Function File: y = zmf (x, params)
Function File: y = zmf ([x1 x2 ... xn], [a b])

For a given domain x and parameters params (or [a b]), return the corresponding y values for the Z-shaped membership function.

The argument x must be a real number or a non-empty vector of strictly increasing real numbers, and a and b must be real numbers, with a < b. This membership function satisfies: 

        1                                if x <= a
f(x) =  1 - 2 * ((x - a)/(b - a))^2      if a < x <= (a + b)/2
        2 * ((x - b)/(b - a))^2          if (a + b)/2 < x < b
        0                                if x >= b

which always returns values in the range [0, 1].

The parameters a and b specify:

  • a == the rightmost point at which f(x) = 1
  • b == the leftmost point at which f(x) = 0

At the midpoint of the segment [a, b], the function value is 0.5:

  • f((a + b)/2) = 0.5

To run the demonstration code, type demo('zmf') at the Octave prompt.

See also: dsigmf, gauss2mf, gaussmf, gbellmf, pimf, psigmf, sigmf, smf, trapmf, trimf, zmf_demo.

Demonstration 1

The following code

 x = 0:100;
 params = [40 60];
 y1 = zmf(x, params);
 params = [25 75];
 y2 = zmf(x, params);
 params = [10 90];
 y3 = zmf(x, params);
 figure('NumberTitle', 'off', 'Name', 'zmf demo');
 plot(x, y1, 'r;params = [40 60];', 'LineWidth', 2)
 hold on;
 plot(x, y2, 'b;params = [25 75];', 'LineWidth', 2)
 hold on;
 plot(x, y3, 'g;params = [10 90];', 'LineWidth', 2)
 ylim([-0.1 1.1]);
 xlabel('Crisp Input Value', 'FontWeight', 'bold');
 ylabel('Degree of Membership', 'FontWeight', 'bold');
 grid;

Produces the following figure

Figure 1

Package: fuzzy-logic-toolkit