Granular Constructs

Abstract

Consider a highly nonlinear function like the one in Fig. 10.1 as the linear fit of experimental points describing the relationship between variables x and y. Willing to understand this relationship we may decide to express it by a 5-th degree polynomial, whose fitting looks like in Fig. 10.1(a). A better way is to identify three fuzzy sets centered at the points x₁ = 1, x₂ = 3.5 and x₃ = 8 each with a membership function $\mu_i=\mathrm e^{-(x-x_i)^2}$ and endowed with its own relation:

$$y=g_i(x)~~~(10.1)$$

that holds between x and y, namely: 1) y = g₁(x) = 2.4x; 2) y = g₂(x) = 0.4 + 1.3x; 3) y = g₃(x) = 10.0 − 0.7x. The mixture of these functions weighted with the above values of μ _i , i.e.

$$y=\sum_{i=1}^k g_i(x)\mu_i(x),~~~(10.2)$$

reproduces the function at hand (see Fig. 10.1(b)). This delineates the way through which we manage in this chapter a complex fuzzy set by computing its membership function: using a few variables quantized into a few elementary granules and elementary functions connecting the variables.