Advanced FIS configuration

This document summarizes the advanced configuration options available for Fuzzy Inference Systems.

For a general background on the components of Fuzzy Inference Systems, you can check out MatWorks’s documentation at https://www.mathworks.com/help/fuzzy/fuzzy-inference-process.html

import zadeh
import matplotlib.pyplot as plt

TODO: This guide is currrently being edited and does reflect the current development status.

Logical operators

By default, the zadeh package uses the most common definitions for the logical operators of fuzzy logic. In terms of membership funtion values:

Python notation	Logical notation	Interpretation
~X	NOT X	x: 1-x
X|Y	X OR Y	x, y: max(x, y)
X&Y	X AND Y	x, y: min(x, y)

However, there are different conventions available to construct a fuzzy logic. We typically want the operators to obey the De Morgan law. A De Morgan triplet is a triplet \((\top, \bot, \neg)\) ensuring such a thing, as well as other important considerations. In most of the cases the standard negator is kept, and the other two are related by a duality transformation. The following table describes the common choices available in the package.

	T-Norm (AND)			T-conorm (OR)
Name	Python	** \[\top(a,b)\] **	Name	Python	** \[\bot(a,b)\] **
Minimum	min	\[\min \{ a,b \}\]	Maximum	max	\[\max\{a,b\}\]
Product	product	\[a\cdot b\]	Probabilistic sum	psum	\[1-(1-a)(1-b)\]
Łukasiewicz	lukasiewicz	\[\max\{0, a + b - 1\}\]	Bounded sum	bsum	\[\min\{1,a+b\}\]

for method in ["max", "psum", "bsum"]:
    with zadeh.set_fuzzy_context(zadeh.FuzzyContext(OR=method)):
        test = zadeh.FuzzyVariable(
            zadeh.FloatDomain("test", 0, 20, 100),
            {
                method: 0.8 * zadeh.TrapezoidalFuzzySet(0, 2, 8, 12)
                | 0.5 * zadeh.TrapezoidalFuzzySet(5, 7, 12, 14)
                | 0.1 * zadeh.TrapezoidalFuzzySet(12, 13, 18, 19),
            },
        )
        test.plot()

plt.title("OR methods")
plt.show()

for method in ["min", "product", "lukasiewicz"]:
    with zadeh.set_fuzzy_context(zadeh.FuzzyContext(AND=method)):
        test = zadeh.FuzzyVariable(
            zadeh.FloatDomain("test", 0, 20, 100),
            {
                method: 0.8 * zadeh.TrapezoidalFuzzySet(0, 2, 8, 12)
                & 0.5 * zadeh.TrapezoidalFuzzySet(5, 7, 12, 14)
            },
        )
        test.plot()

plt.title("AND methods")
plt.show()

Implication

The current version of zadeh supports only the most common definitions for the logical implication output. In terms of membership funtion values:

Python notation	Logical notation	Interpretation
X >> Y	X -> Y	x, y: min(x, y)

Aggregation

The current version of zadeh supports only the most common aggregation method for multiple rules, which the usage of the maximum function

Python notation	Logical notation	Interpretation
[X, Y, …]	{X, Y, …}	x, y, …: max(x, y)

Defuzzification

Defuzzification is the last stage of the Fuzzy Inference System, where the fuzzy output is mapped into a crisp output using one of the available procedures. Available options are:

• centroid: Analog to the center of mass.

• bisector: Value separating two portions of equal area under the membership function.

• mom (middle of maximum): Middle (median) of the set of values whose images are the maximum of the membership function.

• som (smaller of maximum): Smaller (min) of the set of values whose images are the maximum of the membership function.

• lom (largest of maximum): Largest (max) of the set of values whose images are the maximum of the membership function.

The following plot illustrates the different methods using a custom Fuzzy Set, matching the one from https://www.mathworks.com/help/fuzzy/defuzzification-methods.html

test = zadeh.FuzzyVariable(
    zadeh.FloatDomain("test", 0, 20, 100),
    {
        "membership": 0.8 * zadeh.TrapezoidalFuzzySet(0, 2, 8, 12)
        | 0.5 * zadeh.TrapezoidalFuzzySet(5, 7, 12, 14)
        | 0.1 * zadeh.TrapezoidalFuzzySet(12, 13, 18, 19),
    },
)
test.plot()
lims=plt.ylim()
plt.vlines(test.domain.centroid(test.values["membership"]), *lims, color="r", ls="--", label="centroid")
plt.vlines(test.domain.bisector(test.values["membership"]), *lims, color="g", ls="-.", label="bisector")
plt.vlines(test.domain.som(test.values["membership"]), *lims, color="gray", ls="-", label="som")
plt.vlines(test.domain.mom(test.values["membership"]), *lims, color="gray", ls="--", label="mom")
plt.vlines(test.domain.lom(test.values["membership"]), *lims, color="gray", ls="-", label="lom")
plt.legend()

plt.show()

To set any of this methods in a FIS, just use the defuzzification kwargs in its construction.