1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
|
## Copyright (C) 2011-2025 L. Markowsky <lmarkowsky@gmail.com>
##
## This file is part of the fuzzy-logic-toolkit.
##
## The fuzzy-logic-toolkit is free software; you can redistribute it
## and/or modify it under the terms of the GNU General Public License
## as published by the Free Software Foundation; either version 3 of
## the License, or (at your option) any later version.
##
## The fuzzy-logic-toolkit is distributed in the hope that it will be
## useful, but WITHOUT ANY WARRANTY; without even the implied warranty
## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with the fuzzy-logic-toolkit; see the file COPYING. If not,
## see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{y} =} zmf (@var{x}, @var{params})
## @deftypefnx {Function File} {@var{y} =} zmf (@var{[x1 x2 ... xn]}, @var{[a b]})
##
## For a given domain @var{x} and parameters @var{params} (or @var{[a b]}),
## return the corresponding @var{y} values for the Z-shaped membership function.
##
## The argument @var{x} must be a real number or a non-empty vector of strictly
## increasing real numbers, and @var{a} and @var{b} must be real numbers, with
## @var{a} < @var{b}. This membership function satisfies:
##
## @verbatim
## 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
## @end verbatim
##
## which always returns values in the range [0, 1].
##
## The parameters a and b specify:
##
## @verbatim
## a == the rightmost point at which f(x) = 1
## b == the leftmost point at which f(x) = 0
## @end verbatim
##
## At the midpoint of the segment [a, b], the function value is 0.5:
##
## @verbatim
## f((a + b)/2) = 0.5
## @end verbatim
##
## To run the demonstration code, type "@t{demo zmf}" (without the quotation
## marks) at the Octave prompt.
##
## @seealso{dsigmf, gauss2mf, gaussmf, gbellmf, pimf, psigmf, sigmf, smf, trapmf, trimf, zmf_demo}
## @end deftypefn
## Author: L. Markowsky
## Keywords: fuzzy-logic-toolkit fuzzy membership z-shaped
## Directory: fuzzy-logic-toolkit/inst/
## Filename: zmf.m
## Last-Modified: 13 Jun 2024
function y = zmf (x, params)
## If the caller did not supply 2 argument values with the correct
## types, print an error message and halt.
if (nargin != 2)
error ("zmf requires 2 arguments\n");
elseif (!is_domain (x))
error ("zmf's first argument must be a valid domain\n");
elseif (!are_mf_params ('zmf', params))
error ("zmf's second argument must be a parameter vector\n");
endif
## Calculate and return the y values of the membership function on the
## domain x.
a = params(1);
b = params(2);
a_b_ave = (a + b) / 2;
b_minus_a = b - a;
y_val = @(x_val) zmf_val (x_val, a, b, a_b_ave, b_minus_a);
y = arrayfun (y_val, x);
endfunction
##----------------------------------------------------------------------
## Usage: y_val = zmf_val (x_val, a, b, a_b_ave, b_minus_a)
##
## zmf_val returns one value of the Z-shaped membership function, which
## 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
##
## zmf_val is a private function, called only by zmf. Because zmf_val
## is not intended for general use -- and because the parameters a and b
## are checked for errors in the function zmf (defined above), the
## parameters are not checked for errors again here.
##----------------------------------------------------------------------
function y_val = zmf_val (x_val, a, b, a_b_ave, b_minus_a)
## Calculate and return a single y value of the Z-shaped membership
## function for the given x value and parameters specified by the
## arguments.
if (x_val <= a)
y_val = 1;
elseif (x_val <= a_b_ave)
y_val = 1 - 2 * ((x_val - a) / b_minus_a)^2;
elseif (x_val < b)
y_val = 2 * ((x_val - b) / b_minus_a)^2;
else
y_val = 0;
endif
endfunction
%!demo
%! 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;
%!test
%! x = 0:10:100;
%! params = [25 75];
%! y = [1 1 1 0.9800 0.8200 0.5000 0.1800 0.020000 0 0 0];
%! z = zmf(x, params);
%! assert(z, y, 1e-4);
## Test input validation
%!error <zmf requires 2 arguments> zmf()
%!error <zmf requires 2 arguments> zmf(1)
%!error <zmf: function called with too many inputs> zmf(1, 2, 3)
%!error <zmf's first argument must be a valid domain> zmf([1 0], 2)
%!error <zmf's second argument must be a parameter vector> zmf(1, 2)
%!error <zmf's second argument must be a parameter vector> zmf(0:100, [])
%!error <zmf's second argument must be a parameter vector> zmf(0:100, [30])
%!error <zmf's second argument must be a parameter vector> zmf(0:100, [90 80 30])
%!error <zmf's second argument must be a parameter vector> zmf(0:100, 'abc')
%!error <zmf's second argument must be a parameter vector> zmf(0:100, '')
|
