed pre- and postfire inventory and quantification of all fuel components, and the PyOM (i.e., all visually charred, blackened materials) produced in each of them. Our results show that, overall, 27.6% of the C affected by fire was retained in PyOM (4.8 ± 0.8 t C ha−1), rather than emitted to the atmosphere (12.6 ± 4.5 t C ha−1). The conversion rates varied substantially between fuel components. For down wood and bark, over half of the C affected was converted to PyOM, whereas for forest floor it was only one quarter, and less than a tenth for needles. If the overall conversion rate found here were applicable to boreal wildfire in general, it would translate into a PyOM production of ∽100 Tg C yr−1 by wildfire in the global boreal regions, more than five times the amount estimated previously. Our findings suggest that PyOM production from boreal wildfires, and potentially also from other fire-prone ecosystems, may have been underestimated and that its quantitative importance as a C sink warrants its inclusion in the global C budget estimates.

Bootstrap approach to the multi-sample test of means with imprecise data

A bootstrap approach to the multi-sample test of means for imprecisely valued sample data is introduced. For this purpose imprecise data are modelled in terms of fuzzy values. Populations are identified with fuzzy-valued random elements, often referred to in the literature as fuzzy random variables. An example illustrates the use of the suggested method. Finally, the adequacy of the bootstrap approach to test the multi-sample hypothesis of means is discussed through a simulation comparative study.

Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data

In previous studies we have stated that the well-known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables, when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one-dimensional Euclidean space. In this paper we show that the one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). This extension is immediately developed by combining some tools in the literature, namely, bootstrap techniques on Banach spaces, a metric between fuzzy sets based on the support function, and an embedding of the space of fuzzy random variables into a Banach space which is based on the support function.

Estimation of a simple linear regression model for fuzzy random variables

A generalized simple linear regression statistical/probabilistic model in which both input and output data can be fuzzy subsets of R^p is dealt with. The regression model is based on a fuzzy-arithmetic approach and it considers the possibility of fuzzy-valued random errors. Specifically, the least-squares estimation problem in terms of a versatile metric is addressed. The solutions are established in terms of the moments of the involved random elements by employing the concept of support function of a fuzzy set. Some considerations concerning the applicability of the model are made.

Estimation of a flexible simple linear model for interval data based on set arithmetic

The estimation of a simple linear regression model when both the independent and dependent variable are interval valued is addressed. The regression model is defined by using the interval arithmetic, it considers the possibility of interval-valued disturbances, and it is less restrictive than existing models. After the theoretical formalization, the least-squares (LS) estimation of the linear model with respect to a suitable distance in the space of intervals is developed. The LS approach leads to a constrained minimization problem that is solved analytically. The strong consistency of the obtained estimators is proven. The estimation procedure is reinforced by a real-life application and some simulation studies.

Least squares estimation of linear regression models for convex compact random sets

Simple and multiple linear regression models are considered between variables whose “values” are convex compact random sets in