"""
intervalops.py : Functions for proper mathematical treatment of intervals.
"""
import numpy as np
from ..defuzzify import lambda_cut_series
[docs]def addval(interval1, interval2):
"""
Add intervals interval1 and interval2.
Parameters
----------
interval1 : 2-element iterable
First interval set.
interval2 : 2-element iterable
Second interval set.
Returns
-------
Z : 2-element array
Sum of interval1 and interval2, defined as::
Z = interval1 + interval2 = [a + c, b + d]
"""
# Handle arrays
if not isinstance(interval1, np.ndarray):
interval1 = np.asarray(interval1)
if not isinstance(interval2, np.ndarray):
interval2 = np.asarray(interval2)
try:
return np.r_[interval1] + np.r_[interval2]
except Exception:
return interval1 + interval2
[docs]def divval(interval1, interval2):
"""
Divide ``interval2`` into ``interval1``, by inversion and multiplication.
Parameters
----------
interval1 : 2-element iterable
First interval set.
interval2 : 2-element iterable
Second interval set.
Returns
-------
z : 2-element array
Interval result of interval1 / interval2.
"""
# Handle arrays
if not isinstance(interval1, np.ndarray):
interval1 = np.asarray(interval1)
if not isinstance(interval2, np.ndarray):
interval2 = np.asarray(interval2)
# Invert interval2 and multiply
interval2 = 1. / interval2
return multval(interval1, interval2)
[docs]def dsw_add(x, mfx, y, mfy, n):
"""
Add two fuzzy variables together using the restricted DSW method [1].
Parameters
----------
x : 1d array
Universe for first fuzzy variable.
mfx : 1d array
Fuzzy membership for universe ``x``. Must be convex.
y : 1d array
Universe for second fuzzy variable.
mfy : 1d array
Fuzzy membership for universe ``y``. Must be convex.
n : int
Number of lambda-cuts to use; a higher number will have greater
resolution toward the limit imposed by input sets ``x`` and ``y``.
Returns
-------
z : 1d array
Output universe variable.
mfz : 1d array
Output fuzzy membership on universe ``z``.
Notes
-----
The Dong, Shah, and Wong (DSW) method requires convex fuzzy membership
functions. The ``dsw_*`` functions return results similar to Matplotlib's
``fuzarith`` function.
References
----------
.. [1] W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and
decision analysis, Civ Eng Syst, 2, 1985, pp 201-208.
"""
# Restricted DSW w/n lambda cuts
x = lambda_cut_series(x, mfx, n)
y = lambda_cut_series(y, mfy, n)
n1, n2 = x.shape
ff = np.zeros((n1, n2))
ff[:, 0] = x[:, 0]
# Compute F = x + y
for n in range(n1):
ff[n, [1, 2]] = addval(x[n, [1, 2]], y[n, [1, 2]])
# Arrange for output or plotting
out = np.zeros((2 * n1, 2))
out[0:n1, 1] = ff[:, 0]
out[n1:2 * n1, 1] = np.flipud(ff[:, 0])
out[0:n1, 0] = ff[:, 1]
out[n1:2 * n1, 0] = np.flipud(ff[:, 2])
# No need for transposes; rank-1 arrays have no transpose in Python
return out[:, 0], out[:, 1]
[docs]def dsw_div(x, mfx, y, mfy, n):
"""
Divide one fuzzy variable by another using the restricted DSW method [1].
Parameters
----------
x : 1d array
Universe for first fuzzy variable.
mfx : 1d array
Fuzzy membership for universe ``x``. Must be convex.
y : 1d array
Universe for second fuzzy variable.
mfy : 1d array
Fuzzy membership for universe ``y``. Must be convex.
n : int
Number of lambda-cuts to use; a higher number will have greater
resolution toward the limit imposed by input sets ``x`` and ``y``.
Returns
-------
z : 1d array
Output universe variable.
mfz : 1d array
Output fuzzy membership on universe ``z``.
Notes
-----
The Dong, Shah, and Wong (DSW) method requires convex fuzzy membership
functions. The ``dsw_*`` functions return results similar to Matplotlib's
``fuzarith`` function.
References
----------
.. [1] W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and
decision analysis, Civ Eng Syst, 2, 1985, pp 201-208.
"""
# Restricted DSW w/n lambda cuts
x = lambda_cut_series(x, mfx, n)
y = lambda_cut_series(y, mfy, n)
n1, n2 = x.shape
ff = np.zeros((n1, n2))
ff[:, 0] = x[:, 0]
# Compute F = x / y
for n in range(n1):
ff[n, [1, 2]] = divval(x[n, [1, 2]], y[n, [1, 2]])
# Arrange for output or plotting
out = np.zeros((2 * n1, 2))
out[0:n1, 1] = ff[:, 0]
out[n1:2 * n1, 1] = np.flipud(ff[:, 0])
out[0:n1, 0] = ff[:, 1]
out[n1:2 * n1, 0] = np.flipud(ff[:, 2])
# No need for transposes; rank-1 arrays have no transpose in Python
return out[:, 0], out[:, 1]
[docs]def dsw_mult(x, mfx, y, mfy, n):
"""
Multiply two fuzzy variables using the restricted DSW method [1].
Parameters
----------
x : 1d array
Universe for first fuzzy variable.
mfx : 1d array
Fuzzy membership for universe ``x``. Must be convex.
y : 1d array
Universe for second fuzzy variable.
mfy : 1d array
Fuzzy membership for universe ``y``. Must be convex.
n : int
Number of lambda-cuts to use; a higher number will have greater
resolution toward the limit imposed by input sets ``x`` and ``y``.
Returns
-------
z : 1d array
Output universe variable.
mfz : 1d array
Output fuzzy membership on universe ``z``.
Notes
-----
The Dong, Shah, and Wong (DSW) method requires convex fuzzy membership
functions. The ``dsw_*`` functions return results similar to Matplotlib's
``fuzarith`` function.
References
----------
.. [1] W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and
decision analysis, Civ Eng Syst, 2, 1985, pp 201-208.
"""
# Restricted DSW w/n lambda cuts
x = lambda_cut_series(x, mfx, n)
y = lambda_cut_series(y, mfy, n)
n1, n2 = x.shape
ff = np.zeros((n1, n2))
ff[:, 0] = x[:, 0]
# Compute F = x * y
for n in range(n1):
ff[n, [1, 2]] = multval(x[n, [1, 2]], y[n, [1, 2]])
# Arrange for output or plotting
out = np.zeros((2 * n1, 2))
out[0:n1, 1] = ff[:, 0]
out[n1:2 * n1, 1] = np.flipud(ff[:, 0])
out[0:n1, 0] = ff[:, 1]
out[n1:2 * n1, 0] = np.flipud(ff[:, 2])
# No need for transposes; rank-1 arrays have no transpose in Python
return out[:, 0], out[:, 1]
[docs]def dsw_sub(x, mfx, y, mfy, n):
"""
Subtract a fuzzy variable from another by the restricted DSW method [1].
Parameters
----------
x : 1d array
Universe for first fuzzy variable.
mfx : 1d array
Fuzzy membership for universe ``x``. Must be convex.
y : 1d array
Universe for second fuzzy variable, which will be subtracted from
``x``.
mfy : 1d array
Fuzzy membership for universe ``y``. Must be convex.
n : int
Number of lambda-cuts to use; a higher number will have greater
resolution toward the limit imposed by input sets ``x`` and ``y``.
Returns
-------
z : 1d array
Output universe variable.
mfz : 1d array
Output fuzzy membership on universe ``z``.
Notes
-----
The Dong, Shah, and Wong (DSW) method requires convex fuzzy membership
functions. The ``dsw_*`` functions return results similar to Matplotlib's
``fuzarith`` function.
References
----------
.. [1] W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and
decision analysis, Civ Eng Syst, 2, 1985, pp 201-208.
"""
# Restricted DSW w/n lambda cuts
x = lambda_cut_series(x, mfx, n)
y = lambda_cut_series(y, mfy, n)
n1, n2 = x.shape
ff = np.zeros((n1, n2))
ff[:, 0] = x[:, 0]
# Compute F = x - y
for n in range(n1):
ff[n, [1, 2]] = subval(x[n, [1, 2]], y[n, [1, 2]])
# Arrange for output or plotting
out = np.zeros((2 * n1, 2))
out[0:n1, 1] = ff[:, 0]
out[n1:2 * n1, 1] = np.flipud(ff[:, 0])
out[0:n1, 0] = ff[:, 1]
out[n1:2 * n1, 0] = np.flipud(ff[:, 2])
# No need for transposes; rank-1 arrays have no transpose in Python
return out[:, 0], out[:, 1]
[docs]def multval(interval1, interval2):
"""
Multiply intervals interval1 and interval2.
Parameters
----------
interval1 : 1d array, length 2
First interval.
interval2 : 1d array, length 2
Second interval.
Returns
-------
z : 1d array, length 2
Interval resulting from multiplication of interval1 and interval2.
"""
# Handle arrays
if not isinstance(interval1, np.ndarray):
interval1 = np.asarray(interval1)
if not isinstance(interval2, np.ndarray):
interval2 = np.asarray(interval2)
try:
crosses = np.r_[interval1[0] * interval2[0],
interval1[0] * interval2[1],
interval1[1] * interval2[0],
interval1[1] * interval2[1]]
return np.r_[crosses.min(), crosses.max()]
except Exception:
return interval1 * interval2
[docs]def scaleval(q, interval):
"""
Multiply scalar q with interval ``interval``.
Parameters
----------
q : float
Scalar to multiply interval with.
interval : 1d array, length 2
Interval. Must have exactly two elements.
Returns
-------
z : 1d array, length 2
New interval; z = q x interval.
"""
# Handle array
if not isinstance(interval, np.ndarray):
interval = np.asarray(interval)
try:
return np.r_[min(q * interval[0], q * interval[1]),
max(q * interval[0], q * interval[1])]
except Exception:
return q * interval
[docs]def subval(interval1, interval2):
"""
Subtract interval interval2 from interval interval1.
Parameters
----------
interval1 : 1d array, length 2
First interval.
interval2 : 1d array, length 2
Second interval.
Returns
-------
Z : 1d array, length 2
Resultant subtracted interval.
"""
# Handle arrays
if not isinstance(interval1, np.ndarray):
interval1 = np.asarray(interval1)
if not isinstance(interval2, np.ndarray):
interval2 = np.asarray(interval2)
try:
return np.r_[interval1[0] - interval2[1], interval1[1] - interval2[0]]
except Exception:
return interval1 - interval2
Source