Machine Learning

1. Computational Intelligence
Parveen Malik
Assistant Professor
School of Electronics Engineering
KIIT University

2. Outline
• Soft Computing Constituents
• Fuzzy Set Theory
• Set Theoretic Operations
• Two Dimensional MFs

3. Soft Computing Constituents
• Neural Network
• Fuzzy Set Theory
• Derivative Free Optimization( Genetic Algorithms)
• Conventional AI
Text Book : Chuen-Tsai Sun, Eiji Mizutani, and Jyh-Shing Roger Jang. “Neuro-Fuzzy and Soft Computing: A
Computational Approach to Learning and Machine Intelligence”

4. Hidden Layer
Output Layer
Input Layer
Neural Network

5. Fuzzy Set Theory
Degree
of
Belief
Fuzzy
Hot
Cold
Degree
of
Belief
Non-Fuzzy (Crisp)
1.0
0.0
Cold
Hot
1.0
0.0
𝟒𝟏𝟎𝑪
𝟑𝟗𝟎𝑪
𝟒𝟎𝟎
𝑪
𝟐𝟎𝟎𝑪
𝟏𝟐𝟎𝑪
𝟑𝟎𝟎𝑪
𝟐𝟓𝟎
𝑪
𝟒𝟏𝟎𝑪
𝟑𝟗𝟎𝑪
𝟒𝟎𝟎
𝑪
𝟐𝟎𝟎𝑪
𝟏𝟐𝟎𝑪
𝟑𝟎𝟎𝑪
𝟐𝟓𝟎
𝑪

6. History of Conventional AI approaches

7. Fuzzy Set Theory Vs Classical Set Theory
Classical Set Theory :
• Crisp, clear or unambiguous boundary
𝐴 = 𝑥 𝑥 > 6}
• Important tool for mathematic and computer science.
• Sharp Transition between “Inclusion” and Exclusion”.
• Incapable of addressing human concepts and thoughts
which tend to be abstract and imprecise.
A=“Tall Person” x =“Height”
B= Water is hot” C=“Temp. is High”

8. Fuzzy Set Theory Vs Classical Set Theory
Fuzzy Set Theory :
• Introduced by “L.A.Zadeh” in 1965 paper “Fuzzy Sets”
• Unclear or ambiguous boundary
• Capable of addressing human concepts and thoughts which
tend to be abstract and imprecise.
A=“Tall Person” x =“Height”
B= Water is hot” C=“Temp. is High”
• Introduction of membership function
• Gradual Transition from “Belong to a set” To “Not belong to
set”.

9. Fuzzy Set Theory
Definition 1.1 : Fuzzy Set and membership functions
If X is a collection of objects denoted generically by x, then a fuzzy set A in X
is defined as a set of ordered pairs:
A= 𝒙, 𝝁𝑨(𝒙) | 𝒙 ∈ 𝑿
Where 𝜇𝐴(𝑥) is the membership function for the fuzzy set A
Definition 1.2 : support
The support of A fuzzy set A is the set of all points x in X such that 𝜇𝐴(𝑥) >0
support(A)= 𝒙 | 𝝁𝑨 𝒙 > 𝟎
Definition 1.3 : core
The core of A fuzzy set A is the set of all points x in X such that 𝜇𝐴(𝑥) =1
core(A)= 𝒙|𝝁𝑨 𝒙 = 𝟎
Definition 1.4 : Normality
A fuzzy set A is normal if its core is non-empty.

10. Fuzzy Set Theory
Definition 1.5 : Cross Over Points
A crossover point of a fuzzy set A is a point 𝑥 ∈ 𝑋 at which 𝜇𝐴(𝑥) = 0.5.
crossover(A)= 𝒙|𝝁𝑨 𝒙 = 𝟎. 𝟓
Where 𝜇𝐴(𝑥) is the membership function for the fuzzy set A.
Definition 1.6 : Fuzzy singleton
A fuzzy set whose support is a single point in X with 𝜇𝐴(𝑥) = 1 is called a Fuzzy
Singleton.
Definition 1.7 : α-cut, strong α-cut
α-cut, α-level set of a fuzzy set A is crisp set defined by
𝑨𝜶= 𝒙|𝝁𝑨 𝒙 ≥ 𝟎
Strong α-cut, strong α-level set of a fuzzy set A is defined by
𝑨𝜶
′
= 𝒙|𝝁𝑨 𝒙 > 𝟎

11. Fuzzy Set Theory
Definition 1.8 : Convexity
A fuzzy set A is convex if and only if for any 𝑥1, 𝑥2 ∈ 𝑋 𝑎𝑛𝑑 for any λ ∈ 0,1
𝝁𝑨(𝝀𝒙𝟏 + (𝟏 − 𝝀)𝒙𝟐) ≥ Min 𝝁𝑨 𝒙𝟏 , 𝝁𝑨 𝒙𝟐
Alternatively, A is convex if all its 𝛼 − 𝑠𝑒𝑡𝑠 are convex.
Definition 1.9 : Fuzzy numbers
A fuzzy number A is a fuzzy set in the real line that satisfies the conditions for
Normality and convexity.
Definition 1.10 : Bandwidths of normal and convex fuzzy sets
For a normal and convex fuzzy set, the bandwidth or width is defined as the distance
between the two unique crossover points:
𝒘𝒊𝒅𝒕𝒉 𝑨 = 𝒙𝟐 − 𝒙𝟏
Where 𝝁𝑨 𝒙𝟏 = 𝝁𝑨(𝒙𝟐)=0.5

12. Fuzzy Set Theory
Definition 1.11 : Symmetry
A fuzzy set A is symmetric if its MF is symmetric around a certain
point x=c ,
𝝁𝑨 𝒄 + 𝒙 = 𝝁𝑨 𝒄 − 𝒙 for all 𝒙 ∈ 𝑿
Definition 1.12 : Open left, Open right, closed
A fuzzy set A is open left
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =1 and 𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 = 0
open right
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =0 and 𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 = 1
Closed
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 =0

13. Set Theoretic Operations
Definition 2.1 : Containment or Subset
Fuzzy set A is contained in Fuzzy set B if and only if
𝑨 ⊆ 𝑩 ⇔ 𝝁𝑨 𝒙 ≤ 𝝁𝑩 𝒙
Definition 2.2 : Union (Disjunction)
The union of two sets A and B is a set C (𝐴 ∪ B) whose MF is
related by
𝝁 𝒙 = max {𝝁𝑨 𝒙 , 𝝁𝑩 𝒙 } = 𝝁𝑨 𝒙 ˅ 𝝁𝑩 𝒙
Definition 2.3 : Intersection (Conjunction)
The intersection of two sets A and B is a set C (𝐴 ∩ B) whose MF
is related by
𝝁 𝒙 = min {𝝁𝑨 𝒙 , 𝝁𝑩 𝒙 } = 𝝁𝑨 𝒙 ˄ 𝝁𝑩 𝒙

14. Set Theoretic Operations
Definition 2.4 : Complement (Negation)
The complement of Fuzzy set A is ഥ
𝐀 ¬𝐀 𝐨𝐫 𝐍𝐎𝐓 𝐀
𝝁ഥ
𝑨 𝒙 = 𝟏 − 𝝁𝑨 𝒙
Definition 2.5 : Cartesian product and co-product
Let A and B be fuzzy sets in X and Y respectively. The Cartesian
product of A and B denoted by 𝑨 × 𝑩 ,
𝝁𝑨×𝑩 𝒙, 𝒚 = min {𝝁𝑨 𝒙 , 𝝁𝑩 𝒚 }
The Co- product of A and B denoted by A + B,
𝝁𝑨+𝑩 𝒙, 𝒚 = max {𝝁𝑨 𝒙 , 𝝁𝑩 𝒚 }

15. Two Dimensional MFs
Definition 3.1 : Cylindrical extensions of 1D fuzzy sets
If A is a fuzzy set in X, then its cylindrical extension in 𝑿 × 𝒀 is a
fuzzy set c(A) defined by
𝒄(𝑨) = න
𝑿×𝒀
𝝁𝑨(𝒙)/(𝒙, 𝒚)

16. Two Dimensional MFs
Definition 3.1 : Projections of fuzzy sets
If R be a two-dimensional fuzzy set on 𝑿 × 𝒀. The projections of R onto X and Y are
defined as
𝑹𝑿 = න
𝑋
𝒎𝒂𝒙 {𝝁𝑹(𝒙, 𝒚)} /𝒙
𝑹𝒀 = න
𝑌
𝒎𝒂𝒙 {𝝁𝑹(𝒙, 𝒚)} /𝒚