Michal Černý and Milan Hladík. Possibilistic linear regression with fuzzy data: Tolerance approach with prior information. Fuzzy Sets Syst., 340:127–144, 2018.
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We introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data (both input and output). The method is very general: the only assumption is that $\alpha$-cuts of the fuzzy data are efficiently computable. We take into account possible prior restrictions of the parameters space: we assume that the restrictions are given by linear and quadratic constraints. The method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy-valued model to an interval-valued model on a given $\alpha$-cut, which is further reduced to a linear-time method (i.e., running in $O(np)$) computing with endpoints of the intervals. The speed of computation makes the method applicable for huge datasets. Unlike various approaches based on mathematical programming formulations, the tolerance-based construction preserves central tendency of the resulting regression coefficients. In addition, we prove further properties: if inputs are crisp and outputs are fuzzy, then the construction preserves piecewise linearity and convex shape of fuzzy numbers. We derive an $O(n^2 p)$-algorithm for enumeration of breakpoints of the membership function of the estimated coefficients. (Here, $n$ is the number of observations and $p$ is the number of regression parameters). Similar results are also derived for the fuzzy input-and-output model. We illustrate the theory for the case of triangular and asymmetric Gaussian fuzzy inputs and outputs of an inflation-consumption model.
@article{CerHla2018a,
author = "Michal {\v{C}}ern\'{y} and Milan Hlad\'{\i}k",
title = "Possibilistic linear regression with fuzzy data: {Tolerance} approach with prior information",
journal = "Fuzzy Sets Syst.",
fjournal = "Fuzzy Sets and Systems",
volume = "340",
pages = "127-144",
year = "2018",
doi = "10.1016/j.fss.2017.10.007",
issn = "0165-0114",
url = "https://www.sciencedirect.com/science/article/pii/S0165011417303718",
bib2html_dl_html = "http://dx.doi.org/10.1016/j.fss.2017.10.007",
abstract = "We introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data (both input and output). The method is very general: the only assumption is that $\alpha$-cuts of the fuzzy data are efficiently computable. We take into account possible prior restrictions of the parameters space: we assume that the restrictions are given by linear and quadratic constraints. The method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy-valued model to an interval-valued model on a given $\alpha$-cut, which is further reduced to a linear-time method (i.e., running in $O(np)$) computing with endpoints of the intervals. The speed of computation makes the method applicable for huge datasets. Unlike various approaches based on mathematical programming formulations, the tolerance-based construction preserves central tendency of the resulting regression coefficients. In addition, we prove further properties: if inputs are crisp and outputs are fuzzy, then the construction preserves piecewise linearity and convex shape of fuzzy numbers. We derive an $O(n^2 p)$-algorithm for enumeration of breakpoints of the membership function of the estimated coefficients. (Here, $n$ is the number of observations and $p$ is the number of regression parameters). Similar results are also derived for the fuzzy input-and-output model. We illustrate the theory for the case of triangular and asymmetric Gaussian fuzzy inputs and outputs of an inflation-consumption model.",
keywords = "Possibilistic regression; Fuzzy regression; Linear regression; Constrained regression; Tolerance quotient",
}
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