Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates

doi:10.1016/S0020-0255(98)00010-3

Information Sciences

Volume 109, Issues 1–4, August 1998, Pages 95-118

https://doi.org/10.1016/S0020-0255(98)00010-3 Get rights and content

Abstract

This paper deals with problems which arise if for well justified statistical models like linear regression only fuzzy data are available. Three approaches are discussed: The first one is an application of Zadeh's extension principle to optimal classical estimators. Here, the result is that they, in general, do not keep their optimality properties. The second one is the attempt to develop a certain kind of linear theory for fuzzy random variables w.r.t. extended addition and scalar multiplication. The main problem in this connection is that fuzzy sets, with these extended operations, do not constitute a linear space. Therefore, in the third approach, the least squares approximation principle for fuzzy data is investigated, which leads to most acceptable results.

References (15)

KwakernaakH.
Fuzzy random variables — I. Definitions and theorems
Inform. Sci.
(1978)
PuriM.L. et al.
Fuzzy random variables
J. Math. Anal. Appl.
(1986)
TanakaH.
Fuzzy data analysis by possibility linear models
Fuzzy Sets and Systems
(1987)
DiamondP.
Fuzzy least squares
Inform. Sci.
(1988)
KruseR. et al.
Statistics with Vague Data
(1987)
SakawaM. et al.
Fuzzy linear regression and its applications
KacprzykJ. et al.
Fuzzy Regression Analysis
(1992)

There are more references available in the full text version of this article.

Cited by (81)

Interval-valued kriging for geostatistical mapping with imprecise inputs
2022, International Journal of Approximate Reasoning
Many geosciences data are imprecise due to various limitations and uncertainties in the measuring process. In other situations, collocated measurements of variables from the same, yet unknown, distribution are characterized with separate models that may not respect the relatedness of the measurements. One way to preserve this imprecision or relatedness in a geostatistical mapping framework is to characterize the measurements as intervals rather than single numbers. To effectively analyze the interval-valued data, this paper develops an interval-valued kriging based on the modification of a previous attempt and the recent development of the random sets theory. Numerical implementation of our interval-valued kriging is provided using a penalty-based constrained optimization algorithm. An interval-valued kriging of design ground snow loads in Ohio, USA, demonstrates the applicability and advantages of the proposed methodology.
The range of uncertainty on the property market pricing: The case of the city of Shanghai
2021, Finance Research Letters
Citation Excerpt :
Therefore, traditional regression methodologies have to be adjusted accordingly, and in the corresponding fuzzy literature, Fuzzy Linear Regression (FLR) models have been put forward (e.g., Wang and Tsaur, 2000; Wang et al., 2015). Tanaka et al. (1995) are among the first who developed them in details, and since then, significant research has been conducted (e.g., Savic and Pedrycz, 1991; Peters, 1994; Redden and Woodall, 1994; Kim et al., 1996; Ko and Na, 1998; Chen and Dang, 2008).2 Further, two kinds of fitting criteria commonly are used in developing FLR.
Property prices around the globe have seen very strong growth over the last two decades. Across various advanced countries, such a rapid and uncontrolled growth in house prices puts their economy in danger, and most importantly, their social integration and interconnectedness at great risk. In this paper, we develop a novel fuzzy linear regression framework using symmetric and asymmetric trapezoidal fuzzy numbers for determining the relationship of particular (non-) policy factors with the house prices. An interviewing questionnaire survey was conducted for collecting real data for the city of Shanghai to illustrate our theoretical treatment.
Fuzzy regression analysis: Systematic review and bibliography
2019, Applied Soft Computing Journal
Statistical regression analysis is a powerful and reliable method to determine the impact of one or several independent variable(s) on a dependent variable. It is the most widely used of all statistical methods and has broad applicability to numerous practical problems. However, various problems can arise, when for instance the sample size is too small, distributional assumptions are not fulfilled, the relationship between independent and dependent variables is vague or when there is an ambiguity of events. Moreover, the complexity of real-life problems often makes the underlying models inadequate, since information is frequently imprecise in many ways. To relax these rigidities, numerous researchers have modified and extended concepts of statistical regression analysis by means of concepts of fuzzy set theory. By now, there is a large number of papers on the topic of fuzzy regression analysis, especially concerning possibilistic, fuzzy least squares or machine learning approaches. Additionally, the variety of approaches includes probabilistic, logistic, type-2 and clusterwise fuzzy regression methods, among many others. Besides papers mainly devoted to advances in methodology, there are also several papers presenting case studies in various research fields. To structure this diversity of papers, proposals and applications we give in this paper a comprehensive systematic review and provide a bibliography on the topic of fuzzy regression analysis. Thus, the paper intends to consolidate the topic in order to aid new researchers in this area, focuses the field’s attention on key open questions, and highlights possible directions for future research.
Asymptotic tests for interval-valued means
2017, Statistics and Probability Letters
The fuzzy characterizing function of the distribution of a random fuzzy number
2015, Applied Mathematical Modelling
Citation Excerpt :
the definition and study of (extended) summary measures for the analysis of fuzzy data and random fuzzy sets, like central tendency measures (see, for instance, [16,1,17,18], etc.), dispersion measures (see, for instance, [19,20], etc.); the least squares regression/correlation analysis with fuzzy data through both descriptive and inferential developments, especially in connection with linear models based on the usual fuzzy arithmetic (see, for instance, [21–30], etc.); the estimation of population parameters/summary measures of the distribution of random fuzzy sets (see, for instance, [31–33,20,34,17,18], etc.);
Characterizing the distribution of random elements is valuable for different purposes. Among them, inferential conclusions about the population distribution can be drawn on the basis of the sample one. When one deals with real-valued random variables this characterization is usually made through the distribution function or other ones, like the moment-generating or the characteristic functions. In case of dealing with random elements taking on fuzzy number values, the distribution function cannot be adequately defined in terms of a total ordering since there is no universally acceptable one for fuzzy numbers. This paper introduces a characterization of the distribution of these random elements by extending the moment-generating function. Properties of this extension are examined, and the notion is illustrated by means of some examples.
Rejoinder on "a distance-based statistical analysis of fuzzy number-valued data"
2014, International Journal of Approximate Reasoning

View all citing articles on Scopus

View full text

Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates

Abstract

Inform. Sci.

J. Math. Anal. Appl.

Fuzzy Sets and Systems

Inform. Sci.

Statistics with Vague Data

Fuzzy linear regression and its applications

Fuzzy Regression Analysis