Multidimensional Least-squares Fitting of Fuzzy Models

We describe a new method for the fitting of differentiable fuzzy model functions to crisp data. The model functions can be either scalar or multidimensional and need not be linear. The data are n-component vectors. An efficient algorithm is achieved by restricting the fuzzy model functions to sets which depend on a fuzzy parameter vector and assuming that the vector has a conical membership function. The fuzzy model function, equated to zero, defines a fuzzy hypersurface in the n-space. The model fitting is done in a least-squares sense by minimizing the squares of the deviations from unity of the membership values of the fitted hypersurface at the observed points. Under the outlined restriction, the problem can be reduced to an ordinary least-squares formulation for which software is available. Application of the new method is illustrated by two examples. In one example, we are concerned with the hazards caused by enemy fire on armor. An important item of information for the assessment of the involved risks is a predictive model for the hole size in terms of physical properties of the projectile and target plate, respectively. We use a non-linear fuzzy model function for this analysis. The second example involves a linear model function and is of theoretical interest because it allows comparison of the new method with a previously developed method.