What is fuzzy membership function – A complete guide

Fuzzy membership function is used to convert the crisp input provided to the fuzzy inference system. Fuzzy logic itself is not fuzzy, rather it deals with the fuzziness in the data. And this fuzziness in the data is best described by the fuzzy membership function.

A fuzzy inference system is the core part of any fuzzy logic system. Fuzzification is the first step in Fuzzy Inference System.

Formally, a membership function for a fuzzy set A on the universe of discourse X is defined as µ_A: X → [0, 1], where each element of X is mapped to a value between 0 and 1. This value, called membership value or degree of membership, quantifies the grade of membership of the element in X to the fuzzy set A. Here, X is the universal set and A is the fuzzy set derived from X.

The fuzzy membership function is the graphical way of visualizing the degree of membership of any value in a given fuzzy set. In the graph, X-axis represents the universe of discourse and the Y-axis represents the degree of membership in the range [0, 1]

In the following discussion, we will see various fuzzy membership functions. These functions are mathematically very simple. Fuzzy logic is meant to deal with the fuzziness, so the use of complex membership functions would not add much precision to the final output.

Fuzzy Membership Function:

Singleton membership function:

The Singleton membership function assigns membership value 1 to a particular value of x and assigns value 0 to the rest of all. It is represented by the impulse function as shown.

Mathematically it is formulated as,

Triangular membership function:

This is one of the most widely accepted and used membership functions (MF) in fuzzy controller design. The triangle which fuzzifies the input can be defined by three parameters a, b and c, where c defines the base and b defines the height of the triangle.

Trivial case:

x is between a and b:

x is between b and c:

Combine all together:

We can combine all the above scenarios in single equation as,

Example: Triangular membership function

Determine μ, corresponding to x = 8.0

Thus, x = 8 will be mapped to a fuzzy value of 0.5 using the given triangle fuzzy membership function

Trapezoidal membership function:

The trapezoidal membership function is defined by four parameters: a, b, c and d. Span b to c represents the highest membership value that element can take. And if x is between (a, b) or (c, d), then it will have a membership value between 0 and 1.

There are two special forms of trapezoidal function based on the openness of function. They are known as R-function (Open right) and L-function (Left open). Shape and parameters of both functions are depicted here:

Example: Trapezoidal membership function

Determine μ, corresponding to x = 3.5

Gaussian membership function:

A Gaussian MF is specified by two parameters {m, σ} and can be defined as follows.

In this function, m represents the mean / center of the gaussian curve and σ represents the spread of the curve. This is a more natural way of representing the data distribution, but due to mathematical complexity, it is not much used for fuzzification.

Example: Gaussian membership function

Determine μ corresponding to x = 9, m = 10 and σ = 3.0

Generalized bell-shaped function:

It is also called Cauchy MF. A generalized bell MF is specified by three parameters {a, b, c} and can be defined as follows.

Example: Generalized bell shape membership function

Determine μ corresponding to x = 8

it is called generalized MF, because by changing the parameters a, b and c, we can produce a family of different membership functions.

The function μ(X) =1 / (1 + x² ) can be modelled by setting a = b = 1 and c = 0. Similarly, we can produce other shapes/functions by setting appropriate a, b and c

Sigmoid Membership function:

Sigmoid functions are widely used in classification tasks in machine learning. Specifically, it is used in logistic regression and neural networks, where it suppresses the input and maps it between 0 and 1.

It is controlled by parameters a and c. Where a controls the slope at the crossover point x = c

Example: Sigmoid function

Determine μ corresponding to x = 8

Test Your Knowledge !

1. What is the use of fuzzy membership functions?
2. Which membership function is used in Machine Learning?
3. State the pros and cons of complex fuzzy membership function

Please post your answer / query / feedback in comment section below !