| Hybrid regression is an integral of classical and fuzzy regression into one model. Hybrid regression becomes classical regression when data fuzziness is removed. While classical regression can process the randomness type of uncertainty, traditional fuzzy regression only deals with fuzziness as an uncertainty type. In many engineering problems, both randomness and fuzziness co-exist, and need to be considered in regression analysis.;For regression analysis involving fuzzy numbers, weighted fuzzy arithmetic is defined and used as a replacement to conventional fuzzy arithmetic. Weighted fuzzy arithmetic defines the arithmetic operations between two fuzzy numbers as operating two corresponding ordinary numbers in each fuzzy set at the same membership level, and integrating each level operation weighted by its membership value for the entire fuzzy sets, and dividing the weighted integration by the total integral of the membership function. Since the concept of defuzzification is used, the result of weighted fuzzy arithmetic is a crisp number, which can be interpreted as the mean value of fuzzy arithmetic operation. Weighted fuzzy arithmetic is used to derive hybrid reliability and uncertainty measures for fuzzy regression and hybrid regression.;A method of hybrid least-squares regression based on the weighted fuzzy arithmetic is proposed and developed. Hybrid least-squares regression analysis can process fuzzy data, crisp data, and their mixture, and integrate both randomness and fuzziness into a regression model. The developed method conveniently uses classical regression programs to solve for the fuzzy centers and fuzzy widths of regression coefficients respectively. The method produces classical regression results when no data fuzziness exists. Based on the evaluation of the hybrid reliability and uncertainty measures, the developed method for hybrid least-squares regression shows the best goodness-of-fit results compared with other fuzzy regression models. The limitations of the developed method are also discussed.;Hybrid least-squares regression analysis can be used to model data containing randomness and fuzziness types of uncertainty, which are traditionally modeled by either classical regression or fuzzy regression. Two case studies are presented; one for modeling measurement errors, and another one for aggregating expert opinions. |
