Interval type-2 fuzzy least-squares estimation to formulate a regression model based on a new outlier detection method using a new distance

Abstract

In modeling, planning, and decision-making under uncertainty, most data available contain some degree of uncertainty. In this context, interval type-2 fuzzy (IT2F) regression models through combining the statistical methods and IT2F techniques can provide a wide range of tools for analyzing fuzziness, vagueness, and randomness in data and measurements. Therefore, the main objective of this study is to propose a new fuzzy least-squares deviation estimator based on the extended Euclidean distance to formulate regression models in the full IT2F environment. For this purpose, we first introduce a new fuzzy distance metric based on extended Euclidean distance and express the properties of the proposed distance with proof, and then propose some new formulas for estimating the parameters of the IT2F regression model. In addition, we propose a new formula to define the mild and extreme outlier cutoffs and then introduce a new algorithm for detecting outliers in fuzzy linear regression based on the distance between the estimated and observed IT2 fuzzy responses. We also introduce a suitable validation approach to investigate the goodness of fit of the proposed regression models and then propose a new algorithm that presents the process of performing this method. Furthermore, to evaluate the performance of the proposed methodology, we introduce some new criteria based on the fuzzy distance and similarity measures on the space of trapezoidal IT2F numbers. Finally, we present two numerical examples. In the first example, we explain the theoretical results of the proposed method and show that how the proposed procedure is applicable to obtain a suitable IT2 fuzzy regression model. In the second example, we compare the performance of the proposed model with some well-known models and show that our proposed model is more efficient in terms of distance and similarity measures.