Learning algorithms when class membership is poorly defined

Let X be a measurement space and f(x) a real function defined on it. If f(x) takes only a small set of discrete values then we have the standard classification problem. Otherwise f (x) can be considered as defining a fuzzy pattern recognition problem. We consider the problem of dividing X into regions xi( = 1,2 .... R) such that on each one of them f(x) is approximated either by a constant or a linear function. The partition is generated for a given R by minimizing the total integral square error. This problem is equivalent to piecewise functional approximation. After the regions Xi. and the approximations have been determined than it is possible to predict the value f(x) for any given measurement x. The computational requirements of this approach are higher than those of the common learning algorithms but it is applicable in cases where (except for extreme cases) class membership is vaguely defined as it is often the case in socio-economic problems, mechanical and medical diagnosis etc.