"""
fuzzy_logic.py : General logical operations on fuzzy membership functions,
a.k.a. fuzzy sets.
"""
import numpy as np
def _resampleuniverse(x, mfx, y, mfy):
"""
Resamples fuzzy universes `x` and `y` to include the full range of either
universe, with resolution of the lowest difference between any two
reported points.
"""
minstep = np.asarray([np.diff(x).min(), np.diff(y).min()]).min()
mi = min(x.min(), y.min())
ma = max(x.max(), y.max())
z = np.r_[mi:ma:minstep]
xidx = np.argsort(x)
mfx = mfx[xidx]
x = x[xidx]
mfx2 = np.interp(z, x, mfx)
yidx = np.argsort(y)
mfy = mfy[yidx]
y = y[yidx]
mfy2 = np.interp(z, y, mfy)
return z, mfx2, mfy2
def fuzzy_norm(x, mfx, y, mfy, norm):
"""
Fuzzy operator, logic operatrion of two fuzzy sets.
Parameters
----------
x : 1d array
Universe variable for fuzzy membership function `mfx`.
mfx : 1d array
Fuzzy membership function for universe variable `x`.
y : 1d array
Universe variable for fuzzy membership function `mfy`.
mfy : 1d array
Fuzzy membership function for universe variable `y`.
norm : Function
T-norm or T-conorm (S-norm)
Returns
-------
z : 1d array
Universe variable for union of the two provided fuzzy sets.
mfz : 1d array
Fuzzy membership function, the result of the operation
of `mfx` and `mfy`.
Notes
-------
See `T-Norm <https://en.wikipedia.org/wiki/T-norm>`_ for t-norms.
"""
# Check if universes are the same
sameuniverse = False
if x.shape == y.shape and (x == y).all():
z = x
mfx2 = mfx
mfy2 = mfy
sameuniverse = True
if not sameuniverse:
z, mfx2, mfy2 = _resampleuniverse(x, mfx, y, mfy)
return z, norm(mfx2, mfy2)
[docs]def fuzzy_and(x, mfx, y, mfy):
"""
Fuzzy AND operator, a.k.a. the intersection of two fuzzy sets.
Parameters
----------
x : 1d array
Universe variable for fuzzy membership function `mfx`.
mfx : 1d array
Fuzzy membership function for universe variable `x`.
y : 1d array
Universe variable for fuzzy membership function `mfy`.
mfy : 1d array
Fuzzy membership function for universe variable `y`.
Returns
-------
z : 1d array
Universe variable for union of the two provided fuzzy sets.
mfz : 1d array
Fuzzy AND (intersection) of `mfx` and `mfy`.
"""
# Check if universes are the same
return fuzzy_norm(x, mfx, y, mfy, norm=np.fmin)
[docs]def fuzzy_or(x, mfx, y, mfy):
"""
Fuzzy OR operator, a.k.a. union of two fuzzy sets.
Parameters
----------
x : 1d array
Universe variable for fuzzy membership function `mfx`.
mfx : 1d array
Fuzzy membership function for universe variable `x`.
y : 1d array
Universe variable for fuzzy membership function `mfy`.
mfy : 1d array
Fuzzy membership function for universe variable `y`.
Returns
-------
z : 1d array
Universe variable for intersection of the two provided fuzzy sets.
mfz : 1d array
Fuzzy OR (union) of `mfx` and `mfy`.
"""
# Check if universes are the same
return fuzzy_norm(x, mfx, y, mfy, norm=np.fmax)
[docs]def fuzzy_not(mfx):
"""
Fuzzy NOT operator, a.k.a. complement of a fuzzy set.
Parameters
----------
mfx : 1d array
Fuzzy membership function.
Returns
-------
mfz : 1d array
Fuzzy NOT (complement) of `mfx`.
Notes
-----
This operation does not require a universe variable, because the
complement is defined for a single set. The output remains defined on the
same universe.
"""
return 1. - mfx
Source