ZShape

ZShape

Bases: Term

Edge Term that represents the ZShape membership function.

Equation

\(\mu(x) = \begin{cases} 1 & \mbox{if } x \leq s \cr h - 2h\left(\dfrac{x - s}{e-s}\right)^2 & \mbox{if } s < x < \dfrac{s+e}{2} \cr 2h \left(\dfrac{x - e}{e-s}\right)^2 & \mbox{if } \dfrac{s+e}{2} \le x < e\cr 0 & \mbox{otherwise} \end{cases}\)

where

• \(h\): height of the Term
• \(s\): start of the ZShape
• \(e\): end of the ZShape

Attributes

end instance-attribute

end = end

start instance-attribute

start = start

Functions

__init__

__init__(name: str = '', start: float = nan, end: float = nan, height: float = 1.0) -> None

Constructor.

Parameters:

Name	Type	Description	Default
name	str	name of the Term	''
start	float	start of the ZShape	nan
end	float	end of the ZShape	nan
height	float	height of the Term	1.0

configure

configure(parameters: str) -> None

Configure the term with the parameters.

Parameters:

Name	Type	Description	Default
parameters	str	start end [height].	required

is_monotonic

is_monotonic() -> bool

Return True because the term is monotonic.

Returns:

Type	Description
bool	True

membership

membership(x: Scalar) -> Scalar

Computes the membership function evaluated at \(x\).

Parameters:

Name	Type	Description	Default
x	Scalar	scalar	required

Returns:

Type	Description
Scalar	\(\mu(x) = \begin{cases} 1 & \mbox{if } x \leq s \cr h - 2h\left(\dfrac{x - s}{e-s}\right)^2 & \mbox{if } s < x < \dfrac{s+e}{2} \cr 2h \left(\dfrac{x - e}{e-s}\right)^2 & \mbox{if } \dfrac{s+e}{2} \le x < e\cr 0 & \mbox{otherwise} \end{cases}\)

parameters

parameters() -> str

Return the parameters of the term.

Returns:

Type	Description
str	start end [height].

tsukamoto

tsukamoto(y: Scalar) -> Scalar

Compute the tsukamoto value of the monotonic term for activation degree \(y\).

Equation

\(y = \begin{cases} 1 & \mbox{if } x \leq s \cr h - 2h\left(\dfrac{x - s}{e-s}\right)^2 & \mbox{if } s < x < \dfrac{s+e}{2} \cr 2h \left(\dfrac{x - e}{e-s}\right)^2 & \mbox{if } \dfrac{s+e}{2} \le x < e\cr 0 & \mbox{otherwise} \end{cases}\)

\(x = \begin{cases} e + (e-s) \sqrt{\dfrac{y}{2h}} & \mbox{if } y \le \dfrac{h}{2} \cr s + (e-s) \sqrt{\dfrac{h-y}{2h}} & \mbox{otherwise} \end{cases}\)

Parameters:

Name	Type	Description	Default
y	Scalar	activation degree	required

Returns:

Type	Description
Scalar	\(x = \begin{cases} e + (e-s) \sqrt{\dfrac{y}{2h}} & \mbox{if } y \le \dfrac{h}{2} \cr s + (e-s) \sqrt{\dfrac{h-y}{2h}} & \mbox{otherwise} \end{cases}\)