This document provides an overview of adaptive neuro-fuzzy inference systems (ANFIS). It begins by explaining fuzzy sets and fuzzy logic. It then describes the key components of a fuzzy inference system, including fuzzification, a rule base, a database of membership functions, and defuzzification. The document notes that ANFIS integrates neural networks and fuzzy inference by using neural networks to optimize a fuzzy inference system. It outlines the architecture of ANFIS, including its five functional layers that take an input, determine membership functions, apply fuzzy rules, normalize membership functions, and generate an output. Finally, it discusses some benefits of combining fuzzy logic and neural networks.

1. Adaptive
Neuro-Fuzzy
Inference
System

2. A set is an unordered collection of different elements
THE CLASSICAL SET THEORY
Example
A set of all positive integers
A set of all the planets in the solar system
A set of all the states in India

3. MATHEMATICAL REPRESENTATION OF A
CLASSICAL SET
Roster or Tabular Form
Set of vowels in English alphabet, A = {a,e,i,o,u}
Set of odd numbers less than 10, B = {1,3,5,7,9}
Set Builder Form
Example 1 − The set {a,e,i,o,u} is written as
A = {x: x is a vowel in English alphabet}
Example 2 − The set {1,3,5,7,9} is written as
B = {x:1 ≤ x < 10 and (x%2) ≠ 0}

4. Cardinality of a Set
MORE ABOUT CLASSICAL/CRISP SETS
Cardinality of a set S, denoted by |S|, is
the number of elements of the set.
Example
|{1,4,3,5}| = 4

5. Types of Sets
Finite Set
Infinite Set
Subset
Proper Subset
Universal Set
Equal Set
Equivalent Set
Overlapping Set

6. A = [1,2,3,4,5,6]
B = [5,6,7,8,9,0]
U = [1,2,3,4,5,6,7,8,9,0]

7. Operations on Classical Sets
UNION INTERSECTION
AUB = [1,2,3,4,5,6,7,8,9,0] A∩B = [5,6]

8. DIFFERENCE
COMPLEMENT OF A SET
A-B = [1,2,3,4] B-A = [7,8,9,0]
A’ = [7,8,9,0]

9. What is Fuzzy Logic?
FUZZY refers to things which are
NOT CLEAR
VAGUE
UNQUANTIFIABLE
INDEFINITE
ROUGH
SUBJECTIVE

10. Words like young, tall, good, shades of a colour,
high are Fuzzy

11. What are Fuzzy Sets?

12. U =
CRISP SET
[ ]
Distinguish Apples

13. U =
CRISP SET
[ ]
Set (Apples) =
CRISP SET
[ ]
Distinguish Apples
ASSIGN DEGREES OF MEMBERSHIP
[1] [0] [0] [0] [1] [0] [1]

14. Distinguish Red Apples
FUZZY SET

15. Distinguish Red Apples
1
2 3
4
5
6
7
8
9
10

16. Distinguish Red Apples
1
2 3
4
5
6
7
8
9
10
(Assign membership functions)

17. MEMBERSHIP FUNCTION
DEGREE OF BELONGINGNESS
PROBABILITY OF AN ELEMENT BELONGING TO A SET
VARY BETWEEN 0 AND 1
DENOTED BY µ

18. SET (RED APPLES)
1
2 3
4
5
6
7
8
9
10
(0.1)
(0.2)
(0.3)
(0.4)
(0.5)
(0.6)
(0.7)
(0.8)
(0.9)
(1.0)

19. [(1,0.1), (2,0.8), (3,1), (4,0.2), (5,0.5), (6,0.3),
(7,0.6), (8,0.7), (9,0.9), (10,4)]
à =
Representation of a FUZZY SET
= {(x, µ(x)), x є X }
Ã
Here, µ = membership function

20. SOME MORE EXAMPLES
OF
FUZZY SETS
AND
SITUATIONS

21. 20, 40, 60, 80, 100

22. Conventional Logic:
HOT ≥ 80
Membership Functions:
HOT = 1
NOT HOT = 0
Let set of temperatures qualified as Hot be H
Crisp Set Representation:
H = [80, 100]
U= [ 20, 40, 60, 80, 100 ]
[0] [0] [0] [1] [1]

23. Fuzzy Logic:
Hotness defined by degrees of membership
Membership Functions:
20 (NOT HOT) = 0.2
40 (SOMEWHAT HOT) = 0.4
60 (HOT) = 0.6
80 (VERY HOT) = 0.8
100 (EXTREMELY HOT) = 1
Let set of temperatures qualified as Hot be H
Fuzzy Set Representation:
{(x, µ(x)), x є U }
H= [ (20,0.2), (40,0.4), (60,0.6), (80,0.8), (100,1) ]
U= [ 20, 40, 60, 80, 100 ]
[0.2] [0.4] [0.6] [0.8] [1]

25. LINGUISTIC VARIABLE
HEIGHT
WEIGHT
AGE
EDUCATION
LINGUISTIC TERM
TALL, VERY TALL, SHORT,
SOMEWHAT SHORT
HEAVY, LIGHT, VERY HEAVY, VERY
LIGHT
YOUNG, OLD, NOT OLD, VERY
YOUNG
PRIMARY, SECONDARY, 10, 10+2,
GRADUATE, POST GRADUATE, PhD

26. FUZZY INFERENCE SYSTEM
key unit of a fuzzy logic system
having decision making as its
primary work
output from FIS is always a fuzzy
set irrespective of its input which
can be fuzzy or crisp
A de-fuzzification unit would be
there with FIS to convert fuzzy
variables into crisp variables

28. Functional Blocks of Fuzzy
Inference System
Fuzzification Interface Unit − converts
the crisp quantities into fuzzy quantities
Rule Base − contains fuzzy IF-THEN rules
Database − defines the membership
functions of fuzzy sets used in fuzzy rules
Decision-making Unit − performs
operation on rules
De-fuzzification Interface Unit −
converts the fuzzy quantities into crisp
quantities

29. Fuzzification unit - converts the
crisp input into fuzzy input
Knowledge base - collection of
rule base and database is formed
upon the conversion of crisp input
into fuzzy input
De-fuzzification unit - fuzzy
input is finally converted into crisp
output
Working of Fuzzy Inference System

30. Determine the best circulation level
Inputs are the current temperature and moisture level
FUZZY RULES
IF the room is hot THEN circulate the air a lot
IF the room is cool THEN do not circulate the air
IF the room is cool and moist THEN circulate the
air slightly
If an input does not precisely correspond to an IF THEN
rule, partial matching of the input data is used to
interpolate an answer
Fuzzy IF - THEN Rules

31. RULE BASE
Rule 1: IF x is low AND y is low THEN z is high
Rule 2: IF x is low AND y is high THEN z is low
Rule 3: IF x is high AND y is low THEN z is low
Rule 4: IF x is high AND y is high THEN z is high

32. Adaptive Neuro-Fuzzy Inference System
Integration system in which neural networks
are applied to optimize the fuzzy inference
system
Constructs a series of fuzzy if–then rules with
appropriate membership functions
Initial fuzzy rules and membership functions
are first set by using human expertise about
the outputs to be modeled
Then, ANFIS can modify these fuzzy if–then
rules and membership functions to minimize
the output error measure

33. Architecture Of ANFIS

34. Two fuzzy if–then rules are considered
Rule 1: If (x is A1) and (y is B1) then (z1 = p1x + q1y+r1)
Rule 2: If (x is A2) and (y is B2) then (z2 = p2x + q2y+r2)
Where,
x and y are the inputs
Ai and Bi are the fuzzy sets
zi (i = 1,2) are the outputs within the fuzzy region
pi, qi, and ri are the parameters determined during
the training process

35. Layer 1: Input membership function
first layer is used to fuzzificate the inputs, and all
the nodes of this layer are adaptive. Its outputs
are the membership grade of the inputs.
Layer 2: Rule
The nodes of this layer are fixed nodes. They
are labeled with M, which indicates that they
perform as multipliers. The outputs of this
layer represent the fuzzy strengths ωi of each
rule.

36. Layer 3: Normalization
the nodes are also fixed. These nodes are
labeled with N, which means that they
play a normalization role to the fuzzy
strengths from the previous layer. The
normalization factor is computed by the
sum of the weight functions. The outputs
of this layer are called normalized fuzzy
strengths

37. Layer 5: Output
Only one single fixed node, labeled with
S, is in this layer. This node performs the
sum of the incoming signals
Layer 4: Output membership
function
The nodes of this layer are adaptive ones

38. Why use Fuzzy Logic in Neural Network?
Fuzzy logic is largely used to define the
weights, from fuzzy sets, in neural
networks.
When crisp values are not possible to
apply, then fuzzy values are used.
We have already studied that training
and learning help neural networks
perform better in unexpected
situations. At that time fuzzy values
would be more applicable than crisp
values.
When we use fuzzy logic in neural
networks then the values must not be
crisp and the processing can be done in
parallel.