A Hybrid Fuzzy Regression-Based Methodology for Normal Distribution (Case Study: Cumulative Annual Precipitation)

Abstract

An advantage of the probabilistic approach is the exploitation of the observed probability values in order to test the goodness-of-fit for the examined theoretical probability distribution function (pdf). Since in fact, the interest of the engineers is to determine the hydrological variable which corresponds to a selected return period, a fuzzy linear relation between the standardized normal variable Z and the examined hydrologic random variable is achieved in condition that the hydrological variable is normally distributed. In this work, for the first time, the implementation of the fuzzy linear regression of Tanaka is proposed, to achieve a fuzzy relation between the standardized variable Z and the annual cumulative precipitation. Thus, all the historical data are included in the produced fuzzy band. The proposed innovative methodology provides the opportunity to achieve simultaneously a fuzzy assessment of the mean value and the standard deviation based on the solution of the fuzzy linear regression. The suitability test of the examined theoretical pdf is founded on the comparison of the spread of the fuzzy band and the distance between the achieved central values of the mean value and the standard deviation with the unbiased statistical estimation of the same variables.

Hybrid Fuzzy—Probabilistic Analysis and Classification of the Hydrological Drought

“…The uncertainty of the matching between the observed probabilities and the adopted theoretical probability distribution can be treated by using the fuzzy regression of Tanaka (1987) and hence, all the observed data will be included in the produced fuzzy band [8].…”