Fuzzy regression

In order to obtain the criterion function, a fuzzy partition A, A must be determined. The fuzzy set A is characterized by the prototype L. With respect to complementary fuzzy set. A, we will consider that the dissimilarity between its hypothetical prototype and the points x is constant and equal to a/1 — a, where a is a constant from [0 1], with a role to be seen later. Denote the dissimilarity between the point x and the prototype L by 

The optimal fuzzy set will be determined using an iterative method, where J is successively minimized with respect to A and L. The proposed algorithm will be called Fuzzy Regression  

Compute the prototype L = v, u) of the fuzzy set A l using the relations given above. 

Determine the new fuzzy set using the relation given above. 

Our results show that a good value of s with respect to the similarity of A and Al + l is s = 10 .  

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The development of new data analysis methods is also an important area of QSAR research. Several methods have been developed in recent years, and these include kernel partial least squares (K-PLS) [92], robust continuum regression [93], local lazy regression [94], fuzzy interval number -nearest neighbor (FINkNN) [95], and fast projection plane classifier (FPPC) [96], These methods have been shown to be useful for the prediction of a wide variety of target properties, which include moisture, oil, protein and starch... 

B. K. Alsberg, R. Goodacre, J.J. Rowland and D.B. Kell, Classification of Pyrolysis Mass Spectra by Fuzzy Multivariate Rule Induction-comparison with Regression, K-nearest Neighbour, Neural and Decision-tree Methods. Analytica Chimica Acta, 348(1-3) (1997), 389 07. 

Alsberg, B. K., Goodacre, R., Rowland, J. J. and Kell, D. B. (1997) Classification of pyrolysis mass spectra by fuzzy multivariate rule induction comparison with regression, K-nearest neighbour, neural and decision-tree methods. Analytica Chimica Acta, in press. 

Objective entities in the process operation system, such as reactor, distillation column units and processes, are used and conducted by the different operation tasks. Modeling methods for the units and processes objects are first principle rules, statistical regression, artificial neural network, and fuzzy logic relation. 

However, the mentioned above single models have some shortages unavoidably. To remedy the defects of single models, hybrid models are actively researched recently. One kind of hybrid models (Qi, 1999) combines part of first principle equations with ANN, in which ANN is used to determine parameters of the first principle models. Fuzzy logic approach (Qian, 1999) is used for representing imprecision and approximation of the relationship among process variables. It is successfully incorporated into conventional process simulators. Several efforts (Baffi, 1999) have been made to combine statistical analysis with non-linear regression, which are polynomial, spline function and ANN. 

A model was developed using both regression and neuro-fuzzy models to predict thrust force and torque in the drilling operation of GFRP composites. The neuro-fuzzy model showed much better accuracy than the regression model. The average absolute errors for thrust force and torque were found to be 5.83 per cent and 4.57 per cent respectively with the neuro-fuzzy model, whereas for the regression model, they were 17.3 per cent and 26.67 per cent respectively (Jayabal and Natarajan, 2010). 

Jayabal, S. and Natarajan, U. (2010) Regression and neuro fuzzy models for prediction of thrust force and torque in drilling of glass fiber reinforced composites, J Sci Ind Res, 69 741-5. 

Linear Regression Analysis with Fuzzy Model. 

The achieved results complement the set of approaches to the indirect observation of a technical condition. The approaches using purely a regression analysis and fuzzy logic, see e.g. Koucky Valis (2011), Valis et al. (2012), have been applied so far. Following the conclusions of modelling with the Wiener process, the results of previous approaches might be completed when searching for ... 

Takagi and Sugeno (1985) proposed models where the consequent part of the rule is described by a linear regression model. These models are easier to identify because each rale describes a fuzzy region in which the output depends on the inputs in a linear maimer. An example of such a model is shown in Eqn. (28.3) ... 

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