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1.7.A. Adding to the Problem of the Automatic Correction of the Width of the Membership Function.

As it is affirmed, the maximum of the function of conformity that is determined by expression

   

is reached at the finite value of parameter D, when the exponent in which it is raised in numerator - g (governing the amplitude growth rate) lays in the interval 1 < g < 2.

Let's prove this statement.

Let's consider an approximated segment (see fig.6b) as a continuous point set. In this case the conformity measure is determined by an integral

   

Here the variable v = d is entered to avoid confusion with differential label. The behaviour of this measure depending on width of membership functions is defined by a derivative on D:

   

   

   

Equating this expression to zero, and introducing the variable gamma = V/D, we obtain:

   

In a range of interest for us (0 < gamma < oo) this function has the maximum value 2 at gamma = 0 and asymptotically comes nearer to 1 at gamma -> oo (see fig.6d).


Fig.6d

Its decrease monotonicity is confirmed by that its derivative everywhere is negative in the interval 0 < gamma < oo:

   

A graph of the inverse function D/V=1/gamma (g), our actual subject of interest, is represented on fig.6c. This graph shows, what will be the width of memberships functions (in comparison with the flowing relative width of the approximated segment) at reaching by the considered conformity measure of its maximum value depending on index g, governing amplitude growth rate of memberships functions.

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