"""
defuzz.py : Various methods for defuzzification and lambda-cuts, to convert
'fuzzy' systems back into 'crisp' values for decisions.
"""
import numpy as np
from ..image.arraypad import pad
[docs]def arglcut(ms, lambdacut):
"""
Determines the subset of indices `mi` of the elements in an N-point
resultant fuzzy membership sequence `ms` that have a grade of membership
>= lambdacut.
Parameters
----------
ms : 1d array
Fuzzy membership sequence.
lambdacut : float
Value used for lambda cutting.
Returns
-------
lidx : 1d array
Indices corresponding to the lambda-cut subset of `ms`.
Notes
-----
This is a convenience function for `np.nonzero(lambdacut <= ms)` and only
half of the indexing operation that can be more concisely accomplished
via::
ms[lambdacut <= ms]
"""
return np.nonzero(lambdacut <= ms)
[docs]def centroid(x, mfx):
"""
Defuzzification using centroid (`center of gravity`) method.
Parameters
----------
x : 1d array, length M
Independent variable
mfx : 1d array, length M
Fuzzy membership function
Returns
-------
u : 1d array, length M
Defuzzified result
See also
--------
skfuzzy.defuzzify.defuzz, skfuzzy.defuzzify.dcentroid
"""
'''
As we suppose linearity between each pair of points of x, we can calculate
the exact area of the figure (a triangle or a rectangle).
'''
sum_moment_area = 0.0
sum_area = 0.0
# If the membership function is a singleton fuzzy set:
if len(x) == 1:
return x[0]*mfx[0] / np.fmax(mfx[0], np.finfo(float).eps).astype(float)
# else return the sum of moment*area/sum of area
for i in range(1, len(x)):
x1 = x[i - 1]
x2 = x[i]
y1 = mfx[i - 1]
y2 = mfx[i]
# if y1 == y2 == 0.0 or x1==x2: --> rectangle of zero height or width
if not(y1 == y2 == 0.0 or x1 == x2):
if y1 == y2: # rectangle
moment = 0.5 * (x1 + x2)
area = (x2 - x1) * y1
elif y1 == 0.0 and y2 != 0.0: # triangle, height y2
moment = 2.0 / 3.0 * (x2-x1) + x1
area = 0.5 * (x2 - x1) * y2
elif y2 == 0.0 and y1 != 0.0: # triangle, height y1
moment = 1.0 / 3.0 * (x2 - x1) + x1
area = 0.5 * (x2 - x1) * y1
else:
moment = (2.0 / 3.0 * (x2-x1) * (y2 + 0.5*y1)) / (y1+y2) + x1
area = 0.5 * (x2 - x1) * (y1 + y2)
sum_moment_area += moment * area
sum_area += area
return sum_moment_area / np.fmax(sum_area,
np.finfo(float).eps).astype(float)
[docs]def dcentroid(x, mfx, x0):
"""
Defuzzification using a differential centroidal method about `x0`.
Parameters
----------
x : 1d array or iterable
Independent variable.
mfx : 1d array or iterable
Fuzzy membership function.
x0 : float
Central value to calculate differential centroid about.
Returns
-------
u : 1d array
Defuzzified result.
See also
--------
skfuzzy.defuzzify.defuzz, skfuzzy.defuzzify.centroid
"""
x = x - x0
return x0 + centroid(x, mfx)
def bisector(x, mfx):
"""
Defuzzification using bisector, or division of the area in two equal parts.
Parameters
----------
x : 1d array, length M
Independent variable
mfx : 1d array, length M
Fuzzy membership function
Returns
-------
u : 1d array, length M
Defuzzified result
See also
--------
skfuzzy.defuzzify.defuzz
"""
'''
As we suppose linearity between each pair of points of x, we can calculate
the exact area of the figure (a triangle or a rectangle).
'''
sum_area = 0.0
accum_area = [0.0] * (len(x) - 1)
# If the membership function is a singleton fuzzy set:
if len(x) == 1:
return x[0]
# else return the sum of moment*area/sum of area
for i in range(1, len(x)):
x1 = x[i - 1]
x2 = x[i]
y1 = mfx[i - 1]
y2 = mfx[i]
# if y1 == y2 == 0.0 or x1==x2: --> rectangle of zero height or width
if not(y1 == y2 == 0. or x1 == x2):
if y1 == y2: # rectangle
area = (x2 - x1) * y1
elif y1 == 0. and y2 != 0.: # triangle, height y2
area = 0.5 * (x2 - x1) * y2
elif y2 == 0. and y1 != 0.: # triangle, height y1
area = 0.5 * (x2 - x1) * y1
else:
area = 0.5 * (x2 - x1) * (y1 + y2)
sum_area += area
accum_area[i - 1] = sum_area
# index to the figure which cointains the x point that divide the area of
# the whole fuzzy set in two
index = np.nonzero(np.array(accum_area) >= sum_area / 2.)[0][0]
# subarea will be the area in the left part of the bisection for this set
if index == 0:
subarea = 0
else:
subarea = accum_area[index - 1]
x1 = x[index]
x2 = x[index + 1]
y1 = mfx[index]
y2 = mfx[index + 1]
# We are interested only in the subarea inside the figure in which the
# bisection is present.
subarea = sum_area/2. - subarea
x2minusx1 = x2 - x1
if y1 == y2: # rectangle
u = subarea/y1 + x1
elif y1 == 0.0 and y2 != 0.0: # triangle, height y2
root = np.sqrt(2. * subarea * x2minusx1 / y2)
u = (x1 + root)
elif y2 == 0.0 and y1 != 0.0: # triangle, height y1
root = np.sqrt(x2minusx1*x2minusx1 - (2.*subarea*x2minusx1/y1))
u = (x2 - root)
else:
m = (y2-y1) / x2minusx1
root = np.sqrt(y1*y1 + 2.0*m*subarea)
u = (x1 - (y1-root) / m)
return u
[docs]def defuzz(x, mfx, mode):
"""
Defuzzification of a membership function, returning a defuzzified value
of the function at x, using various defuzzification methods.
Parameters
----------
x : 1d array or iterable, length N
Independent variable.
mfx : 1d array of iterable, length N
Fuzzy membership function.
mode : string
Controls which defuzzification method will be used.
* 'centroid': Centroid of area
* 'bisector': bisector of area
* 'mom' : mean of maximum
* 'som' : min of maximum
* 'lom' : max of maximum
Returns
-------
u : float or int
Defuzzified result.
See Also
--------
skfuzzy.defuzzify.centroid, skfuzzy.defuzzify.dcentroid
"""
mode = mode.lower()
x = x.ravel()
mfx = mfx.ravel()
n = len(x)
assert n == len(mfx), 'Length of x and fuzzy membership function must be \
identical.'
if 'centroid' in mode or 'bisector' in mode:
zero_truth_degree = mfx.sum() == 0 # Approximation of total area
assert not zero_truth_degree, 'Total area is zero in defuzzification!'
if 'centroid' in mode:
return centroid(x, mfx)
elif 'bisector' in mode:
return bisector(x, mfx)
elif 'mom' in mode:
return np.mean(x[mfx == mfx.max()])
elif 'som' in mode:
tmp = x[mfx == mfx.max()]
return tmp[tmp == np.abs(tmp).min()][0]
elif 'lom' in mode:
tmp = x[mfx == mfx.max()]
return tmp[tmp == np.abs(tmp).max()][0]
else:
raise ValueError('The input for `mode`, %s, was incorrect.' % (mode))
def _interp_universe(x, xmf, mf_val):
"""
Find the universe variable corresponding to membership `mf_val`.
Parameters
----------
x : 1d array
Independent discrete variable vector.
xmf : 1d array
Fuzzy membership function for x. Same length as x.
mf_val : float
Discrete singleton value on membership function mfx.
Returns
-------
x_interp : float
Universe variable value corresponding to `mf_val`.
"""
slope = (xmf[1] - xmf[0]) / float(x[1] - x[0])
x_interp = (mf_val - xmf[0]) / slope
return x_interp
[docs]def lambda_cut_series(x, mfx, n):
"""
Determine a series of lambda-cuts in a sweep from 0+ to 1.0 in n steps.
Parameters
----------
x : 1d array
Universe function for fuzzy membership function mfx.
mfx : 1d array
Fuzzy membership function for x.
n : int
Number of steps.
Returns
-------
z : 2d array, (n, 3)
Lambda cut intevals.
"""
x = np.asarray(x)
mfx = np.asarray(mfx)
step = (mfx.max() - mfx.min()) / float(n - 1)
lambda_cuts = np.arange(mfx.min(), mfx.max() + np.finfo(float).eps, step)
z = np.zeros((n, 3))
z[:, 0] = lambda_cuts.T
z[0, [1, 2]] = _support(x, mfx)
for ii in range(1, n):
xx = _lcutinterval(x, mfx, lambda_cuts[ii])
z[ii, [1, 2]] = xx
return z
def _lcutinterval(x, mfx, lambdacut):
"""
Determine upper & lower interval limits of the lambda-cut for membership
function u(x) [here mfx].
Parameters
----------
x : 1d array
Independent variable.
mfx : 1d array
Fuzzy membership function for x.
lambdacut : float
Value used for lambda-cut.
Returns
-------
z : 1d array
Lambda-cut output.
Notes
-----
Membership function mfx must be convex and monotonic in rise or fall.
"""
z = x[lambdacut - 1e-6 <= mfx]
return np.hstack((z.min(), z.max()))
[docs]def lambda_cut(ms, lcut):
"""
The crisp (binary) lambda-cut set of the membership sequence `ms`
with membership >= `lcut`.
Parameters
----------
ms : 1d array
Fuzzy membership set.
lcut : float
Value used for lambda-cut, on range [0, 1.0].
Returns
-------
mlambda : 1d array
Lambda-cut set of `ms`: ones if ms[i] >= lcut, zeros otherwise.
"""
if lcut == 1:
return (ms >= lcut) * 1
else:
return (ms > lcut) * 1
[docs]def lambda_cut_boundaries(x, mfx, lambdacut):
"""
Find exact boundaries where `mfx` crosses `lambdacut` using interpolation.
Parameters
----------
x : 1d array, length N
Universe variable
mfx : 1d array, length N
Fuzzy membership function
lambdacut : float
Floating point value on range [0, 1].
Returns
-------
boundaries : 1d array
Floating point values of `x` where `mfx` crosses `lambdacut`.
Calculated using linear interpolation.
Notes
-----
The values returned by this function can be thought of as intersections
between a hypothetical horizontal line at ``lambdacut`` and the membership
function ``mfx``. This function assumes the end values of ``mfx`` continue
on forever in positive and negative directions. This means there will NOT
be crossings found exactly at the bounds of ``x`` unless the value of
``mfx`` at the boundary is exactly ``lambdacut``.
"""
# Pad binary set two values by extension
mfxx = pad(mfx, [2, 2], 'edge')
# Find binary lambda cut set
lcutset = lambda_cut(mfxx, lambdacut)
# Detect crossings with convolution, cutting off one padded value
crossings = np.convolve(lcutset, [1, -1])[1:-1]
argcrossings = np.where(np.abs(crossings) > 0)[0] - 1
# Calculate exact crossing points, removing the last padded value
boundaries = []
for cross in argcrossings:
idx = slice(cross - 1, cross + 1)
boundaries.append(
x[cross - 1] + _interp_universe(x[idx], mfx[idx], lambdacut))
# Eliminate degenerate points at peaks with np.unique
return np.unique(np.r_[boundaries])
def _support(x, mfx):
"""
Determine lower & upper limits of the support interval.
Parameters
----------
x : 1d array
Independent variable.
mfx : 1d array
Fuzzy membership function for x; must be convex, continuous,
and monotonic (rise XOR fall).
Returns
-------
z : 1d array, length 2
Interval representing lower & upper limits of the support interval.
"""
apex = mfx.max()
m = np.nonzero(mfx == apex)[0][0]
n = len(x)
xx = x[0:m + 1]
mfxx = mfx[0:m + 1]
z = xx[mfxx == mfxx.min()].max()
xx = x[m:n]
mfxx = mfx[m:n]
return np.r_[z, xx[mfxx == mfxx.min()].min()]
Source