A nonlinear hybrid fuzzy least-squares regression model

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A nonlinear hybrid fuzzy least-squares regression
model

• Olga Poleshchuk, Evgeniy Komarov
• Moscow State Forest University, Russia

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The approaches under the heading of Fuzzy
Regression

• (a) Methods proposed by H.Tanaka and investigated
  by H.Tanaka, A.Celmins, D.Savic, W.Pedrycz,
  Y.-H.O.Chang, B.M.Ayyub, H.Ishibuchi. The
  coefficients of input variables are assumed to be
  fuzzy numbers.
• (b) Method proposed by R.J. Hathaway and J.C.
  Bezdek, where first the fuzzy clusters determined
  by fuzzy clustering define how many ordinary
  regressions are to be constructed, one for each
  cluster. Next each fuzzy cluster is used to
  determine the most appropriate ordinary
  regression that is to be applied for a new input
  from the ordinary regressions determined in the
  first place.
• (c) Methods proposed by I.B.Turksen, D.H.Hong,
  C.H.Hwang, where the fuzzy functions approach to
  system modeling was developed. These methods are
  based on a fuzzy clustering together with the
  least squares estimation techniques and approach
  that identifies the fuzzy functions using support
  vector machines.

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A quadratic hybrid fuzzy least-squares regression
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The method for formalization the meanings of
qualitative characteristic
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A quadratic hybrid fuzzy least-squares regression
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A weighted interval
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Distance between fuzzy numbers
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Weighted intervals and distances between initial
output fuzzy numbers and model fuzzy numbers
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Optimization problem
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Identifying a model fuzzy number with meanings of
qualitative characteristic
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Hybrid standard deviation, hybrid correlation
coefficient, hybrid standard error of estimate
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Numerical example
TABLE I Students grades
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Numerical example
TABLE II Membership functions of grades
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Numerical example
TABLE III Membership functions of grades
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Numerical example Linear hybrid fuzzy
least-squares regression
(1)
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Numerical exampleQuadratic hybrid fuzzy
least-squares regression
(2)
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Numerical example Ordinary regression
(3)
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Numerical example
TABLE IV Predicted and observed data
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Conclusions

• Quadratic hybrid fuzzy least-squares regression
  based on weighted intervals was developed.
• The method for formalization qualitative
  characteristics meanings was developed.
• The numerical example has demonstrated that
  developed hybrid regression model can be used for
  analysis relations among linguistic variables
  with success.