Linear Z-shaped¶

The Linear Z-shaped Membership Function (LinZShapedMF) is a type of membership function in fuzzy logic that represents a smooth linear transition from a full degree of membership (1.0) to zero. Its shape is ideal for modeling concepts like "cold" or "low," where membership is high up to a certain point and then decreases linearly to zero.

The formula for the LinZShapedMF is a piecewise linear function:

$$ \mu(x) = \begin{array}{ll} 1, & x \le a \\[6pt] \frac{b - x}{b - a}, & a < x < b \\[6pt] 0, & x \ge b \end{array} $$

Where $\mu(x)$ is the degree of membership of element $x$ in the fuzzy set.

The function is defined by two parameters that delimit the linear transition region:

• a (Left Shoulder): The point where the membership value begins to transition from 1.0. For input values less than or equal to a, the membership is always 1.0.
• b (Right Foot): The point where the membership value reaches and stays at zero. For input values greater than or equal to b, the membership is always 0.0.

It is crucial that the parameter a is less than b for the linear transition to occur correctly.

Partial Derivatives (Gradients)¶

The partial derivatives are essential for optimizing the parameters a and b in adaptive fuzzy systems.

Derivative with respect to `a`

$$\frac{\partial \mu}{\partial a} = \frac{b - x}{(b - a)^2}$$

Derivative with respect to `b`

$$\frac{\partial \mu}{\partial b} = -\frac{x - a}{(b - a)^2}$$

Python Example¶

In [ ]:

import numpy as np
import matplotlib.pyplot as plt

from anfis_toolbox.membership import LinZShapedMF

mf = LinZShapedMF(a=15, b=35)

x = np.linspace(0, 50, 100)
y = mf.forward(x)

plt.plot(x, y)
plt.show()

import numpy as np import matplotlib.pyplot as plt from anfis_toolbox.membership import LinZShapedMF mf = LinZShapedMF(a=15, b=35) x = np.linspace(0, 50, 100) y = mf.forward(x) plt.plot(x, y) plt.show()

Visualization¶

Below is a comprehensive visualization showing how the LinZShapedMF shape changes with different parameter combinations.