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 achieve a relation between the hydrological variable and the corresponding probability 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 article, primary, the implementation of the fuzzy linear regression of Tanaka is proposed regarding the annual cumulative precipitation. Thus, all the historical data will be included in the produced fuzzy band. However, since many times the question is about the inverse process, that is, the determination of the return period for a given hydrological value, then, for this purpose, a fuzzy bi-sector regression is developed. The proposed bi-sector fuzzy regression incorporates the inclusion property regarding the produced fuzzy band as the fuzzy regression of Tanaka does. 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. To test the suitability of the produced fuzzy band, several measures are proposed which incorporate the magnitude of the produced fuzzy band and the comparison between the estimated fuzzy mean value and standard deviation with the unbiased crisp estimation of the same variables and the median of the sample.
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Spiliotis, M., Angelidis, P. & Papadopoulos, B. A hybrid probabilistic bi-sector fuzzy regression based methodology for normal distributed hydrological variable. Evolving Systems 11, 255–268 (2020). https://doi.org/10.1007/s12530-019-09284-7
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DOI: https://doi.org/10.1007/s12530-019-09284-7