Multi-objective fuzzy regression applied to the calibration of conceptual rainfall-runoff models

Abstract

The purpose of this research is (1) to develop a multi-objective fuzzy regression (MOFR) tool to overcome the shortcomings of the existing fuzzy regression approaches while keeping the good characteristics, and (2) to study systems with uncertain elements, using the example of rainfall-runoff process to illustrate the approach. Previous research has shown that fuzzy regression performs superior compared to statistical regression in some cases. On the other hand, fuzzy regression has also been criticized because it does not allow all data points to influence the estimated parameters, it is sensitive to data outliers, and the prediction intervals become wider as more data are collected. Here, several MOFR techniques are developed to overcome these problems by enabling the decision maker select a non-dominated solution based on the tradeoff between data outliers and prediction vagueness. It is shown that MOFR provides superior results to existing fuzzy regression techniques, and the existing fuzzy regression approaches and classical least squares regression are specific cases of the MOFR framework. The methodology is illustrated with examples from rainfall-runoff modeling, more specifically, conceptual rainfall-runoff (CRR) models are analyzed here. One of the main problems in CRR modeling is dealing with the uncertainty associated with the model parameters which is related to data and/or model structure. A fuzzy CRR (FCRR) framework is proposed herein where every element of the CRR is assumed to be uncertain, taken here as fuzzy. Parameter calibration of FCRR models using newly developed fuzzy regression techniques is also investigated. Applications are provided for a linear CRR model, the experimental two-parameter (TWOPAR) and the six-parameter (SIXPAR) models. The major findings can be summarized as follows: (1) FCRR enables the decision maker to gain insight about the CRR model sensitivity to uncertainty of the model elements, (2) using MOFR for the calibration of FCRR leads to non-convex, constrained, non-linear optimization problems, (3) fuzzy least squares regression model yields to more stable parameter estimates than the non-fuzzy regression model, (4) the methodology is applicable to any dynamic system with discrete modes.