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Data Analysis Machine Learning and Applications Episode 3 Part 8

Tham khảo tài liệu 'data analysis machine learning and applications episode 3 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 692 Palumbo et al. In the early 80 s Tanaka proposed the first fuzzy linear regression model moving on from fuzzy sets theory and possibility theory Tanaka et al. 1980 . The functional relation between dependent and independent variables is represented as a fuzzy linear function whose parameters are given by fuzzy numbers. Tanaka proposed the first Fuzzy Possibilistic Regression FPR using the following fuzzy linear model with crisp input and fuzzy parameters yn P0 p 1 xn 1 p pxnp Ppx P 4 where the parameters are symmetric triangular fuzzy numbers denoted by Pp cp wp L with cp and wp as center and the spread respectively. Differently from statistical regression the deviations between data and linear models are assumed to depend on the vagueness of the parameters and not on measurement errors. The basic idea of Tanaka s approach was to minimize the uncertainty of the estimates by minimizing the total spread of the fuzzy coefficients. Spread minimization must be pursued under the constraint of the inclusion of the whole given data set which satisfies a degree of belief a 0 a 1 defined by the decision maker. The estimation problem is solved via a mathematical programming approach where the objective function aims at minimizing the spread parameters and the constraints guarantee that observed data fall inside the fuzzy interval N minimize 2 Wp xnp 5 subject to the following constraints c0 52P 1 cpxnpj 1 - tt w0 52P 1 wp xnp j yn c0 52P 1 cpxnp - 1 - tt w0 52P 1 wp xnp Ộ yn Wp 0 cp R Xn0 1 n 1 N p 1 P where xn0 1 n 1 N wp 0 and cp R p 1 P . 2.2 The F-PLSPM algorithm The F-PLSPM follows the component based approach SEM-PLS alternatively defined PLS Path Modeling PLS-PM Tenenhaus et al. 2005 . The reason is that fuzzy regression and PLS path modeling share several characteristics. They are both soft modeling and data oriented approaches. Specifically fuzzy regression joins PLS-PM in its final step allowing for a fuzzy structural model see Figure 1 but a still crisp .