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kuliah 12 - Fuzzy Logic by qingyunliuliu

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```									Fuzzy Logic

Yeni Herdiyeni
http://www.cs.ipb.ac.id/~yeni
Departemen Ilmu Komputer IPB

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Introduction

   “Mathematics that refers to reality is not certain and mathematics
that is certain does not refer to reality”
Albert Einstein

   “While the mathematician constructs a theory in terms of
´perfect´objects, the experimental observes objects of which the
properties demanded by theory are and can, in the very nature of
measurement, be only approximately true”
Max Black

   “What makes society turn is science, and the language of science
is math, and the structure of math is logic, and the bedrock of logic
is Aristotle, and that is what goes out with fuzzy logic”
Bart Kosko

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Introduction (cont.)
   Uncertainty is produced when a lack of information exists.
   The complexity also involves the degree of uncertainty.
   It is possible to have a great deal of data (facts collected from
observations or measurements) and at the same time lack of
information (meaningful interpretation and correlation of data
that allows one to make decisions.)

Data                                 Information
Knowledge & Intelligence

Database                             Intelligent information systems
Knowledge base & AI

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Introduction (cont.)
Knowledge is information at a higher level of abstraction.
Ex: Ali is 10 years old   (fact)
Ali is not old         (knowledge)
   Our problems are:
   Decision
   Management
   Prediction
   Solutions are:
   Understanding – utilizing information available
   Managing with information not avaliable
   Large amount of information with large amount of uncertainty lead to
complexity.
   Avareness of knowledge (what we know and what we do not know) and
complexity goes together.
Ex: Driving a car is complex, driving in an iced road is more compex, since more knowledge is
needed for driving in an iced road.

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Introduction (cont.)
   Fuzzy logic provides a systematic basis for representation of uncertainty,
imprecision, vagueness, and/or incompletenes.
   Uncertain information: Information for which it is not possible to
determine whether it is true or false. Ex: a person is “possibly 30 years old”
   Imprecise information: Information which is not available as precise as it
should be. Ex: A person is “around 30 years old.”
   Vague information: Information which is inherently vague.
Ex: A person is “young.”
   Inconsistent information: Information which contains two or more
assertions that cannot be true at the same time. Ex: Two assertions are given:
“Ali is 16” and “Ali is older than 20”
   Incomplete information: information for which data is missing or data is
partially available. Ex: A person’s age is “not known” or a person is
“between 25 and 32 years old”
   Combination of the various types of such information may also exist. Ex:
“possibly young”, “possibly around 30”, etc.

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Introduction (cont.)

UNCERTAINTY
(Uncertainty-based information)

COMPLEXITY                                CREDIBILITY
(Description-algorithmic infor.)                   (relevance)

USEFULNESS
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Introduction (cont.)
Example: When uncertainties like heavy traffic,
unfamiliar roads, unstable wheather conditions, etc.
increase, the complexity of driving a car increases.
How do we go with the complexity?
   We try to simplify the complexity by making a
satisfactory trade-off between information available
to us and the amount of uncertainty we allow.
   We increase the amount of uncertainty by replacing
some of the precise information with vague but more
useful information.

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Introduction (cont.)

Examples:
   Travel directions: try to do it in mm terms (or turn the wheel % 23 left,
etc.), which is very precise and complex but not very useful. So replace
mm information with city blocks, which is not as precise but more
meaningful (and/or useful) information.
   Parking a car: doing it in mm terms, which is very precise and complex
but difficult and very costly and not very useful. So replace mm
information with approximate terms (between two lines), which is not as
precise but more meaningful (or useful) information and can be done in
less cost.
    Describing wheather of a day: try to do it in % cloud cover, which is very
precise and complex but not very useful. So replace % cloud information
with vague terms (very cloudy, sunny etc.), which is not as precise but
more meaningful (or useful) information.

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Introduction (cont.)

   Fuzzy logic has been used for two different senses:
   In a narrow sense: refers to logical system generalizing crisp
logic for reasoning uncertainty.
   In a broad sense: refers to all of the theories and technologies
that employ fuzzy sets, which are classes with imprecise
boundaries.
   The broad sense of fuzzy logic includes the narrow sense of
fuzzy logic as a branch.
   Other areas include fuzzy control, fuzzy pattern recongnition,
fuzzy arithmetic, fuzzy probability theory, fuzzy decision
analysis, fuzzy databases, fuzzy expert systems, fuzzy
computer SW and HW, etc.

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Introduction (cont.)

With Fuzzy Logic, one can accomplish two things:

   Ease of describing human knowledge involving vague concepts

   Enhanced ability to develop a cost-effective solution to real-
world

   In another word, fuzzy logic not only provides a cost effective
way to model complex systems involving numeric variables but
also offers a quantitative description of the system that is easy
to comprehend.

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Introduction (cont.)

Fuzzy Logic was motivated by two objectives:

   First, it aims to minimize difficulties in developing and analyzing
complex systems encountered by conventional mathematical
tools. This motivation requires fuzzy logic to work in quantitative
and numeric domains.

   Second, it is motivated by observing that human reasoning can
utilize concepts and knowledge that do not have well defined,
sharp boundaries (i.e., vague concepts). This motivation
enables fuzzy logic to have a descriptive and qualitative form.
This is related to AI.

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Introduction (cont.)
Components of Fuzzy Logic

 Fuzzy Predicates: tall, small, kind, expensive,...

 Predicates modifiers (hedges): very, quite, more or less,
extremely,..

 Fuzzy truth values: true, very true, fairly false,...

 Fuzzy quantifiers: most, few, almost, usually, ..

 Fuzzy probabilities: likely, very likely, highly likely,...

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Introduction (cont.)
Applications

 Control: “If the temperature is very high and the presure is
decreasing rapidly, then reduce the heat significantly.”

 Database: “Retrieve the names of all candidates that are fairly
young, have a strong background in algorithms, and a modest

 Medicine: Hepatitis is characterized by the statement, „Total
proteins are usually normal, albumin is decreased, -globulins
are slightly decreased, -globulins are slightly decreased, -
globulins are increased‟

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Introduction (cont.)

   Probability theory vs fuzzy set theory:
   Probability measures the likelihood of a future event,
based on something known now. Probability is the
theory of random events and is not capable of
capturing uncertainty resulting from vagueness of
linguistic terms.
   Fuzziness is not the uncertainty of expectation. It is
the uncertainty resulting from imprecision of
meaning of a concept expressed by a linguistic term
in NL, such as “tall” or “warm” etc.

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Introduction (cont.)

   Probability theory vs fuzzy set theory (cont):
   Fuzzy set theory makes statements about one concrete object;
therefore, modeling local vagueness, whereas probability theory
makes statements about a collection of objects from which one is
selected; therefore, modeling global uncertainty.
   Fuzzy logic and probability complement each other.
Example: “highly probable” is a concept that involves both
randomness and fuziness.
   The behaviour of a fuzzy system is completely deterministic.
   Fuzzy logic differs from multivalued logic by introducing concepts
such as linguistic variables and hedges to capture human
linguistic reasoning.

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Introduction (cont.)
   Even though the broad sense of fuzzy logic covers a wide range
of theories and techniques, its core technique is based on four
basic concepts:
   Fuzzy sets: sets with smooth boundaries;
   Linguistic variables: variables whose values are both qualitatively
and quantitatively described by a fuzzy set;
   Possibility distribution: constraints on the value of a linguistic
variable imposed by assigning it a fuzzy set; and
   Fuzzy if-then rules: a knowledge representation scheme for
describing a functional mapping (fuzzy mapping rules) or a
logical formula that generalizes an implication in two-valued logic
(fuzzy implication rules).
   The first three concepts are fundamental for all subareas in fuzzy
logic, but the fourth one is also important.

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Introduction

   Most of the phenomena we encounter everyday are imprecise -
the imprecision may be associated with their shapes, position,
color, texture, semantics that describe what they are
   Fuzziness primarily describes uncertainty (partial truth) and
imprecision
   The key idea of fuzziness comes from the multivalued logic:
Everything is a matter of degree
   Imprecision raises in several faces, e.g. as a semantic ambiguity

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Introduction
   By fuzzifying crisp data obtained from
measurements, FL enhances the robustness of a
system
   Imprecision raises in several faces - for example, as
a semantic ambiguity
   the statement “the soup is HOT” is ambiguous, but not fuzzy
Definition of
The temperature of the soup         the domain
Hot                                                  of
The amount of spices used            discourse

       e.g. [20º,80º]              Transaction to Fuzziness

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   The word “fuzzy” can be defined as “imprecisely
defined, confused, vague”
   Humans represent and manage natural language terms
(data) which are vague. Almost all answers to questions
raised in everyday life are within some proximity of the
absolute truth

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Probability vs uncertainty vs Fuzziness
   Probability theory is one of the most traditional
theories for representing uncertainty in mathematical
models
   Nature of uncertainty in a problem is a point which
should be clearly recognized by engineer - there is
uncertainty that arises from chance, from
imprecision, from a lack of knowledge, from
vagueness, from randomness…
   probability theory deals with the expectation of an
event (future event, its outcome is not known yet),
i.e. it is a theory of random events

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   Fuzziness deals with the impression of
meaning of concepts expressed in natural
language - it is not concerned with events at all
   Fuzzy theory handles nonrandom uncertainty

Random

Uncertain                      Certain

Fuzzy, imprecise, vague

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   a Fuzzy System (FS) is defined as a system with operating
principles based on fuzzy information processing and
decision making
   There are several ways to represent knowledge, but the
most commonly used has a form of rules:
IF (premise)A THEN (conclusion)B

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   From a knowledge representation viewpoint, a fuzzy IF-
THEN rule is a scheme for capturing knowledge that
involves imprecision - if we know a premise (fact), then we
can infer another fact (conclusion)
   A fuzzy system (FS) is constructed from a collection of
fuzzy IF-THEN rules
   Acquisition of knowledge captured in IF-THEN rules is
NOT a trivial task (expert knowledge, systems
measurements, etc.)
   The building blocks for fuzzy IF-THEN rules are FUZZY
SETS

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   The rule
“IF the air is cool THEN set the motor speed to slow”
has a form:
IF x is A THEN y is B,
where fuzzy sets “cool” and “slow” are labeled by A and B,
correspondingly
   A and B characterize fuzzy propositions about variables x
and y
   Most of the information involved in human communication uses
natural language terms that are often vague, imprecise,
ambiguous by their nature, and fuzzy sets can serve as the
mathematical foundation of natural language

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   A Fuzzy Set is a set with a smooth boundaries
    Fuzzy Set Theory generalizes classical set theory to allow
partial membership
   Fuzzy Set A is a universal set U is determined by a
membership function A(x) that assigns to each element
xU a number A(x) in the unit interval [0,1]
   Universal set U (Universe of Discourse) contains all
possible elements of concern for a particular application
   Fuzzy set has a one-to-one correspondence           with its
membership function

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   Fuzzy set A is defined as
A = { (x, A(x)) }, xU, A(x)[0,1]
   A(x) = Degree(xA) is a grade of membership of element
xU in set A

X1
X2
X3
.                  0        1/2         1
.
.
unit interval
xN

.
.
U (universe of discourse)

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   The membership functions themselves are NOT fuzzy - they are
precise mathematical functions; once a fuzzy property is
represented by a membership function, nothing is fuzzy
anymore
   Suppose U is the interval [0,85] representing the age of ordinary
human beings, and the linguistic term “young” as a function of
age (value of the variable age) can be defined as
25       85                       -1
  x - 25        2
A  " young"

0
1  1 
x    5  
25
 

x

[see the graphical representation on the next slide]
[ !! pay attention to the usage of the symbol “ / “ ]

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1                           (x)            Universe of
A
discourse U is
0.41                                   continuos
31

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0        U (universe of discourse)    85
   If U is a set of integers from 1 to 10 ( U={1,2,…,10} ), then
“small” is a fuzzy subset of U, and it can be defined using
enumeration (summation notation):
A = “small” = 1/1+1/2+0.85/3+0.75/4+0.5/5+0.3/6+0.1/7

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   In the previous example elements of U (universal set) with
zero membership degrees are not included into
enumeration
   A notion of a fuzzy set provides a convenient way of
defining abstraction - a process which plays a basic role in
human thinking and communication
   All theories that use the basic concept of fuzzy set can be
called in a whole Fuzzy Theory
   Rough classification of Fuzzy Theory can be depicted as
follows [note that dependencies between the branches are not
shown] :

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Fuzzy Theory

Fuzzy                     Fuzzy                   Uncertainty &
Mathematics             Decision-Making              Information

Fuzzy Systems               Fuzzy Logic
& AI

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crisp (classical) set A   A = set of TALL people      fuzzy set A

1.0                                     1.0

0.65

0.0                                     0.0

1.75m               height                1.75m

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Membership Functions
  Main types of membership functions (MF):
(a) Triangular MF is specified by 3 parameters {a,b,c}:

 0,                     if x  a
 (x - a) (b - a),     if a  x  b

trn(x : a, b, c)  
 (c - x) (c - b),     if b  x  c
 0,
                        if x  c

(b) Trapezoidal MF is specified by 4 parameters {a,b,c,d}:

   0,                   if x  a
   (x - a) (b - a), if a  x  b


trp(x : a,b,c, d)     1,              if b  x  c
   (d - x) (d - c),   if c  x  d


   0,                     if x  d

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(c) Gaussian MF is specified by 2 parameters {a,}:
 - (x - a)2 
gsn(x : a, )  e xp            
 2         
            

(d) Bell-shaped MF is specified by 3 parameters {a,b,}:

1
bll(x : a,b,)                     2b
x-
1
a

(e) Sigmoidal MF is specified by 2 parameters {a,b}:

1
sgm(x : a,b) 
1  e- a(x -b)

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Terima Kasih

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