How to design a Fuzzy classifier (inference system) in three hours without knowing the mathematics (part 3/6)

Link for part (1/6): https://medium.com/@reh.yawar2/how-to-design-a-fuzzy-classifier-in-three-hours-without-knowing-the-mathematics-6ff552132bbe

Link for part (2/6): https://medium.com/@reh.yawar2/how-to-design-a-fuzzy-classifier-in-three-hours-without-knowing-the-mathematics-0ff605dffdd9

Link for part (4/6): https://medium.com/@reh.yawar2/how-to-design-a-fuzzy-classifier-in-three-hours-without-knowing-the-mathematics-d1b5d5d2e5f0

Link for part (5/6): https://medium.com/@reh.yawar2/how-to-design-a-fuzzy-classifier-in-three-hours-without-knowing-the-mathematics-part-5-6-9392ac902d45

Link for part (6/6): https://medium.com/@reh.yawar2/how-to-design-a-fuzzy-classifier-inference-system-in-three-hours-without-knowing-the-mathematics-76ef7ff4a0a1

part (3/6)

In this section, we will discuss about a few Fuzzy operations on the fuzzy sets, visually! In the following section all the fuzzy sets have been represented visually. We can also represent a fuzzy set by making a graph. We just need to plot the variable or quantity or any process on the x-axis and its membership values on the y-axis. The membership values would be in the range of [0, 1].

Now lets consider the following membership functions. Height is the variable that has been distributed in the two classes. First class is About 6 feet, and other class is Tall. Both of these classes on the y-axis have been mapped to the fuzzy membership values in the range of [0, 1].

Figure 1: Height membership values

Now lets apply the fuzzy operations that we learned in the previous section. Lets find out the maximum of two fuzzy membership function for both the classes of variable Height. For every value on the x-axis, the maximum fuzzy operation would give the maximum of two membership function values on the y-axis as indicated in the following figure.

Figure 2: Maximum fuzzy operation on the two classes for height values

This maximum fuzzy operation is indicating the ‘or’ operation between the membership values of two classes. The ‘or’ operation or the ‘union’ operation between two sets in the set theory represents the collection of all the elements of those sets. In the fuzzy domain, the ‘or’ operation or the ‘max’ operation also selects the maximum of two value so that all the membership values can be accommodated in the selected range. For example max(0.4, 0.8) would be 0.8. This implies that the membership value of 0.4 can be accommodated in the range of [0, 0.8].

Now lets apply the minimum fuzzy operation on the both classes of same variable Height. For every value on the x-axis, the minimum fuzzy operation would give the minimum of two membership function values on the y-axis as indicated in the following figure.

Figure 3: Minimum fuzzy operation on the two classes for Height variables

This minimum fuzzy operation is indicating the ‘and’ operation between the membership values of two classes. The ‘and’ operation or the ‘intersection’ operation between two sets in the set theory represents the common elements from those sets. In the fuzzy domain, the ‘and’ operation or the ‘min’ operation selects the minimum of two values so that a set of common values can be identified in between two fuzzy sets. For example min(0.4, 0.8) would be 0.4. This implies that the membership value of 0.4 is common in between both the fuzzy sets.

The complement operation in the fuzzy set is achieved by taking the difference of the membership value from 1. This inverts the fuzzy set because in fuzzy sets, the membership values range between [0, 1]. This operation is shown in the following figure.

Figure 4: Complement operation in the fuzzy set

In this figure the complement of fuzzy set FS yields a fuzzy set FS’, which is the inverted version of the original input fuzzy set.

So, until now we have learned about the basic operations and their behavior and outputs on the fuzzy sets. Lets now discuss a few definitions about the fuzzy sets. Understanding of these terms and definition will allow us to visualize the properties of fuzzy functions.

Properties of membership functions

Core: The core of a membership function for a fuzzy set A is defined as the region of the universe that is characterized by complete and full membership in the set A. That is, the core is comprised of elements x of the universe whose membership values are µA(x) = 1.

Support: The support of a membership function for a fuzzy set A is defined as the region of the universe that is characterized by nonzero membership in the set A. That is, the support is comprised of elements x of the universe whose membership values are µA(x) > 0.

Boundaries: The boundaries of a membership function for a fuzzy set A are defined as the region of the universe containing elements that have a nonzero membership but not complete membership. That is, the boundaries are comprised of elements x of the universe whose membership values are 0 < µA(x) < 1.

This can be seen in the following figure.

Figure 5: Core, support, and boundaries of a fuzzy function

In Figure 5, the core is considered to be the values of the variable X that have the membership value equal to 1. Support are those values of the variable X, whose membership values are greater than 0. And the boundary values have membership values greater than 0 but smaller than

Normal Fuzzy set: A normal fuzzy set is one whose membership has at least one element x in the universe whose membership value is unity.

Convex Fuzzy Set: It is a fuzzy set described by a membership function whose values are strictly monotonically increasing, or whose membership values are strictly monotonically decreasing, or whose membership values are strictly monotonically increasing then strictly monotonically decreasing with increasing values for elements in the universe.

Crossover Points: The crossover points of a membership function are defined as the elements in the universe for which a particular fuzzy set A has values equal to 0.5.

Height: The height of a fuzzy set A is the maximum value of the membership function.

In this article we learned about the graphical representation of the fuzzy set. We also learned about the basic fuzzy operations of the fuzzy sets and their visual representation. At last we discussed about some properties of the fuzzy membership function. This is it for this article. If you have any question you may post it in the comment section. I will be glad to answer. The next article in this fuzzy logic series, we will focus on understanding the fuzzification and defuzzification of the membership functions. This will provide us the ground knowledge of developing a fuzzy system. Stay tuned … thank you.

* From the lecture notes of my respected teacher Professor Dr. Mukhtar Ali Unar Sahib