Introduction To Fuzzy Logic

1

• Introduction To Fuzzy Logic

• Dr. Emad A. El-Sebakhy
• Room 22-316, Phone [phone]
• Email dodi05_at_ccse_at_kfupm.edu.sa

2
What Is Fuzzy Logic  ?

• Fuzzy logic is a powerful problem-solving
  methodology with a lot of applications in
  embedded control and information processing .

Fuzzy provides a remarkably simple way to draw
definite conclusions from vague, ambiguous or
imprecise information. In a sense, fuzzy logic
resembles human decision making with its ability
to work from approximate data and find precise
solutions.
3
What Is Fuzzy Logic  ?

• Unlike classical logic which requires a deep
  understanding of a system, exact equations, and
  precise numeric values.
• Fuzzy logic incorporates an alternative way of
  thinking, which allows modeling complex systems
  using a higher level of abstraction originating
  from our knowledge and experience.
• Fuzzy Logic allows expressing this knowledge with
  subjective concepts such as very hot, bright red,
  and a long time which are mapped into exact
  numeric ranges.

4
What Is Fuzzy Logic  ?

• Fuzzy Logic has been gaining increasing
  acceptance during the past few years. There are
  over two thousand commercially available products
  using Fuzzy Logic, ranging from washing machines
  to high speed trains.
• Nearly every application can potentially realize
  some of the benefits of Fuzzy Logic, such as
  performance, simplicity, lower cost, and
  productivity.

• Fuzziness vs. Probability
• Fuzziness is deterministic uncertainty
  probability is nondeterministic.
• Fuzziness describes event ambiguity probability
  describes event occurrence. Whether an event
  occurs is random. The degree to which it occurs
  is fuzzy.
• Probabilistic uncertainty dissipates with
  increasing number of occurrences fuzziness does
  not.

5
Why Use Fuzzy Logic ?

• An Alternative Design Methodology Which Is
  Simpler, And Faster
• Fuzzy Logic reduces the design development cycle
• Fuzzy Logic simplifies design complexity
• Fuzzy Logic improves time to market
• A Better Alternative Solution To Non-Linear
  Control
• Fuzzy Logic improves control performance
• Fuzzy Logic simplifies implementation
• Fuzzy Logic reduces hardware costs

6
Some Historical Developments

• Fuzzy systems have been around since the
  1920s, when they where first proposed by
  Lukaciewicz. He proposed to modify traditional
  false (0) true (1) reasoning to include some
  truth such as 0.5.
• Since then the approach has been further
  develop and systems have incorporated the
  methodologies

• 1965 - Fuzzy Set Theory (Prof. Lofti A. Zadeh, U.
  of Cal. at Berkely)
• 1966 - Fuzzy Logic (Dr. Peter N. Morinos, Bell
  Labs.)
• 1972 - Fuzzy measure (Prof. Michio Sugeno, TIT)
• 1974 - Fuzzy logic controller (Prof. E.H.
  Mamdani, London Univ.)
• 1980 - Control of cement0kiln with monitor
  capability (Denmark)
• 1987 - Automatic train operation for Sendai
  Subway (Hitachi, Japan)
• 1988 - Stock Trading Expert System (Yamaichi
  Security, Japan).

7
Fuzzy Logic vs Boolean Logic
What is Fuzziness? Fuzziness is deterministic
uncertainty. It is concerned with the degree to
which events occurrence rather than the
likelihood of their occurrence. For example, the
degree to which a person is tall is a fuzzy event
rather than a random event.

• Two fundamental assumptions of traditional logic
• An element belongs to a set or its complement
• An element cannot belong to both a set and its
  complement -- law of excluded middle
• Traditional logic is crisp.
• Fuzzy logic violates both the above assumptions.

8
Fuzzy Logic generalizes Boolean Logic

• It is very useful that the Boolean Logic is
  included in the Fuzzy Logic
• If we think of x and y as crisp values 0 and 1,
  Fuzzy logic gets back to Boolean logic.

9
Fuzzy Vs. Crisp Values

• A crisp value is a precise number that
  represents the exact status of the associated
  phenomenon.
• Example The fastest land mammal,
  cheetahs can accelerate from 0 to 70 mph in 3
  seconds. A Lamborghini Diablo sports car
  accelerates from 0 to 62 mph in 4 seconds! Of
  course, you can use the car to go on an
  appointment.
• A fuzzy value is an ambiguous term that
  characterizes an imprecise or not very well
  understood phenomenon.
• Example
  Cheetahs can run very fast.
• Fuzzy Set Theory provides the means for
  representing uncertainty using set theory.
• Example Any velocity between 45 and 70 mph is a
  very fast velocity.
• Any velocity between 25 and 57 mph is a
  fast velocity.

10
Fuzzy Sets

• Sets with fuzzy boundaries

A Set of tall people
Fuzzy set A
1.0
.9
Membership function
.5
510
62
Heights
11
Fuzzy Sets

• Fuzzy sets and set membership is the key
  to decision making when faced with uncertainty
  (Zadeh, 1965).
• Classical sets contain objects that
  satisfy precise properties.
• Fuzzy sets contain objects that satisfy
  imprecise properties of membership (membership of
  an object can be approximate).
• Example The set of heights from 5 to
  7 feet is crisp (classical set)
• The set of heights in
  the region around 6 feet is fuzzy.
• A fuzzy set can be defined as a set of
  crisp values that can be group together with an
  associated fuzzy term.
• Example A person between 5 ft and 6
  ft belongs to the set of tall people.

NOTE Because a crisp value can belong to a fuzzy
set in one context and to another set in a
different context a crisp value can be associated
to more than one fuzzy set. For example, the set
of tall people can overlap with the set of
non-tall people (an impossibility in the world of
binary logic).
12
Fuzzy Sets

• Formal definition
• A fuzzy set A in X is expressed as a set of
  ordered pairs

Membership function (MF)
Universe or universe of discourse
Fuzzy set
A fuzzy set is totally characterized by
a membership function (MF).
13
Types of fuzzy sets

• There are basically two types of fuzzy sets
  normal and subnormal.

?(x)
?(x)
A
1
1
Height of the fuzzy set
A
Height of the fuzzy set
0
0
x
x
Normal fuzzy set
Subnormal fuzzy set
A normal fuzzy set is one whose membership
function has at least one element x in the
universe whose membership is unity.
A subnormal fuzzy set is one whose membership
function doesnt have an element x in the
universe whose membership is unity.
14
Fuzzy Sets with Discrete Universes

• Fuzzy set C desirable city to live in
• X SF, Boston, LA (discrete and nonordered)
• C (SF, 0.9), (Boston, 0.8), (LA, 0.6)
• Fuzzy set A sensible number of children
• X 0, 1, 2, 3, 4, 5, 6 (discrete universe)
• A (0, .1), (1, .3), (2, .7), (3, 1), (4, .6),
  (5, .2), (6, .1)

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Fuzzy Sets with Cont. Universes

• Fuzzy set B about 50 years old
• X Set of positive real numbers (continuous)
• B (x, mB(x)) x in X

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How to represent fuzzy sets

• There are two common ways to represent fuzzy sets
  depending if the set is discrete or continuous.
• Discrete fuzzy sets
• A notation convention for fuzzy sets when the
  universe of disclosure, X, is discrete and
  finite, is as follows for a fuzzy set A
• A ?A(x1)/x1 ?A(x2)/x2 ?i
  ?A(xi)/xi
• Continuous fuzzy sets
• When the universe, X, is continuous and
• finite, the fuzzy set A is denoted by
• A ?
  ?A(xi)/xi

The ? is not for algebraic summation but
rather denotes the collection or aggregation of
each element hence the signs are
not algebraic add but are a function-theoretic
union.

• In both notations
• Not a quotient bar but a delimiter
• The numerator in each term is the
• membership value in set A associated
• with the element of the universe
• indicated in the denominator.

The ? sign is not an algebraic integral but
a continuous function- theoretic union
notation for continuous variables.
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Alternative Notation

• A fuzzy set A can be alternatively denoted as
  follows

X is discrete
X is continuous
Note that S and integral signs stand for the
union of membership grades / stands for a
marker and does not imply division.
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Fuzzy Partition

• Fuzzy partitions formed by the linguistic values
  young, middle aged, and old

lingmf.m
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Fuzzy Set Operations

• Standard Complement of a fuzzy set A
• A(x) 1- A(x)
• Elements for which A(x) A(x) are called
  equilibrium points.
• Standard intersection A?B
• minA(x),B(x)
• Standard union A?B
• (A ? B)(x) maxA(x),B(x)

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Note that the membership degree is not the same
as a probability although the values that it may
take are the same (0 to 1). The chance of a 25
year old being young or not young is not 50 / 50
- rather the degree to which a 25 year old
exemplifies young is about half (0.50). Set
operators can be defined on fuzzy sets similarly
to those on crisp sets.
Set A
Complement of Set A Not A
21
Operations on Fuzzy sets

• If we define 3 fuzzy sets, A, B and C on the
  universe X, for a given element x of the universe
  the following are examples of operations defined
  on A, B and C.

Some operations on fuzzy sets Union ?A?B(x)
?A(x) \/ ?B(x) Intersection ?A?B(x) ?A(x)
/\ ?B(x) Complement ?A(x) 1 - ?A(x) etc.
?
A
1
1
1
A
A
B
B
0.7
0.3
0.3
0
0
0
x
x
x
14
10 14 15
5 10 15 17
Union Intersection Complement
The resulting fuzzy set may be represented
as A?B 0/10 0.3/14 0/15
The fuzzy sets A and B may be represented as A
0/5 1/10 0.3/14 0/15 B 0/10 1/14
0.7/15 0/17
How would A?B be represented?
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Membership functions

• All information contained in a fuzzy set is
  described by its membership function.
• The Core of a membership function is defined as
  the region of the universe that is characterized
  by complete and full membership in the set. The
  core comprises those elements for which, ?(x) 1
• The support is defined as that region of the
  universe that is characterized by nonzero
  membership in the set ?(x) gt 0
• The boundaries are defined as that region of the
  universe containing elements that have nonzero
  membership but not complete membership 0 lt ?(x)
  lt 1

core
?
1
Membership functions can be symmetrical
or asymmetrical.
0
x
boundary
boundary
support
23
Membership when using fuzzy sets

• For crisp sets an element x in the universe
  X is either a member of some crisp set, say A on
  the universe. This binary issue of membership
  can be represented mathematically as
• 
  1, ? A
• 0, ? A
• where the symbol x A(x) gives the
  indication of an unambiguous membership of
  element x in set A.
• Fuzzy membership extends the notion of
  binary membership to accommodate various degrees
  of membership on the real continuous interval
  0, 1, where the endpoints conform to no
  membership and full membership, respectively.
  The sets on the universe X that can accommodate
  degrees of membership are referred as fuzzy
  sets.

x A(x)
24
More Definitions
MF Terminology

• Support
• Core
• Normality
• Crossover points
• Fuzzy singleton
• a-cut, strong a-cut
• Convexity
• Fuzzy numbers
• Bandwidth
• Symmetricity
• Open left or right, closed

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The Common Membership Functions
Each fuzzy set has a membership function. These
are normally trapezoidal, triangular or Gaussian
(normal). These are usuallynormalized, that is,
have a maximum value of 1. In the picture
above, the fuzzy set young is described with a
trapezoidal membership function. All ages less
than 20 have full membership (?1) and all
ages greater than 30 have no membership (?0).
In between is a linear relationship between age
and membership. Age 25 has ? 0.5.
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Set-Theoretic Operations
subset.m
fuzsetop.m
Common MF Formulation

• Triangular MF

Trapezoidal MF
Gaussian MF
Generalized bell MF
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MF Formulation
disp_mf.m

• Sigmoidal MF

Extensions
Absolute difference of two MF
Product of two MF
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Membership Functions (MFs)

• Characteristics of MFs
• Subjective measures
• Not probability functions

?tall in Asia
MFs
.8
?tall in the US
.5
.1
510
Heights
29
Fuzzy Rules and Inference

• Fuzzy rules are simply rules where the
  premises and conclusions are fuzzy. But crisp
  values can be incorporated as well.
• When using rules you have to select a fuzzy
  inference methodology and applied it to the
  conclusion. The two most popular fuzzy inference
  methodologies are

Max-Product Inference
Max-Min Inference
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Fuzzy rules and their results
Rule Distance (D) Velocity (V)
Acceleration (A)
Medium
Slow
Medium
IF D Medium OR V Slow THEN A Medium
Inputs
Medium
Slow
Medium
IF D Medium OR V Medium THEN A Slow
Final acceleration Centroid
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Fuzzy Logic/Sets Example

• Let Y be the set of all flowers that are yellow.
  Let X, the universe of discourse, be the set of
  flowers in my backyard. In standard set theory,
  every flower x in X is either an element of Y or
  not. In fuzzy set theory, every flower x has a
  degree of yellowness mY(x).
• Let F be the set of all flowers that are perfumed
  .
• Let x be a flower in my backyard. If mY(x) 0.8
  and mF(x) 0.9 then mYF(x) min0.8,0.9 0.8.

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Fuzzification

• Fuzzification is the process of making a crisp
  quantity fuzzy
• We do this by simply recognizing that many of the
  quantities that we consider to be crisp and
  deterministic are actually nondeterministic at
  all They carry considerable uncertainty.
• If the form of uncertainty happens to arise
  because of imprecision, ambiguity, noise or
  vagueness, then the variable is probably fuzzy
  and can be represented by a membership function.

Example In the real world, hardware such as a
digital scale generates crisp data, but these are
subject to experimental error. The information
shown in the figure below shows one possible
range of errors for a typical weight measure
and the associated membership function that might
represent such imprecision.
Reading
1
0
x
-1 1
33
Fuzzification

• The representation of imprecise data as fuzzy
  sets is a useful but not mandatory step when
  those data are used in fuzzy systems.
• This idea is shown in the following figure where
  we can consider the data as a crisp or as a fuzzy
  reading (Figures a and b).

1
1
Reading (fuzzy)
Reading (crisp)
Low voltage
Medium voltage
Membership
0.4
Membership
0.3
x
voltage
0
0
-1 1
b)
a)
In Fig b the intersection of the fuzzy
set Medium voltage and a fuzzified
voltage reading occurs at a membership of 0.4.
We can see that the intersection of the two fuzzy
sets is a small triangle, whose largest
membership occurs at 0.4.
In Fig a we might want to compare A crisp
voltage reading to a fuzzy set, say Low
voltage. In the figure we see that the crisp
reading intersects the fuzzy set at a membership
of 0.3, i.e., the fuzzy set and the reading
can be said to agree at a membership value of
0.3.
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Defuzzification methods
Weighted average
Centroid
Max-membership or Height method
? ( x ) gt (? ( x ))
Center of sums
Mean-max membership
x (a b) / 2
35
Fuzzy logic operations Summary

• Fuzzy math involves in general three operations
• Fuzzyfication membership function
• Rule evaluation
• Defuzzyfication

36
Fuzzyfication

• It makes the translation from real world values
  to Fuzzy world values using membership functions.
  The membership functions in Fig.1, translate a
  speed 55 into fuzzy values (Degree of
  membership)  SLOW0.25, MEDIUM0.75 and  FAST0.

37
Rule Evaluation

• Rule1 If SpeedSlow and HomeFar then
  GasIncrease
• Supose SLOW0.25 and  FAR0.82. The rule strength
  will be 0.25 (The minimum value of the
  antecedents) and the fuzzy variable INCREASE
  would be also 0.25.
• Rule2 If SpeedMedium and HigherSecure then
  GasIncrease
• Suppose in this case, MEDIUM0.75 and SECURE0.5.
  Now the rule strength will be 0.5 and the fuzzy
  variable INCREASE would be also 0.5.
• So, we have two rules involving fuzzy variable
  INCREASE. The "Fuzzy OR" of the two rules will be
  0.5 (The maximum value between the two proposed
  values).
• INCREASE0.5

38
Defuzzyfication

• After computeing the fuzzy rules and evaluating
  the fuzzy variables, we will need to translate
  these results back to the real world. We need now
  a membership function for each output variable
  like in Fig. 2.
• Let the fuzzy variables be
• DECREASE0.2, SUSTAIN0.8, and INCREASE0.5 

39
Defuzzyfication

• Each membership function will be clipped to the
  value of the correspondent fuzzy variable as
  shown in fig.3.

40
Defuzzyfication

• Defuzzification is the process of making a fuzzy
  quantity crisp.
• There are different ways to do this and the
  deffuzification process to be used greatly
  depends on the degree of uncertainty within the
  fuzzy set
• A new output membership function is built, taking
  for each point in the horizontal axis, the
  maximum value between the three membership
  values.
• Then take the centroid. Here, Engine2.6

2.6
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Steps Needed for Building a Fuzzy System

• Step 1.- Determine the values of the input and
  output variables.
• Step 2.- Fuzzify the variables create fuzzy sets
  to represent the different values of the input
  variables. Fuzzification is the process of making
  a crisp quantity fuzzy.
• Step 3.- Create fuzzy sets for the output
  variables of the system.
• Step 4.- Generate a set of fuzzy rules based on
  the input and output fuzzy sets.
• Step 5.- Choose a deffuzification method and
  apply it to the results obtained from the rules
  that are satisfied.
• Step 6.- The crisp value obtained from Step 5 is
  the answer to your problem.

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Example

• The inverted pendulum
• Inputs the angle ? and d?/dt input values

43
The fuzzy regions for the input values ? (a) and
d?/dt (b).
The fuzzy regions of the output value u,
indicating the movement of the pendulum base.
44
The fuzzification of the input measures x11, x2
-4.
The Fuzzy Associative Matrix (FAM) for the
pendulum problem. The input values are on the
left and top.
45
The fuzzy consequents (a) And their union
(b). The centroid of the union (-2) is the crisp
output.
46
Characteristics and Comparison of Four AI
Techniques
47
Fuzzy Logic Applications Area
48
Application areas

• Fuzzy Control
• Subway trains
• Cement kilns
• Washing Machines
• Fridges
• Video cameras
• Electric shavers

49
Fuzzy Sets Review

• Extension of Classical Sets
• Not just a membership value of in the set and out
  the set, 1 and 0
• but partial membership value, between 1 and 0

50
Example Height

• Tall people say taller than or equal to 1.8m
• 1.8m , 2m, 3m etc member of this set
• 1.0 m, 1.5m or even 1.79999m not a member
• Real systems have measurement uncertainty
• so near the border lines, many misclassifications

51
Member Functions

• Membership function
• better than listing membership values
• e.g.
• Tall(x) 1 if x gt 1.9m ,0 if x lt 1.7m, else (
  x - 1.7 ) / 0.2

52
Example Fuzzy Short

• Short(x) 0 if x gt 1.9m ,
• 1 if x lt 1.7m
• else ( 1.9 - x ) / 0.2

53
Fuzzy Set Operators Again

• Fuzzy Set
• Union
• Intersection
• Complement
• Many possible definitions
• we introduce one possibility

54
Fuzzy Set Union

• Union ( fA(x) and fB(x) ) max (fA(x) , fB(x) )
• Union ( Tall(x) and Short(x) )

55
Fuzzy Set Intersection

• Intersection ( fA(x) and fB(x) ) min (fA(x) ,
  fB(x) )
• Intersection ( Tall(x) and Short(x) )

56
Fuzzy Set Complement

• Complement( fA(x) ) 1 - fA(x)
• Not ( Tall(x) )

57
Fuzzy Logic Operators Summary

• Fuzzy Logic
• NOT (A) 1 - A
• A AND B min( A, B)
• A OR B max( A, B)

58
Fuzzy Logic NOT
59
Fuzzy Logic AND
60
Fuzzy Logic OR
61
Fuzzy Controllers

• Used to control a physical system

62
Structure of a Fuzzy Controller
63
Fuzzification

• Conversion of real input to fuzzy set values
• e.g. Medium ( x )
• 0 if x gt 1.90 or x lt 1.70,
• (1.90 - x)/0.1 if x gt 1.80 and x lt 1.90,
• (x- 1.70)/0.1 if x gt 1.70 and x lt 1.80

64
Inference Engine

• Fuzzy rules
• based on fuzzy premises and fuzzy consequences
• e.g.
• If height is Short and weight is Light then feet
  are Small
• Short( height) AND Light(weight) gt Small(feet)

65
Fuzzification Inference Example

• If height is 1.7m and weight is 55kg
• what is the value of Size(feet)

66
Defuzzification

• Rule base has many rules
• so some of the output fuzzy sets will have
  membership value gt 0
• Defuzzify to get a real value from the fuzzy
  outputs
• One approach is to use a centre of gravity method

67
Defuzzification Example

• Imagine we have output fuzzy set values
• Small membership value 0.5
• Medium membership value 0.25
• Large membership value 0.0
• What is the deffuzzified value

68
Fuzzy Control Example
69
Input Fuzzy Sets

• Angle- -30 to 30 degrees

70
Output Fuzzy Sets

• Car velocity- -2.0 to 2.0 meters per second

71
Fuzzy Rules

• If Angle is Zero then output ?
• If Angle is SP then output ?
• If Angle is SN then output ?
• If Angle is LP then output ?
• If Angle is LN then output ?

72
Fuzzy Rule Table
73
Extended System

• Make use of additional information
• angular velocity -5.0 to 5.0 degrees/ second
• Gives better control

74
New Fuzzy Rules

• Make use of old Fuzzy rules for angular velocity
  Zero
• If Angle is Zero and Angular velocity is Zero
• then output Zero velocity
• If Angle is SP and Angular velocity is Zero
• then output SN velocity
• If Angle is SN and Angular velocity is Zero
• then output SP velocity

75
Table format
76
Complete Table

• When angular velocity is opposite to the angle do
  nothing
• System can correct itself
• If Angle is SP and Angular velocity is SN
• then output ZE velocity
• etc

77
Example

• Inputs10 degrees, -3.5 degrees/sec
• Fuzzified Values
• Inference Rules
• Output Fuzzy Sets
• Defuzzified Values

78
Example of a Fuzzy Controller
A cart on a 4-meter long track. The goal is to
return the cart to the center of the track with 0
velocity. The available control is to push or
pull on the cart.
2m
-2m
0m
79
Cart Position
80
Cart Velocity
81
Cart Force
82
Simple Control Rules

• If left then push
• If right then pull
• If middle then none
• If moving left then push
• If standing still then none
• If moving right then pull
• If left and moving left then push
• If right and moving right then pull

83
Fuzzy Control Algorithms

• Find the sensor values
• For example, the position might be x -0.5
  meters and v 0.
• Calculate the fuzzy membership
• For example, mmiddle(x -0.5) 0.5 and mleft(x
  -0.5) 0.5.
• Calculate the membership of the rule antecedents
  for all control rules.
• Apply the rules
• Aggregate the results from all control rules
• De-fuzzify to arrive at a single-valued action
  recommendation.

84
Dempster-Shafer Theory

• Dempster-Shafer considers sets of propositions
  and provides an interval within which the belief
  must lie.
• interval Belief, Plausibility
• Belief brings together all the evidence that
  would lead us to believe in the proposition with
  some certainty.
• Plausibility brings together the evidence that is
  compatible with the proposition and is not
  inconsistent with it. pl(p) 1 bel(?p)
• So..the interval is a measure of our belief in
  the proposition and the amount of information we
  have to support this belief.

85
Coin Toss Example

• Lets say Bart goes up to Homer and bets him 20
  that the coin he has in his hand will be heads on
  a coin toss. Homer is like, Yeah right, you
  trying to trick me boy?! Its a two headed coin
  isnt it?

86
Coin Toss Example

• Belief(Heads) 0
• Belief(not Heads) 0
• Should Homer take the bet?

All of a sudden Lisa comes in and tells Bart to
give her back her quarter. Homer, knowing Lisa
to be honest, now thinks that maybe the coin
isnt a two-headed coin. Homer is 80 sure about
this. This now increases the belief functions.
87
Coin Toss Example

• Belief(Heads) (Homers deduced certainty
• probability of it coming up heads)
• 0.8 0.5 0.4
• Belief(not Heads)
• (Homers deduced certainty
• probability of it not coming up heads)
• 0.8 0.5 0.4

88
Coin Toss Example

• The probability interval when being ignorant
  would be 0,1 for the probability of heads
  coming up on a coin toss.
• After Lisa comes into the scene, Homer deduces
  the coins probability, increasing the
  uncertainty, so the interval becomes 0.4, 0.6.
• Smaller intervals allows the reasoning system to
  make decisions, based from new information.

89
Example

• Melissa is 90 reliable.
• She said, the computer is broken into
• Belief in computer being broken into 0.9
• Belief in computer not being broken into 0
• Pl(broken) 1-0 1
• belief,plausibility(broken) 0.9,1
• Bill is 80 reliable.
• He said, the computer is broken into
• Belief in computer being broken into 0.8
• Belief in computer not being broken into 0
• Pl(broken) 1-0 1
• belief,plausibility(broken) 0.9,1
• Probability that both of them are unreliable is
  0.02
• Combined belief,plausibility(broken) 0.98,1

90
Dempster-Shafer Theory

• Let ? represent our frame of discernment, which
  is the set of all hypothesis. We want to attach a
  measure of belief to each of these hypothesis
  after we have been presented with some evidence.
• But the evidence may support subsets of ?.
• Also evidence supporting one hypothesis may alter
  our belief in other hypothesis.
• Dempster-Shafer allows us to handle these
  interactions.

91
Dempster-Shafer Theory

• If ? contains n elements then there are 2n
  subsets of ? (including the empty set ?).
• m(p) is the current belief for each of the
  subsets of ?.
• Dempster-Shafer allows us to combine ms that
  arise from multiple sources of evidence.

92
Example

• A patient may be suffering from Cold, Flue,
  migraine Headache or Meningitis
• Call this set of hypothesis Q C,F,H,M
• Patient has fever, which supports C,F,M at 0.6
• Thats, m1(C,F,M) 0.6, m1(Q)0.4
• Patient has extreme nausea, which supports
• m2(C,F,H) 0.7, m2(Q)0.3
• We can combine these two belief distributions

• All the sets in m3 are non-empty and unique

93
Example

• Third evidence, lab culture supports
• m4(M) 0.8 and m4(Q) 0.2
• We can combine this with m3
• The denominator is 1- (0.336 0.224) 0.44
• m5(M) (0.144 0.096)/0.44 0.545,
  m5(C,F)0.191
• m5(C,F,H) 0.127, m5(C,F,M) 0.082,
  m5(Q)0.055
• m5() 0.56

94
Summary on Dempster-Shafer

• A large belief assigned to empty set (as 0.56 in
  the previous example) indicates that there is
  conflicting evidence in belief sets.
• When there are large hypothesis sets and complex
  sets of evidence, calculations can get
  cumbersome.
• But complexity is still less than Bayesian
  approach.
• Very useful tool when stronger Bayesian
  conclusions may not be justified.
• A distinction is made between probability of a
  proposition given uncertain evidence, and
  probability of proposition given no evidence.

95
Default Reasoning

• A gentle introduction
• Simulates human nature of qualitative reasoning.
• All birds fly.
• Emu is a bird.
• Therefore, Emu flies.
• Went to Australia and saw it does nor fly.
• Update your belief !
• Jumping to conclusions
• Making assumptions
• To believe one thing until a reason is found to
  believe otherwise.

96
Default Reasoning Example

• Homer is at work at the Springfield nuclear
  power plant. Now hes having some coffee and
  doughnuts and accidentally spills it over some
  controls. All of a sudden there are explosions
  and alarms are going off in the plant. Homer is
  frantic and is saying, Oh my god, oh my god, Mr.
  Burns is going to fire me. Oh no, what do I do,
  what do I do?

Then Smithers comes and tells Homer that Mr.
Burns wants to see him.
97
Default Reasoning

• Homers assumption that he is going to be fired
  by Mr. Burns is an example of default reasoning.
• Later Homer finds out that he wasnt even at his
  own workstation, he was at someone elses
  workstation, so Homer doesnt have to worry about
  being fired now. He retracts his initial
  assumption, and gives a big sigh and relaxes.

98
Default Reasoning Problems

• What are good default rules to have?
• What to do in the case where some evidence
  matches two default rules with different
  conclusions?
• What conclusions should be kept and which ones
  should be retracted?
• How can beliefs that have default status be used
  to make decisions?

99
Fuzzy Logic (FL) vs CF

• We use both FL and CF to handle incomplete
  knowledge
• In FL, Precision/vagueness is expressed by
  membership function to a set
• mF(20,adult)0.6, mF(20,young)0.4, mF(20,old)0
• Fuzzy Logic is not concerned how these
  distribution are created but how they are
  manipulated.
• There are many interpretations, similar to
  Certainty Algebra

100
Exercises
   Given the fuzzy sets-        Tall(X)    
0 if X lt 1.6m                (X - 1.6m) /
0.2, if 1.6m lt X lt 1.8m                1, if
X gt 1.8m         Short(X)     1 if X lt
1.6m                (1.8m - X) / 0.2, if 1.6m
lt X lt 1.8m                0, if X gt 1.8m
    a). Sketch the graphs of Tall(X) and
Short(X).    b).    i. Calculate the Union of
the fuzzy sets Tall(X) and Short(X). ii.
Calculate the Intersection of the fuzzy sets
Tall(X) and Short(X).    c). Show that the
complement of Tall(X) is Short(X).
101
Given additional fuzzy sets-        Strong(Y)
    0 if Y lt 30kg                (Y -
30kg) / 20, if 30kg lt Y lt 50kg               
1, if Y gt 50kg         Weak(Y)     1 if
Y lt 30kg                (50kg - Y) / 20, if
30kg lt Y lt 50kg                0, if Y gt
50kg     and the fuzzy rules-         If
Tall(X) OR Strong(Y) then Heavy(Z)        If
Short(X) AND Weak(Y) then Light(Z) Calculate
the membership values of Heavy(Z) and Light(Z)
where      i.    X 1.65m, Y 30kg     
ii.    X 1.70m, Y 45kg
102
Complexity of the system Vs. precision in its
model
Mathematical equations
Model-free Methods (e.g., ANNs)
Precision in the model
Fuzzy Systems
Complexity (uncertainty) of the system

• For systems with little complexity, hence little
  uncertainty, closed-form mathematical expressions
  provide precise description of the system.
• For systems that are a little more complex, but
  for which significant data exists, model free
  methods such as artificial NNs, provide a
  powerful and robust means to reduce uncertainty
  through learning, based on patterns in the
  available data.
• For most complex systems where few numerical data
  exists and where only ambiguous or imprecise
  information may be available, fuzzy reasoning
  provides a way to understand system behavior by
  allowing us to interpolate approximately between
  observed input and output situations.