Table of Contents

• Triangular fuzzy numbers
• Fuzzy sets
• Plotting

To-Do

• implementar Cauchyset [Done]
  1. Try with many codes []
  2. create a code with example at README []
• implementar tnorma( DeMorgan tripdrástica, produto e soma probabilista, min max ) obs: qualquer definição de uma tnorma -> automaticamente no t-conorma [Done]
  1. Verify the Tnorm product
  2. Create code example of Tnorm product
  3. Verify the Tconorm probabilistic sum
  4. Create code example of probabilistic sum
  5. verify the Tnorm min max
  6. Create code with Tnorm min and Tconorm max
• implementar takagi-sugeno, mandanam -> dados de saída são em dataframe com todas as variáveis

fuzzysets

fuzzysets is a simple Python package that is well suited for interactive experiments with fundamental concepts in fuzzy set theory.

Installation

The package is available on PYPI and can be installed with pip:
pip install initfuzzy
pip install git+https://github.com/thigs0/initfuzzy.git

Notation

In the following sections, code snippets will be highlighted like this.

import fuzzysets as fs

Examples involving a read-evaluate-print loop will show both the Python statement, which will be preceded by a >>> prompt, and its output, if any.

Triangular fuzzy numbers

One of the concepts implemented by fuzzysets is a triangular fuzzy number (TFN), represented with the TriangularFuzzyNumber class.

Each TFN can be uniquely represented as a 3-tuple of real numbers (left, peak, right) (l, n and r below) where:

• peak is the number whose membership degree is 1, that is, the number being modeled
• left (< peak) and right (> peak) determine the fuzzy number's membership function:
  • mu(x) = 0, x ∈ (-inf, l) U (r, +inf)
  • mu(x) = (x - l) / (n - l), l <= x <= n
  • mu(x) = (r - x) / (r - n), n <= x <= r

The TriangularFuzzyNumber class offers a more complex abstraction than that.

We first need to import the fuzzysets package:

>>> import fuzzysets as fs

The class is available through an alias - TFN. The default value of the class is 0:

import matplotlib.pyplot as plt
import numpy as np
from fuzzysets import TriangularFuzzyNumber as tfn

plt.style.use('ggplot')
a = tfn()
b = tfn(3,2,4)
plt.plot(x, a.mu(x), label="Triangular Fuzzy Set (a)", color="red")
plt.plot(x, b.mu(x), label="Cauchy Fuzzy Set (b)", color="blue")
x = np.linspace(-5,10,200)

>>> fs.TFN()
TriangularFuzzyNumber(l=-1.0, n=0.0, r=1.0)

As you can see, the left and right properties are offset by 1 by default. We could create a TFN that models any number like this:

>>> fs.TFN(11.5)
TriangularFuzzyNumber(l=10.5, n=11.5, r=12.5)

It is also possible to set one or both of the other properties:

>>> fs.TFN(10, r=12.8)
TriangularFuzzyNumber(l=9.0, n=10.0, r=12.8)
>>> fs.TFN(10.6, 8, 12)
TriangularFuzzyNumber(l=8.0, n=10.6, r=12.0)

We can also obtain the alpha cut of a TFN:

>>> n
TriangularFuzzyNumber(l=1.0, n=2.0, r=4.0)
>>> n.alpha_cut
AlphaCut(1.0 + alpha * 1.0, 4.0 + alpha * -2.0)

It can be converted to a more user-friendly string than the one above and can compute the interval for a fixed alpha:

>>> cut = n.alpha_cut
>>> str(cut)
'[1.0 + alpha * 1.0, 4.0 + alpha * -2.0]'
>>> cut.for_alpha(0.5)
(1.5, 3.0)

Cauchy fuzzy Set

import matplotlib.pyplot as plt
import numpy as np
from fuzzysets import CauchyFuzzySet as cfs

a = cfs(1,2,3)
b = cfs(2,4,5)

x = np.linspace(-5,10,200)
plt.plot(x, a.mu(x))
plt.style.use('ggplot')
plt.plot(x, a.mu(x), label="Cauchy Fuzzy Set (a)", color="red")
plt.plot(x, b.mu(x), label="Cauchy Fuzzy Set (b)", color="blue")

Tnorm and Tconorm

Example

import matplotlib.pyplot as plt
import numpy as np
from cauchyfuzzyset import CauchyFuzzySet as cfn
from tfn import TriangularFuzzyNumber as tfn
from tnorm import Tnorm
plt.style.use('ggplot')

c = cfn(1,2,3)
d = cfn(2,4,5)

x = np.linspace(-1,10,200)
sets = [c,d]
colors = ["yellow", "green"]
for i in range(2): plt.plot(x, [sets[i].mu(j) for j in x ], label=f"Fuzzy Set)", color=colors[i])

out = Tnorm("minimum", c,d)
plt.plot(x, [out.tnorm(i) for i in x], "r--", label="Tnorm")
plt.plot(x, [out.tconorm(i) for i in x], "*", label="tconorm", color="gray")
plt.title("tnorm and tconorm about two cauchy fuzzy set")
plt.legend(loc="center left")
plt.savefig('test.png', dpi=300)

Mandani inference method

Example

import numpy as np
from typing import Callable, List, Tuple
from fuzzysets import CauchyFuzzySet, TriangularFuzzyNumber, Tnorm, AND, MamdaniInference

#heavy of cloths
ml = TriangularFuzzyNumber(0, -20, 20)
l = TriangularFuzzyNumber(30,10, 50)
p = TriangularFuzzyNumber(65, 40, 90)
mp = TriangularFuzzyNumber(90, 75,100)

#dirt cloths
ql = TriangularFuzzyNumber(0, -20, 20)
s = TriangularFuzzyNumber(30, 10, 50)
ms = TriangularFuzzyNumber(70, 40, 100)
es = TriangularFuzzyNumber(100, 80, 120)

# Define regras com funções anônimas ou conjuntos fuzzy
muito_pouco = TriangularFuzzyNumber(10, 0, 20)
pouco = TriangularFuzzyNumber(30, 20, 40)
moderado = TriangularFuzzyNumber(50, 40 ,60)
exagerado = TriangularFuzzyNumber(70, 60 ,80)
maximo = TriangularFuzzyNumber(100, 80 ,120)

#tnorm
p1,p2 = 10,15
ps = [p1, p2]
w1 = AND([ml, ql])
w2 = AND([ml, s])
w3 = AND([ml, ms])
w4 = AND([ml, es])

w5 = AND([l, ql])
w6 = AND([l, s])
w7 = AND([l, ms])
w8 = AND([l, es])

w9  = AND([p, ql])
w10 = AND([p, s])
w11 = AND([p, ms])
w12 = AND([p, es])

w13 = AND([mp, ql])
w14 = AND([mp, s])
w15 = AND([mp, ms])
w16 = AND([mp, es])

# Sistema de inferência
mi = MamdaniInference()
mi.add_rule(antecedent = w1, consequent = muito_pouco)
mi.add_rule(antecedent = w2, consequent = pouco)
mi.add_rule(antecedent = w3, consequent = moderado)
mi.add_rule(antecedent = w4, consequent = moderado)
mi.add_rule(antecedent = w5, consequent = pouco)
mi.add_rule(antecedent = w6, consequent = pouco)
mi.add_rule(antecedent = w7, consequent = moderado)
mi.add_rule(antecedent = w8, consequent = exagerado)
mi.add_rule(antecedent = w9, consequent = moderado)
mi.add_rule(antecedent = w10, consequent = moderado)
mi.add_rule(antecedent = w11, consequent = exagerado)
mi.add_rule(antecedent = w12, consequent = exagerado)
mi.add_rule(antecedent = w13, consequent = moderado)
mi.add_rule(antecedent = w14, consequent = exagerado)
mi.add_rule(antecedent = w15, consequent = maximo)
mi.add_rule(antecedent = w16, consequent = maximo)

# Inferência
saida = mi.infer(ps, np.linspace(0, 100, 1000))
print("Saída defuzzificada:", saida)

takagi-sugeno inference method

Example

import numpy as np
from typing import Callable, List, Tuple
from fuzzysets import CauchyFuzzySet, TriangularFuzzyNumber, Tnorm, AND, TagagiSugenoInference

#heavy of cloths
ml = TriangularFuzzyNumber(0, -20, 20)
l = TriangularFuzzyNumber(30,10, 50)
p = TriangularFuzzyNumber(65, 40, 90)
mp = TriangularFuzzyNumber(90, 75,100)

#dirt cloths
ql = TriangularFuzzyNumber(0, -20, 20)
s = TriangularFuzzyNumber(30, 10, 50)
ms = TriangularFuzzyNumber(70, 40, 100)
es = TriangularFuzzyNumber(100, 80, 120)

# Define regras com funções anônimas ou conjuntos fuzzy
def muito_pouco(x):return 10
def pouco(x): return 30
def moderado(x): return 50
def exagerado(x): return 70
def maximo(x): return 95

#tnorm
p1,p2 = 10,15
ps = [p1, p2]
w1 = AND([ml, ql])
w2 = AND([ml, s])
w3 = AND([ml, ms])
w4 = AND([ml, es])

w5 = AND([l, ql])
w6 = AND([l, s])
w7 = AND([l, ms])
w8 = AND([l, es])

w9  = AND([p, ql])
w10 = AND([p, s])
w11 = AND([p, ms])
w12 = AND([p, es])

w13 = AND([mp, ql])
w14 = AND([mp, s])
w15 = AND([mp, ms])
w16 = AND([mp, es])

# Sistema de inferência
mi = TakagiSugenoInference()
mi.add_rule(antecedent = w1, consequent = muito_pouco)
mi.add_rule(antecedent = w2, consequent = pouco)
mi.add_rule(antecedent = w3, consequent = moderado)
mi.add_rule(antecedent = w4, consequent = moderado)
mi.add_rule(antecedent = w5, consequent = pouco)
mi.add_rule(antecedent = w6, consequent = pouco)
mi.add_rule(antecedent = w7, consequent = moderado)
mi.add_rule(antecedent = w8, consequent = exagerado)
mi.add_rule(antecedent = w9, consequent = moderado)
mi.add_rule(antecedent = w10, consequent = moderado)
mi.add_rule(antecedent = w11, consequent = exagerado)
mi.add_rule(antecedent = w12, consequent = exagerado)
mi.add_rule(antecedent = w13, consequent = moderado)
mi.add_rule(antecedent = w14, consequent = exagerado)
mi.add_rule(antecedent = w15, consequent = maximo)
mi.add_rule(antecedent = w16, consequent = maximo)

# Inferência
saida = mi.infer(ps)
print("Saída defuzzificada:", saida)