Multilinear

class floulib.Multilinear(*args, label='', color=None)

Bases: Plot

Contains methods to perform operations on multilinear fuzzy subsets.

Note

Multilinear is a subclass of Plot, therefore all methods in Plot may be used.

__init__(*args, label='', color=None)

Constructor

Parameters:

  • *args (Tuple[float, float]) – Several tuple containing the point of the universe of discourse and the grade of membership for this point.

  • label (str, optional) – Label associated with the multilinear fuzzy subset. The default is ‘’.

  • color (matplotlib.colors, optional) – Color associated with the multilinear fuzzy subset. The default is None.

Return type:

None.

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (1.0, 1.0), (1.5, 1.0), (2.0, 0.5), (3.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> A.plot()
_images/Multilinear.__init__.png
cut(alpha)

Computes the alpha-cut of the multilinear fuzzy subset.

Parameters:

alpha (float) – The level.

Returns:

alpha_cut – The alpha-cut.

Return type:

numpy.ndarray

Example

>>> from floulib import Multilinear
>>> alpha = 0.8
>>> A = Multilinear((0.0, 0.0), (1.0, 1.0), (1.5, 1.0), (2.0, 0.5), (3.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> B = Multilinear((0.0, alpha), (4.0, alpha))
>>> A.plot().add_plot(B, linestyle = '--')
>>> print(A.cut(alpha))
[[1.  1.7]
 [2.6 3.6]]
_images/Multilinear.cut.png
kernel()

Returns the kernel of the multilinear fuzzy subset.

Returns:

The kernel.

Return type:

numpy.ndarray

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (1.0, 1.0), (1.5, 1.0), (2.0, 0.5), (3.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> A.plot()
>>> print(A.kernel())
[[1.  1.5]
 [3.  3.5]]
_images/Multilinear.kernel_1.png 
>>> B = Multilinear((0.0, 0.0), (1.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> B.plot()
>>> print(B.kernel())
[[1.  3.5]]
_images/Multilinear.kernel_2.png
max()

Computes the maximum grade of membership.

Returns:

The maximum grade of membership.

Return type:

float

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 0.0), (0.5, 0.3), (1.5, 0.0))
>>> A.plot()
>>> print(A.max())
0.3
_images/Multilinear.max.png
membership(x)

Computes the grade of membership for x.

Parameters:

x (float) – Point where the grade of membership is computed.

Returns:

Grade of membership.

Return type:

float

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (1.0, 1.0), (1.5, 1.0), (2.0, 0.5), (3.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> print(A.membership(1.8))
0.7
mf(x)

Computes the grades of membership for all points in x.

This method can be used as an interface with other libraries.

Parameters:

x (numpy.ndarray) – The array of points.

Returns:

y – The grades of membership array.

Return type:

numpy.ndarray

Example

>>> from floulib import LR
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> A = LR(1, 0.5, 1)
>>> x = np.linspace(0, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, A.mf(x))
_images/Multilinear.mf.png
min()

Computes the minimim grade of membership.

Returns:

The maximum grade of membership.

Return type:

float

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 0.0), (0.5, 0.3), (1.5, 0.0))
>>> (~A).plot()
>>> print((~A).min())
0.7
_images/Multilinear.min.png
necessity(dpi)

Computes the necessity of the multilinear fuzzy subset knowing the distribution of possibility dpi.

Parameters:

dpi (Multilinear) – The possibility distribution.

Raises:

TypeError – Raised if the parameter is not an instance of Multilinear.

Returns:

The necessity.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = LR(1, 0.25, 0.25, label = 'B')
>>> A.plot().add_plot(B)
_images/Multilinear.necessity.png 
>>> print(A.necessity(B))
0.40000000000000036
possibility(dpi)

Computes the possibility of the multilinear fuzzy subset knowing the distribution of possibility dpi.

Parameters:

dpi (Multilinear) – The possibility distribution.

Raises:

TypeError – Raised if the parameter is not an instance of Multilinear.

Returns:

The possibility.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = LR(1, 0.25, 0.25, label = 'B')
>>> A.plot().add_plot(B)
_images/Multilinear.possibility.png 
>>> print(A.possibility(B))
0.6
support()

Returns the support of the multilinear fuzzy subset.

Returns:

The kernel.

Return type:

numpy.ndarray

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (1.0, 1.0), (1.5, 1.0), (2.0, 0.0), (2.5, 0.0), (3.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> A.plot()
>>> print(A.support())
[[0.  2. ]
 [2.5 4. ]]
_images/Multilinear.support_1.png 
>>> B = Multilinear((0.0, 0.0), (1.0, 1.0), (3.5, 1.0), (4.0, 0.0))
>>> B.plot()
>>> print(B.support())
[[0. 4.]]
_images/Multilinear.support_2.png
translate(delta)

Translation over the x-axis all the points whose x-coordinates are greater than the smallest one and smaller than the greatest one.

Parameters:

delta (float) – The value for the translation.

Returns:

The translated multilinear fuzzy subset.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = A.translate(0.5).label('B')
>>> A.plot().add_plot(B)
_images/Multilinear.translate.png

Points (0.5, 1.0) and (1.5, 0.0) are translated by 0.5 but points (0.0, 1.0) and (2.0, 0.0) are not translated.

universe(x)

Sets the universe of discourse.

Parameters:

x (numpy.ndarray) – The universe of discourse.

Returns:

The multilinear fuzzy subset.

Return type:

Multilinear

Certainty(level)

Adds a certainty level to the multilinear fuzzy subset.

This method is generally used with variables in rules. For this reason, it starts with the capital letter C.

Parameters:

level (float) – The certainty level.

Returns:

The multilinear fuzzy subset with certainty level.

Return type:

Multilinear

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (0.5, 0.0), (0.75, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = A.Certainty(0.3).label('A with 0.3 certainty level')
>>> A.plot(nrows = 2).add_plot(B, index = 1)
_images/Multilinear.Certainty.png
Uncertainty(level)

Adds an uncertainty level to the multilinear fuzzy subset.

This method is generally used with variables in rules. For this reason, it starts with the capital letter U.

Parameters:

level (float) – The uncertainty level.

Returns:

The multilinear fuzzy subset with uncertainty level.

Return type:

Multilinear

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 0.0), (0.5, 0.0), (0.75, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = A.Uncertainty(0.3).label('A with 0.3 uncertainty level')
>>> A.plot(nrows = 2).add_plot(B, index = 1)
_images/Multilinear.Uncertainty.png
__and__(other)

Special method for using the operator & as the intersection of two multilinear fuzzy subsets.

Parameters:

other (Multilinear) – The RHS multilinear fuzzy subset.

Raises:

TypeError – Raised if the RHS operand is not an instance of Multilinear.

Returns:

The intersection.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = LR(1, 0.25, 0.25, label = 'B')
>>> C = (A & B).label('C = A & B')
>>> A.plot(nrows = 2).add_plot(B).add_plot(C, index = 1)
_images/Multilinear.__and__.png
__call__(x)

Special method to transform a multilinear fuzzy subset into a discrete fuzzy subsets.

Parameters:

x (numpy.ndarray) – The universe of discourse on which the transformation is performed.

Raises:

TypeError – Raised if the parameter is not an instance of numpy.ndarray.

Returns:

The discrete fuzzy subset.

Return type:

Discrete

Example

>>> from floulib import Multilinear
>>> import numpy as np
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0))
>>> C = A(np.linspace(0, 2, 50))
>>> A.plot(nrows = 2).add_plot(C, index = 1)
_images/Multilinear.__call__.png
__invert__()

Special method for using the unary operator ~ as the complement of a multlinear fuzzy subsets.

Returns:

The complement.

Return type:

Multilinear

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = (~A).label('B = ~Ã')
>>> A.plot().add_plot(B)
_images/Multilinear.__invert__.png
__neg__()

Special method for using the unary operator - as the opposite of a multlinear fuzzy subsets.

Returns:

The opposite.

Return type:

Multilinear

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0))
>>> C = (-A ).label('C = -A')
>>> C.plot()
_images/Multilinear.__neg__.png
__or__(other)

Special method for using the operator | as the union of two multilinear fuzzy subsets.

Parameters:

other (Multilinear) – The RHS multilinear fuzzy subset.

Raises:

TypeError – Raised if the RHS operand is not an instance of Multilinear.

Returns:

The union.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = LR(1, 0.25, 0.25, label = 'B')
>>> C = (A | B).label('C = A | B')
>>> A.plot(nrows = 2).add_plot(B).add_plot(C, index = 1)
_images/Multilinear.__or__.png
__rmul__(other)

Special method for using the operator * with a number as the LHS operand. Its multiplies the grades of membership of the multilinear fuzzy subset by the LHS operand and truncates them to 1.

Parameters:

other (int | float) – The LHS operand.

Raises:

TypeError – Raised the the LHS in not an instance of int or float.

Returns:

The result.

Return type:

Multilinear

Example

>>> from floulib import Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> C = (1.5*A).label('C')
>>> A.plot(nrows = 2).add_plot(C, index = 1)
_images/Multilinear.__rmul__.png
__xor__(other)

Special method for using the operator ^ as the symetric difference of two multilinear fuzzy subsets.

Parameters:

other (Multilinear) – The other multilinear fuzzy subset.

Raises:

TypeError – Raised if other is not an instance of Multilinear.

Returns:

The symetric difference.

Return type:

Multilinear

Example

>>> from floulib import LR, Multilinear
>>> A = Multilinear((0.0, 1.0), (0.5, 1.0), (1.5, 0.0), (2.0, 0.0), label = 'A')
>>> B = LR(1, 0.25, 0.25, label = 'B')
>>> C = (A ^ B).label('C = A ^ B')
>>> A.plot(nrows = 2).add_plot(B).add_plot(C, index = 1)
_images/Multilinear.__xor__.png