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<!DOCTYPE html>
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<title>Octave Fuzzy Logic Toolkit: gbellmf</title>
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Octave Fuzzy Logic Toolkit
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Membership Functions
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Function Reference: <b><code>gbellmf</code></b>
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<dl>
<dt><u>Function File:</u> <var>y</var> = <b>gbellmf</b><i> (<var>x</var>, <var>params</var>)</i></dt>
<dt><u>Function File:</u> <var>y</var> = <b>gbellmf</b><i> (<var>[x1 x2 ... xn]</var>, <var>[a b c]</var>)</i></dt>
</dl>
<p> For a given domain <var>x</var> and parameters <var>params</var> (or <var>[a b c]</var>),
return the corresponding <var>y</var> values for the generalized bell-shaped
membership function.
</p>
<div class="ms-5">
<p> The argument <var>x</var> must be a real number or a non-empty vector of strictly
increasing real numbers, <var>a</var>, <var>b</var>, and <var>c</var> must be real numbers,
<var>a</var> must be non-zero, and <var>b</var> must be an integer. This membership
function satisfies the equation:
</p>
<pre class="verbatim"> f(x) = 1/(1 + (abs((x - c)/a))^(2 * b))
</pre>
<p> which always returns values in the range [0, 1].
</p>
<p> The parameters <var>a</var>, <var>b</var>, and <var>c</var> give:
</p>
<pre class="verbatim"> a == controls the width of the curve at f(x) = 0.5;
f(c-a) = f(c+a) = 0.5
b == controls the slope of the curve at x = c-a and x = c+a;
f'(c-a) = b/2a and f'(c+a) = -b/2a
c == the center of the curve
</pre>
<p> This membership function has a value of 0.5 at the two points c - a and
c + a, and the width of the curve at f(x) == 0.5 is 2 * |a|:
</p>
<pre class="verbatim"> f(c - a) == f(c + a) == 0.5
2 * |a| == the width of the curve at f(x) == 0.5
</pre>
<p> The generalized bell-shaped membership function is continuously
differentiable and is symmetric about the line x = c.
</p>
<p> To run the demonstration code, type "<tt>demo gbellmf</tt>" (without the quotation
marks) at the Octave prompt.
</p>
<p> <strong>See also: </strong>
<a href="dsigmf.html">dsigmf</a>,
<a href="gauss2mf.html">gauss2mf</a>,
<a href="gaussmf.html">gaussmf</a>,
<a href="pimf.html">pimf</a>,
<a href="psigmf.html">psigmf</a>,
<a href="sigmf.html">sigmf</a>,
<a href="smf.html">smf</a>,
<a href="trapmf.html">trapmf</a>,
<a href="trimf.html">trimf</a>,
<a href="zmf.html">zmf</a>
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Example: 1
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<td> </td>
<td><pre class="example">
x = 0:255;
params = [20 4 100];
y1 = gbellmf(x, params);
params = [30 3 100];
y2 = gbellmf(x, params);
params = [40 2 100];
y3 = gbellmf(x, params);
figure('NumberTitle', 'off', 'Name', 'gbellmf demo');
plot(x, y1, 'r;params = [20 4 100];', 'LineWidth', 2)
hold on;
plot(x, y2, 'b;params = [30 3 100];', 'LineWidth', 2)
hold on;
plot(x, y3, 'g;params = [40 2 100];', 'LineWidth', 2)
ylim([-0.1 1.1]);
xlabel('Crisp Input Value', 'FontWeight', 'bold');
ylabel('Degree of Membership', 'FontWeight', 'bold');
grid;
</pre></td></tr></tbody>
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