Artificial Intelligence lecture notes. AI summarized notes on uncertainty and handling it through fuzzy logic, tipping problem scenarios are seen in it, for reading and may be for self-learning, I think.

1. Handling Uncertainty with
Fuzzy Logic
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2. Outline
• Uncertainty and imprecisions
• Introduction to Fuzzy Logic
• Fuzzy Sets
• Membership Functions
• Logical Operations
• Rules
• Fuzzy Inference Process
• Example
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17. 17

18. Modeling Real World
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19. Fuzzy Logic
• Fuzzy logic is superset of conventional (Boolean) logic that has been
extended to handle the concept of partial truth.
• Central notion of fuzzy systems is that truth values (in fuzzy logic) or
membership values (in fuzzy sets) are indicated by a value on the
range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0
representing absolute Truth.
• Deals with real world vagueness
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20. Fuzzy Logic
• Fuzzy logic is all about the relative importance of precision: How
important is it to be exactly right when a rough answer will do?
`
• Vague world:
• Medical diagnosis is an excellent example of vagueness and uncertainty.
• Applications: Medical instrumentation, decision-support systems
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21. Classical Sets
Law of the Excluded Middle:
• any element X, must be either in set A or in set not-A.
• It cannot be in both.
• And these two sets, set A and set not-A should contain the entire universe
between them.
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22. Classical Sets
 X = universe of discourse = the set of all objects with the same characteristics.
 Let nx = cardinality = total number of elements in X.
 For crisp sets A and B in X, we define:
 x A  x belongs to A.
 x  A  x does not belong to A.
 For sets A and B on X:
 A  B  xA, xB.
 A  B  A is fully contained in B.
 A = B  A  B and B  A.
 The null set, , contains no elements.
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23. Operations on Classical Sets
 Union:
 AB = {x | x  A or x  B}.
 Intersection:
 AB = {x | x  A and x  B}.
 Complement:
 Ac = {x | x  A, x  X}.
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24. Fuzzy Sets
• Fuzzy sets, unlike classical sets, do not restrict themselves to
something lying wholly in either set A or in set not-A.
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25. Fuzzy Sets
 Transition between membership and non-membership can be gradual.
 Fuzzy set contains elements which have varying degrees of
membership.
 Degree of membership measured by a function.
 Function maps elements to a real numbered value on the interval 0 to 1,
A[0,1].
 Elements in a fuzzy set can also be members of other fuzzy sets on the
same universe.
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26. Fuzzy Sets
• Classical set theory
– An object is either in or not in the set
• Sets with smooth boundary
– Not completely in or out – somebody 6” is 80% tall
• Fuzzy set theory
– An object is in a set by matter of degree
– 1.0 => in the set
– 0.0 => not in the set
– 0.0 < object < 1.0 => partially in the set
• Provides a way to write symbolic rules with terms like “medium” but
evaluate them in a quantified way
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27. Fuzzy logic represents partial truth
• In fuzzy logic, the truth of any statement becomes matter of degree
-- Q: Is Saturday a weekend day?
-- A: 1 (yes, or true)
– Q: Is Tuesday a weekend day?
– A: 0 (no, or false)
– Q: Is Friday a weekend day?
– A: 0.7 (for the most part yes, but not completely)
– Q: Is Sunday a weekend day?
– A: 0.9 (yes, but not quite as much as Saturday)
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28. Membership Function
• A membership function (MF) is a curve that defines how each point in
the input space is mapped to a membership value (or degree of
membership) between 0 and 1.
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29. Membership Function
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30. Membership Functions
• The only condition a membership function must really satisfy is that it
must vary between 0 and 1.
• The function itself can be an arbitrary curve whose shape we can
define as a function that suits us from the point of view of simplicity,
convenience, speed, and efficiency.
• A classical set might be expressed as
A = {x | x > 6}
• If X is the universe of discourse and its elements are denoted by x,
then a fuzzy set A in X is defined as a set of ordered pairs.
A = {x, µA(x) | x ∈ X}
• µA(x) is called the membership function (or MF) of x in A.
• The membership function maps each element of X to a membership
value between 0 and 1.
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31. Membership Functions
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32. Membership Functions
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33. Summary of Membership Functions
• Fuzzy sets describe vague concepts (e.g., fast runner, hot weather,
weekend days).
• A fuzzy set admits the possibility of partial membership in it. (e.g.,
Friday is sort of a weekend day, the weather is rather hot).
• The degree an object belongs to a fuzzy set is denoted by a
membership value between 0 and 1. (e.g., Friday is a weekend day to
the degree 0.8).
• A membership function associated with a given fuzzy set maps an
input value to its appropriate membership value.
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34. Example Fuzzy Variable
• Each function tells us how
much we consider a
character in the set if it has
a particular aggressiveness
value
• Or, how much truth to
attribute to the statement:
“The character is nasty (or
meek, or neither)?”
Aggressiveness
Membership
(Degree of Truth)
1.0
0.0 10.5
Medium
NastyMeek
-1 -0.5
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35. Fuzzy vs. probability
• both operate over the same range [0.0 to 1.0].
• In probability theory:
There is a 20% chance that Amber belongs to the set of old people,
there’s an 80% chance that she doesn’t belong to the set of old people.
• In fuzzy terminology:
Amber is definitely not old or some other term corresponding to the
value 0.2. But there are certainly no chances involved, no guess work
left for the system to classify Amber as young or old.
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36. Fuzzy vs. probability
• Crisp Facts – distinct boundaries
• Fuzzy Facts – imprecise boundaries
• Probability - incomplete facts
• Example – Scout reporting an enemy
– “Two tanks at grid NV 123456“ (Crisp)
– “A few tanks at grid NV 123456” (Fuzzy)
– “There might be 2 tanks at grid NV 54 (Probabilistic)
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37. Logical and Fuzzy Operators
• Logical operators
• Fuzzy Operators
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38. Logical and Fuzzy Operators
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39. The Basic Tipping Problem
• Given a number between 0 and 10 that represents the quality of service at
a restaurant (where 10 is excellent), and another number between 0 and 10
that represents the quality of the food at that restaurant (again, 10 is
excellent), what should the tip be?
• The three golden rules of tipping:
If the service is poor or the food is rancid, then tip is cheap.
If the service is good, then tip is average.
If the service is excellent or the food is delicious, then tip is generous.
• Assume that an average tip is 15%, a generous tip is 25%, and a cheap tip
is 5%.
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40. Fuzzy Rules (If-Then Rules)
• Fuzzy sets and fuzzy operators are the subjects and verbs
of fuzzy logic.
• if-then rule statements are used to formulate the conditional
statements that comprise fuzzy logic.
• A single fuzzy if-then rule assumes the form
if x is A then y is B
where A and B are linguistic values defined by fuzzy sets on the ranges
(universes of discourse) X and Y, respectively.
The if-part of the rule "x is A" is called the antecedent or premise, while
the then-part of the rule "y is B" is called the consequent or conclusion.
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41. Fuzzy Rules (If-Then Rules)
• Example: If service is good then tip is average
• the input to an if-then rule is the current value for the input variable (in this case,
service) and the output is an entire fuzzy set (in this case, average). This set will
be defuzzified, assigning one value to the output.
• Interpreting an if-then rule involves distinct parts:
• evaluating the antecedent (which involves fuzzifying the input and applying any
necessary fuzzy operators) and
• applying that result to the consequent (known as implication).
• In binary logic: p → q (p and q are either both true or both false.)
• In fuzzy logic: 0.5 p → 0.5 q (Partial antecedents provide partial implication.)
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42. Fuzzy Rules (If-Then Rules)
• The consequent specifies a fuzzy set be assigned to the output.
• The implication function then modifies that fuzzy set to the degree specified by
the antecedent.
• The most common ways to modify the output fuzzy set are truncation using the
min function (where the fuzzy set is truncated) or scaling using the prod
function (where the output fuzzy set is squashed).
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43. Fuzzy Rules (If-Then Rules)
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44. Summary of If-Then Rules
Interpreting if-then rules is a three-part process.
• Fuzzify inputs:
• Resolve all fuzzy statements in the antecedent to a degree of membership between 0 and 1.
• If there is only one part to the antecedent, then this is the degree of support for the rule.
• Apply fuzzy operator to multiple part antecedents:
• If there are multiple parts to the antecedent, apply fuzzy logic operators and resolve the
antecedent to a single number between 0 and 1.
• This is the degree of support for the rule.
• Apply implication method:
• Use the degree of support for the entire rule to shape the output fuzzy set.
• The consequent of a fuzzy rule assigns an entire fuzzy set to the output.
• This fuzzy set is represented by a membership function that is chosen to indicate the
qualities of the consequent.
• If the antecedent is only partially true, (i.e., is assigned a value less than 1), then the output
fuzzy set is truncated according to the implication method.
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45. Steps by Step Approach
• The above steps are summarized into three main stages
• Fuzzification
• Membership functions used to graphically describe a situation
• Evaluation of Rules
• Application of the fuzzy logic rules
• Diffuzification
• Obtaining the crisp results
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46. Fuzzy Inference Process
• Fuzzy inference is the process of formulating the mapping from a given input
to an output using fuzzy logic.
• The mapping then provides a basis from which decisions can be made, or
patterns discerned.
• The process of fuzzy inference involves all of the pieces that are described in
Membership Functions, Logical Operations, and If-Then Rules.
• To implement a Fuzzy inference system, five steps must be followed:
1.Fuzzification using input membership functions
2.Apply Fuzzy Operand
3.Apply Implication method on each rule
4.Aggregate all Outputs
5.Defuzzification of output set 46

47. Fuzzy Inference System
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48. Example: Tipping
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49. Step 1. Fuzzify Inputs
• The first step is to take the inputs and determine the degree to which
they belong to each of the appropriate fuzzy sets via membership
functions.
• Fuzzification of the input amounts to either a table lookup or a
function evaluation.
• each input is fuzzified over all the qualifying membership functions
required by the rules.
𝑓 𝑥 =
𝑥 − 5.5
5
, 5.5 ≤ 𝑥 ≤ 9
1, 𝑥 > 0
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
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50. Step 2. Apply Fuzzy Operator
• If the antecedent of a given rule has more than one part, the fuzzy
operator is applied to obtain one number that represents the result of
the antecedent for that rule
• This number is then applied to the output function.
• The input to the fuzzy operator is two or more membership values
from fuzzified input variables.
• The output is a single truth value.
• Logical operators: AND, OR
• AND methods: minimum, product
• OR methods: Maximum, probabilistic OR
• probor(a,b) = a + b - ab
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51. Apply Fuzzy Operator
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52. 3. Apply Implication Method
• Before applying the implication method, you must determine the rule's weight.
• Every rule has a weight (a number between 0 and 1), which is applied to the number given
by the antecedent.
• Generally, this weight is 1 (as it is for this example) and thus has no effect at all on the
implication process.
• After proper weighting has been assigned to each rule, the implication method is
implemented.
• A consequent is a fuzzy set represented by a membership function, which weights
appropriately the linguistic characteristics that are attributed to it.
• The consequent is reshaped using a function associated with the antecedent (a single
number).
• The input for the implication process is a single number given by the antecedent, and the
output is a fuzzy set.
• Implication is implemented for each rule.
• Two methods are widely used, and they are the same functions that are used by the AND
method: min (minimum), which truncates the output fuzzy set, and prod (product), which
scales the output fuzzy set.
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53. Apply Implication Method
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54. Step 4. Aggregate All Outputs
• Aggregation is the process by which the fuzzy sets that represent the outputs of
each rule are combined into a single fuzzy set.
• Aggregation only occurs once for each output variable, just prior to the fifth and
final step, defuzzification.
• The input of the aggregation process is the list of truncated output functions
returned by the implication process for each rule.
• The output of the aggregation process is one fuzzy set for each output variable.
• As long as the aggregation method is commutative (which it always should be),
then the order in which the rules are executed is unimportant.
• Three methods are widely used:
• max (maximum)
• probor (probabilistic OR)
• sum (simply the sum of each rule's output set)
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55. Aggregate All Outputs
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56. Step 5. Defuzzify
• The input for the defuzzification process is a fuzzy set (the aggregate output fuzzy
set) and the output is a single number.
• As much as fuzziness helps the rule evaluation during the intermediate steps, the
final desired output for each variable is generally a single number.
• However, the aggregate of a fuzzy set encompasses a range of output values, and so
must be defuzzified in order to resolve a single output value from the set.
• There are five defuzzification methods:
• centroid,
• bisector,
• middle of maximum (the average of the maximum value of the output set),
• largest of maximum,
• smallest of maximum.
• Perhaps the most popular defuzzification method is the centroid calculation, which
returns the center of area under the curve
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57. Fuzzy Inference Diagram
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58. 58

59. Types of Fuzzy System
• two types of fuzzy inference systems:
• Mamdani
• Sugeno
• Fuzzy inference systems have been successfully applied in fields such
as automatic control, data classification, decision analysis, expert
systems, and computer vision.
• Because of its multidisciplinary nature, fuzzy inference systems are
associated with a number of names, such as fuzzy-rule-based systems,
fuzzy expert systems, fuzzy modeling, fuzzy associative memory,
fuzzy logic controllers, and simply (and ambiguously) fuzzy systems.
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60. Sugeno-Type Fuzzy Inference
• The main difference between Mamdani and Sugeno is that the Sugeno output
membership functions are either linear or constant.
• A typical rule in a Sugeno fuzzy model has the form:
If Input 1 = x and Input 2 = y, then Output is z = ax + by + c
For a zero-order Sugeno model, the output level z is a constant (a = b = 0).
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61. 61

62. Comparison of Sugeno and Mamdani Systems
• Advantages of the Sugeno Method
• It is computationally efficient.
• It works well with linear techniques (e.g., PID control).
• It works well with optimization and adaptive techniques.
• It has guaranteed continuity of the output surface.
• It is well suited to mathematical analysis.
• Advantages of the Mamdani Method
• It is intuitive.
• It has widespread acceptance.
• It is well suited to human input.
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63. Advantages of FLSs
• Mathematical concepts within fuzzy reasoning are very simple.
• You can modify a FLS by just adding or deleting rules due to
flexibility of fuzzy logic.
• Fuzzy logic Systems can take imprecise, distorted, noisy input
information.
• FLSs are easy to construct and understand.
• Fuzzy logic is a solution to complex problems in all fields of life,
including medicine, as it resembles human reasoning and
decision making.
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64. Disadvantages of FLSs
• There is no systematic approach to fuzzy system designing.
• They are understandable only when simple.
• They are suitable for the problems which do not need high
accuracy.
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65. Applications in Medicine
1. Fuzzy control of a total artificial heart (TAH) , Rau et al. (1995).
• Fuzzy rules are used to control the filling phase and the ejection phase of the heart.
• The fuzzy rule base consists of twenty-five rules
• A sample rule is
IF Filling is fast AND Pump rate is good
THEN Pump rate a little faster
2. Intelligent alarms for use in cardioanesthesia.
• monitor vital parameters such as blood
pressure, temperature, and blood gases.
• The alarm sounds if the parameters are outside
of an acceptable state, but it is up to the
anesthesiologist to make the decision regarding
changes in treatment (Becker et al., 1997).
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