Fuzzy logic and fuzzy time series edited

1
Fuzzy Logic and Fuzzy Set
Theory & Time Series Modeling
Prof. Dr. S.M. Aqil Burney
College of Computer Science and
Information System (IoBM)
aqil.burney@iobm.edu.pk
Former Meritorious Professor and Chairman Dept. of Computer Science
University of Karachi (1973-20110) burney@uok.edu.pk
www.burney.net

Reasoning
• Inductive and Deductive Reasoning ……
• AI and Logic
see
2

Vageness & Uncertainty
• It is evident that as we are using ICT and
other technologies and generating huge data
we come across complexity which consists
Vagueness and Uncertainty which could not
be handled amicably without logic which ,
helps us to formulate & model and solve the
problems.
3

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Fuzzy Logic and Fuzzy Set Theory
e.g. A={1,2,3,4}, so 1 belongs to A but 5 is not
member of set A.
• Fuzzy sets are a natural outgrowth and
generalization of crisp sets.
• It tells us besides “belongs to” and “not belongs
to” way, other possibilities exist in the relation
between an element and a set emerging in
various practical processes.
• A crisp set defines only two possibilities
“belongs to=(1)” or “not belongs to=(0)”.
Lotfi Zadeh

Fuzzy Logic and Fuzzy Set Theory: (Cont…)
• Unlike crisp sets with sharp boundaries as discussed above, a
fuzzy set is a set with smooth (un-sharp) boundaries.
• e.g. A set whose elements PARTIALLY belongs to that set.
• e.g 40% belongs to and 60% not belongs to some particular set.
• Binary sets contains only two values 0 and 1, Whereas fuzzy
sets consider all the values in the interval [0,1].
• Classical/crisp sets are suitable for various applications and
have proven to be important tool. But they do not reflect the
nature of human concepts and thoughts, which tends to be
abstract and imprecise.
• Fuzzy logic is the study of imperfect, imprecise and ambiguous
knowledge. This knowledge includes the linguistic chaos as well.
5

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Some Examples of fuzzy data (Linguistic Chaos)
•Today is very hot day.
(Degree of hotness is not defined. Inexact value)
• He is very intelligent
(Here intelligence is a matter of degree (%) and differ from
person to person)
• If you work hard then you will get the success
(Inference based on qualitative data)

Fuzzy Logic : Degree of Relation
• We know many things in life are degree
rather than present or not present.
• A green and red apple is not just green and
red; there many levels of green and red
shades, computer scientists ,technologist
and engineers and to some extend
statistician have accepted this theory.
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Example (Degree of Relation)
• A pixel can have a bright ness level between
0 and 255.
• 0 value = Black 255 value = White ,
• While every value between 0 and 255 gives a
gray level.
8

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• Fuzzy theory holds that many things in life are matter of degree.
• Thus the association of each element in a universe of discourse
is a matter of degree, which is a number between 0 and 1.
e.g its 60% cold in this hall and 40% not cold. So degree of
coldness is a fuzzy concept.
• This is represented by where A= fuzzy set and X is the
universe of discourse.
• Relation of an element with its set A is partially true and partially
false .
• Law Contradiction(excluded middle) needs revisit.
• Law Included Middle
( )xAµ

10
• Fuzzy sets are a natural outgrowth and generalization of crisp
sets.
• A fuzzy set can be defined in two ways.
 Enumerating membership values of those elements in the
set (completely or partially).(Discrete membershop function)
 Defining membership function mathematically for the given
universe of discourse. The universe of discourse may be
discrete or continuous or may be mixture of the two types.

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• A crisp set can be generalized to multiple categories and each
category is assigned its relevant value called the membership
value.
• Larger values denote high degrees of set membership
• For simplicity and completeness, a membership function (MF)
maps every element of a universe of discourse X into real
numbers [0,1]

12
• For notational purpose, membership function of a fuzzy set is given
by
( ) ( ) ( ) ( ) nnAAAi
i
iA xxxxxxxxA /.../// 2211 µµµµ +++== ∑
where ix X∈ In case of continuous universe of discourse,
( )∫=
x
iiA xxA /µ

13
• Formally, a fuzzy set A along with its MF ( ) Xxx iA ∈;µ is defined as
( )( ){ }XxxxA ∈= A, µ
Where ( ) Xxx iA ∈;µ can take any of the function that
satisfies the conditions of a fuzzy membership function

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Example: Discrete Universe of Discourse
. One poAn insurer wants to classify the types of plan for late
premium paymentsssible reason is the possible number of months of
delay in premium payments (X). Let
{ }10,...,2,1=X
be the set of available types of offers to the customers by the
insurance company. Then the fuzzy set “ease of payment to the
customer” may be described as follows:
A={(1,0.2), (2,0.5), (3,0.6), (4,0.7), (5,0.8), (6,0.9)}

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1 2 3 4 5 6 7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Premium payements
Membershipvalues
Ease of Payment for a Customer

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Properties of Fuzzy Sets
The fuzzy logic provides us an intuitively pleasing method of
representing one form of uncertainty.
In designing fuzzy inference systems, some preliminary concepts
need to be defined properly.
Some definitions necessary for designing of fuzzy systems are given

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Given a fuzzy set A defined on X and any number [ ]0,1α ∈ , the
α -cut, Aα
and the strong α -cut, A+α
are crisp sets given by
( ){ }αα
≥= xAxA |
( ){ }αα
>=+
xAxA |

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Example: Level Set Representation of Fuzzy Sets Providing Discrete
Approximation to Continuous Membership Functions
Age Fuzzy Set
Young
Middle Age
Old
A discrete approximation of A2 (Middle Age) is presented and the MF values of
A2 are denoted by D2.
( ) ( )





≥
<<−
≤
=
350
352015/35
201
1
x
xx
x
xA
( )
( )
( )







≤≤
<<−
<<−
≥≤
=
45351
604515/60
352015/20
60200
2
x
xx
xx
xorx
xA
( ) ( )





≥
<<−
≤
=
601
604515/45
450
3
x
xx
X
xA

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Discrete Approximation of Continuous Universe of Discourse
0 10 20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
1.2
YoungYoung Middle Aged Old

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Age Factor and Related Risk Membership Function
0 10 20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
1.2
X= Age
MembershipGrades
Young Middle Aged Old

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Support ( ) of a Fuzzy Set
( ){ }0|)( >= xxASupport Aµ
The support of a fuzzy set A denoted by supp(A), within a universe
of discourse X is the crisp set that contains all the elements of X that
have nonzero membership grades in A.
A+0

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Fuzzy convexity
( )( ) ( ) ( )1 2 1 21 min ,A x x A x A xµ λ λ  + − ≥  
A set A in n
R is called convex iff, for every pair of points
is also in A
Here the sets A1-A5 are convex and A6-A9 are non-convex sets.

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Fuzzy symmetry
A fuzzy set A is symmetric if its MF is symmetric around a certain
point , x=c, namely,
( ) ( ) ,A Ac x c x x Xµ µ+ = − ∀ ∈

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Some Membership Functions
A membership function provides a gradual transition form regions
completely outside a set to regions completely inside the set.
Its usefulness depends critically on our capability to construct
appropriate membership functions for various given concepts in
various contexts.
Even for similar contexts, the representation of a system using
fuzzy logic may vary considerably.

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Triangular Membership Function
( )









≤
≤≤
−
−
≤≤
−
−
≤
=
xc
cxb
bc
xc
bxa
ab
ax
ax
cbaxtriangle
0
0
,,;
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
trim f(x, [3 4 5]); trim f(x, [2 4 7]); trim f(x, [1 4 9]);
Triangular M em bership Function
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
Triangular M em bership Function
0.6
0.8
1
Different shapes of Triangular function for varying parameters

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Trapezoidal Membership Function
( )











≤
≤≤
−
−
≤≤
≤≤
−
−
≤
=
xc
dxc
cd
xd
cxb
bxa
ab
ax
ax
dcbaxtrapezoid
0
1
0
,,,;
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
trapmf(x, [2 3 7 9]); trapmf(x, [3 4 6 8]); trapmf(x, [4 5 5 7]); trapmf(x, [5 6 4 6]);
Trapezoidal Membership Function

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A Gaussian membership function is specified by two parameters.
Gaussian Membership Function
( ) 












 −−
=
2
2
2
2
1
exp,;
σ
σ
cx
cxgaussian
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
gaussm f(x, [0.5 5]); gaussm f(x, [1 5]); gaussm f(x, [2 5]); gaussm f(x, [3 5]);
G aussian C urve M em bership Function
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
gaussm f(x, [0.5 5]); gaussm f(x, [1 5]); gaussm f(x, [2 5]); gaussm f(x, [3 5]);
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
gaussm f(x, [1 2]); gaussm f(x, [1 4]); gaussm f(x, [1 6]); gaussm f(x, [1 8]);
Varying c and fixed variance Fixed c and varying variance

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Sigmoidal Membership Function
( )
( )[ ]
,
exp1
1
,;
cxa
caxsig
−−+
=
Depending on the sign of a, a sigmoidal function is inherently open
right or open left and thus is appropriate for representing concepts
such as “very risky” or “very negative”.
Sigmoidal functions of this type are used in training of neural
networks

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Sigmoidal Membership Function (Cont…)
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
sigmf(x, [-1 5]); sigmf(x, [-3 5]); sigmf(x, [4 5]); sigmf(x, [8 5]);
Sigmoid Curve Mmbership Function
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
sigmf(x, [5 2]); sigmf(x, [5 4]); sigmf(x, [5 6]); sigmf(x, [5 8]);
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
sigmf(x, [-1 5]); sigmf(x, [-3 5]); sigmf(x, [4 5]); sigmf(x, [8 5]);
Sigmoid Curve Mmbership Function
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
sigmf(x, [5 2]); sigmf(x, [5 4]); sigmf(x, [5 6]); sigmf(x, [5 8]);
Varying a and fixed c
Fixed a and varying c

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Fuzzy Set Theoretic Operations
• Corresponding to the crisp set operations, we can define fuzzy
set operations for fuzzy sets.
• The set operations intersection and union correspond to logic
operations, conjunction (AND) and disjunction (OR), respectively.
• Normally, for fuzzy intersection, we use ‘min’ or ‘AND’ operator
and for fuzzy union, we usually apply ‘max’ or ‘OR’ operators.

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Fuzzy Intersection (conjunction)
The intersection of two fuzzy sets A and B is specified in general
by a function [ ] [ ] [ ]1,01,01,0: →×T
If membership values of A and B are ( )xAµ and ( )xBµ then fuzzy
conjunction is given by
( ) ( ) ( ) ( ), ,A B x T A x B x x X ∩ = ∀ ∈ 
where T represents a binary operation for the fuzzy intersection.

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T-norm
This class of fuzzy intersection operators are usually referred to as
T-norm (triangular norm) operators.
T-norm operator satisfies the fuzzy arithmetic axioms like boundary
condition, monotonicity, commutativity, associativity, continuity and
etc.
Examples of frequently used fuzzy intersection.
( ) ( )babaT ,min, =
( ) abbaT =,
Standard intersection:
Algebraic product:

33
Fuzzy Union (Disjunction)
Like fuzzy intersection, fuzzy union operator is defined by a function
[ ] [ ] [ ]1,01,01,0: →×S
( ) ( ) ( ) ( ), ,A B x S A x B x x X ∪ = ∀ ∈ 
where S represents a binary operation for the fuzzy union.
Properties that a function S satisfies to be intuitively acceptable as
a fuzzy union are exactly the same as properties of t-conorm.
Some frequently used t-conorm operation
( ) ( )babaS ,max, =
( ) abbabaS −+=,
Standard disjunction:
Algebraic product:

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Membership Functions of Two Dimensions
In multivariate studies, we need to define membership functions of
higher dimensions.
For example, let X and Y be two fuzzy numbers and R be the 2-D
fuzzy set on Z X Y= ×
Cylindrical extension is a natural way to extent one-dimensional
MFs to two-dimensional MFs define below.

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Cylindrical Extension
If A is a fuzzy set in X, then its cylindrical extension in
is a fuzzy set defined by
YX ×
( ) ( ) ( )∫×
=
YX
A yxxAc ,µ
The concept of cylindrical extension extends the dimensions of a
given MFs.

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Fuzzy Projection
• Projects a fuzzy relation to a subset of selected dimensions.
• Often used to extract marginal possibility distribution of a few
selected variables from a given fuzzy relation.
• It decreases the dimension of a given (multidimensional) MF.
• Let R be a two-dimensional fuzzy set on .Then the
projection of R onto X and Y are defined as
YX ×
( ) yyxR R
Y
x
Y ,max µ∫=
( ) xyxR R
X
y
X ,max µ∫=

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Extension Principle
• Provides a general procedure for extending crisp domains of
mathematical expressions to fuzzy domains.
• Plays fundamental role in extending any point-to-point
operations to fuzzy operations.

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Extension Principle For One-to-one Fuzzy Relations
Assume X and Y are two crisp sets and let be a mapping
from X into Y, ,YXf →:
f
( ),x X f x y Y∈ = ∈
Assume A is a fuzzy subset of X, using extension principle, we can
define as a fuzzy subset of Y such that,( )Af
( ) ( ) ( ) ( ) ( ) ( ) nnAAAiiA
Xx
B yxyxyxyxAfy /.../// 2211 µµµµµ +++=∪==
∈
( ) .,...,2,1, nixfy ii ==where

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Extension Principle for Many-to-one Fuzzy Relations
If is a many-to-one relation, then we may have more than
one possibility at each value of x. that is ,
Therefore,
f(x)y =
( ) ( ) Yyxfxf ∈== *
21 2121 ,, xxXxx =∈
( ) ( )
( )xy A
yfx
B µµ 1
max−
=
=

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Generalized Extension Principle
iA
Suppose that f is a mapping from an n-dimensional Cartesian space
nXXX ××× ...21 to a single universe of discourse Y such that ( ) ,,...,, 21 yxxxf n =
and for each , 1,2,...,iX i n= a fuzzy membership is defined.
Thus total n fuzzy membership functions are defined. Then by the
extension principle, we can define
( ) ( )
( )[ ] ( )
( )



=
≠
=
−
−
= −
φ
φµ
µ
yfif
yfifx
y
iAi
yfxxx
B
i
n
1
1
,...,,
,0
minmax 1
21

41
Where to Apply
In fuzzy neural networks, fuzzy projection and extension principle
are used to generalize backpropagation algorithm to fuzzy
backpropagation learning algorithm.

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2. Fuzzy Relations
Topics to be discussed in this section
• Binary Relations
• Linguistic Hedges
• Fuzzy If-Then Rules
• Fuzzy Reasoning

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Fuzzy Relations
• Generalizes the notion of crisp relation into one that allows
partial membership.
• Degree of association can be represented by membership
grades in a fuzzy relation in the same way as the degrees of set
membership are represented in the fuzzy sets.
• A relation defined between two objects is represented by a
binary relation. Similarly, we can form tern-ary, quartern-ary,
quin-ary or n-ary relation between three, four, five or n objects,
respectively.

44
Binary Crisp Relationship
Mathematically speaking, if x and y be two variables from two
domains X and Y respectively, then the binary relation between x
and y, R(x,y) is a subset of Cartesian space of X and Y.
define the relationX<y, as below
( ){ }RyxyxyxR ∈<= ,,|,
( ) YXyxR ×⊆,Where

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Crisp n-Dimensional Relations
( )1 2 1 2, ,..., ...n nR X X X X X X⊆ × × ×
In general, for n-dimensional arguments taken from the
domains ,then
nxxx ,...,, 21
1 2, ,..., nX X X
We see that a relation is again a set and thus follows the same
rules as the domain of Cartesian product of nXXX ,...,, 21

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Binary Fuzzy Relations
If x and y are two fuzzy variables with domains X and Y then the
binary fuzzy relation R is
( ) ( ) ( ){ }YXyxyxyxR R ×∈= ,|,,, µ
[ ]1,0: →×YXRwhere

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Fuzzy Relations
• The binary fuzzy relation can be extended for n-arguments.
• This allows the characteristic function of a crisp relation to allow
tuples to have degree of membership within the relation.
• The membership grades indicate the strength of the relation
present between the elements of the tuple.
• A fuzzy relation is a fuzzy set defined on the Cartesian product
space of crisp sets where the tuples may have
varying degrees of membership within the relation.
( )nxxx ,...,, 21

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Fuzzy Composition Rule
• Fuzzy relationship in different product spaces can be combined
through a composition operation.
• Different composition operations have been suggested for
fuzzy relations
• The best known is the max-min composition operation by
Zadeh (1965).

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Max-Min Fuzzy Composition Rule of Inference
Let and be two fuzzy relation defined on and
,respectively. The max-min composition of and is a fuzzy set
defined by
1R 2R YX × ZY ×
1R 2R
( ) ( ) ( )( ){ }ZzYyXxzyyxzxRR RR
y
∈∈∈= ,,|,,,minmax,, 2121 µµ
or equivalently,
( ) ( ) ( )( )zyyxzx RR
y
RR ,,,minmax, 2121
µµµ =
( ) ( )[ ]zyyx RR
y
,, 21
µµ ∧∨
The composition rule of inference is not uniquely defined. By choosing different
fuzzy conjunction and disjunction operators, we can get different composition
rules of inference.

50
Max-Product Fuzzy Composition Rule of Inference
An alternate to max-min composition called max-product
composition is used due to its higher mathematical tractability then
max-min composition and can be defined as same as max-min
composition:
( ) ( ) ( )( )zyyxzx RR
y
RR ,,,max, 2121
µµµ =

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Properties of Fuzzy Relations
The fuzzy composition rules follow several properties common to
binary relations. If A, B and C are binary relations on and
then
ZYYX ×× ,
WZ ×
Associativity
Distributive over union
Weak distributivity over intersection
Monotonicity
( ) ( )A B C A B C=   
( ) ( ) ( )CABACBA  ∪=∪
( ) ( ) ( )CABACBA  ∩⊆∩
CABACB  ⊆⇒⊆

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3. Fuzzy IF-Then Rules
Topics to be discussed in this section
• Linguistic variables
• Linguistic Hedges
• Fuzzy If-Then Rules
• Fuzzy Reasoning

53
Fuzzy IF-Then Rules
Fuzzy If-then rules or fuzzy inferencing is an extension of crisp
propositional statements.
They allow human knowledge and common sense representation
using modes-ponen rule of inference and are able to make
conclusions in the presence of uncertainty and chaos.
To deal with variety of decisions on a single problem, give place to
include fuzzy hedging operators that enables a fuzzy inference
system to deal with extremities.

54
Linguistic variables
The concept of fuzzy numbers plays a fundamental role in
formulating quantitative fuzzy variables, i.e., the variables whose
states are fuzzy numbers.
When in addition, the fuzzy numbers represents linguistic
concepts, such as very low, high, extreme and so on, as
interpreted in a particular context, the resulting constructs are
usually called linguistic variables.
A linguistic variable is characterized by a quintuple ( )( ), , , ,x T x X G M
x =is the name of variable; X is the universe of discourse.
T(x) =is the linguistic term set of x;
G =is a syntactic rule which generates the terms in T(x) and
M =is the semantic rule which associates with each linguistic
value A its meaning M(A), where A denotes a fuzzy set in X.

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Linguistic Hedges
A linguistic variable enables its values to be described qualitatively
by a linguistic term and quantitatively by a corresponding
membership function.
The linguistic term is used to express concepts and knowledge in
human communication, whereas membership function is useful for
processing numerical input data.
For example, very, more or less, fairly, or extremely are all hedges
defined for linguistic variables.

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Linguistic Hedges (Cont...)
Let A be a linguistic value characterized by a fuzzy set with
membership function , Then is interpreted as modified
version of the original value expressed as
( ).Aµ k
A
( )[ ]∫==
X
k
A
k
xxA µ
The linguistic hedges can either concentrate (increase) or dilate
(decrease) the significance of a fuzzy set.
( ) 0,ACON ≥= kAk
( ) 10, <<= kAADIL k
Using linguistic hedges, we can define composite linguistic
terms in fuzzy reasoning.

57
Fuzzy If-Then Rules
A fuzzy if-then rule (or fuzzy implication) defines a relation between
x and y.
A fuzzy if-then rule be defined as a binary fuzzy relation R on the
product space YX ×
If A and B are two fuzzy sets defined over X and Y and
then the implication is given as
YyXx ∈∈ ,
A B→
if x is A then y is B
Here "x is A" is the antecedent or premise, while "y is B" is called
the consequent or conclusion.

58
Fuzzy If-Then Rules (Cont…)
Some examples of fuzzy if-then rules are:
• if risk is high then premium is high
• if interest rate is high then liquidity is low.
• if rate of return is adequate then investment will increase

59
Fuzzy Reasoning/Fuzzy Expert System
An inferential procedure that derives conclusions from a set of if-
then rules and known facts.
Using compositional rules of inference and generalized modes-
ponens (GMP) rule, we can define three possible cases in fuzzy
reasoning
• Single rule with single antecedent. (SRSA)
• Single rule with multiple antecedents. (SRMA)
• Multiple rules and multiple antecedents. (MRMA)

60
4. Fuzzy Aggregation Operations

61
Fuzzy Aggregation/Averaging Operations
• A class of types of defuzzification in fuzzy inference/expert
systems.
• Using fuzzy aggregation operations on fuzzy sets, we can obtain
appropriate single fuzzy set.
• In fuzzy inference engines, these operations allow us to combine
multiple rules using single rule of inference.

62
Axioms of Fuzzy Aggregation Operations (Cont…)
Formally, we define
[ ] [ ]1,01,0: →
n
h
nAAA ,...,, 21 are fuzzy sets defined on X.
Thus the aggregated fuzzy set A defined over X will be
( ) ( ) ( ) ( )( )xAxAxAhxA n...,,, 21= for each Xx∈
Fuzzy Aggregation operations are necessary in defuzzification of
ordered/unordered fuzzy knowledge.

63
Axioms of Fuzzy Aggregation Operations
In order to qualify as an intuitive and meaningful aggregation
function (h), it must satisfy at least following three requirements.
Axiom 1: ( ) ( ) 11,...,1,100,...,0,0 == handh (Boundary condition)
Axiom 2: ( )naaa ,...,, 21 and ( )nbbb ,...,, 21 such that, if ii ba ≤ ,then
( ) ( )nn bbbhaaah ,...,,...,, 2,121 ≤ (Monotonically increasing)
Axiom 3: Fuzzy aggregation operation, h, is a continuous function.

64
Axioms of Fuzzy Aggregation Operations (Cont…)
Other two additional axioms are:-
h is symmetric function in all its arguments; i.e.,
Axiom 5:
Axiom 4:
( ) ( ) ( ) ( )( )npppn aaahaaah ,...,,,...,, 2121 =
For any permutation of p on N
h is an idempotent function; that is,
( ) [ ]1,0,...,, ∈∀= aaaaah

65
Fuzzy Idempotency for Aggregation Operations
Note that, if any aggregation operation satisfies axioms 2-5, then
it also satisfy the inequality
( ) ( ) ( )nnn aaaaaahaaa ,...,,max,...,,,...,,min 212121 ≤≤
( ) [ ]n
naaa 1,0,...,, 21 ∈for all n-tuples

66
Generalized Fuzzy Aggregation Operation
** All aggregation operations between the standard fuzzy
intersection and union are idempotent.
These aggregation operations are usually called averaging
operations.
The generalized mean is defined as:
( )
αααα
α
1
21
21
...
,...,, 




 +++
=
n
aaa
aaah n
n

67
5. Fuzzy Time Series
Pioneer Researchers and practitioners:
1. Zadeh (1975)
2. Yager R. R. (2005)
3. Kacpryzk (Germany) (Springer-Verlag)
4. Klir (2005)
5. Chen S. M. and Lee L.W. (2004, 2006,2007,2008)
6. Huang K. (2001, 2003, 2004, 2007, 2008)
7. Zimmermann (2002)
8. Oscar Castillo (2007)
9. Burney and Jilani (2006, 2007, 2008)

68
Each observation is assumed interval based fuzzy variable
alongwith associated membership function.
Less than one and half decade of history of fuzzy time series.
Based on fuzzy relation and fuzzy inference rules, efficient modeling
and forecasting of fuzzy time series is possible.
Fuzzy Times Series

69
Fuzzy Times Series
Time series analysis plays vital role in most of the actuarial related
problems.
Based on fuzzy relation, section 2.2.7 and fuzzy inference rules,
section 2.2.8, efficient modeling and forecasting of fuzzy time
series is possible.
This field of fuzzy time series analysis is not very mature due to
the time and space complexities in most of the actuarial related
issue.

70
Review of Fuzzy Time Series
Song and Chissom (1993a; 1993b; 1994) presented the concept of fuzzy time series
based on the concepts of fuzzy set theory to forecast the historical enrollments of the
University of Alabama.
Huarng (2001b) presented the definition of two kinds of intervals in the universe of
discourse to forecast the TAIFEX.
Chen (2002) presented a method for forecasting based on high-order fuzzy time series.
Lee et. al. (2004) presented a method for temperature prediction based on two-factor
high-order fuzzy time series.
Melike and Konstsntin (2004) proposed forecasting method using first order fuzzy time
series.
Lee, Wang and Chen (2006) presented handling of forecasting problems using two-factor
high order fuzzy time series for TAIFEX and daily temperature in Taipei, Taiwan.

71
Jilani T. A. and Aqil Burney S. M. (2008), A Refined Fuzzy Time Series Model for
Enrollments Problem, Physica A, Elsevier Publishers.
Jilani T. A. and Aqil Burney S. M. (2008), Multivariate Stochastic Fuzzy Forecasting
Models, Expert Systems with Applications 37(2), Elsevier Publishers.
Jilani T. A. and Aqil Burney S. M. (2007), M-Factor High Order Fuzzy Time Series
Forecasting for Road Accident Data, In IEEE-IFSA 2007, World Congress, Cancun,
Mexico, June 18-21, In Castillo, O.; Melin, P.; Montiel Ross, O.; Sepúlveda Cruz, R.;
Pedrycz, W.; Kacprzyk, J. (Eds.), Design and Analysis of Intelligent Systems
usingFuzzy Logic and Soft Computing vol. 41, Advances in Soft Computing, Berlin:
Springer-Verlag.
Jilani T. A., Aqil Burney S. M. and Ardil C.(2008), Multivariate High Order Fuzzy Time
Series Forecasting for Car Road Accidents. International Journal of Computational
Intelligence, Vol. 4, no. 1. pp. 7-16.
Review of Fuzzy Time Series (Cont…)

72
Jilani T. A., Aqil Burney S. M. and Ardil C. (2007), Fuzzy metric approach for fuzzy
time series forecasting based on frequency density based partitioning, International
Conference on Machine Learning and Pattern Recognition. August 24-26 (2007),
Berlin, Germany.
Jilani T. A. and Aqil Burney S. M. (2007), A New Quantile Based Fuzzy Time Series
Forecasting Model, submitted in Computers and Mathematics With Applications
(CMWA), Elsevier Publishers.
Jilani T. A. and Aqil Burney S. M. (2007), Fuzzy Time Series Forecasting Using
Frequency Density Based Partitioning for Enrollments Problem, submitted in Expert
Systems with Applications (ESWA), Elsevier Publishers.
Review of Fuzzy Time Series (Cont…)

73
Two-Factor kth-Order Fuzzy Time Series Model
We can extend the concept of single antecedent and single
consequent (one-to-one) to many antecedents and single
consequent (many-to-one).
For example, in designing two-factor kth-order fuzzy time series
model with X be the primary and Y be second fact.
We assume that there are k antecedent
and one consequent
( ) ( ) ( )( )kk YXYXYX ,,...,,,, 2211
1+kX
( ) ( ) ( ) ( )1122221111 ,,...,,,, ++ =→====== kkkkkk xXyYxXyYxXyYxXIf

74
Two- Factor Fuzzy inferencing
We can extend the concept of single antecedent and single
consequent (one-to-one) to many antecedents and single
consequent (many-to-one).
For example, in designing two-factor kth-order fuzzy time series
model with X be the primary and Y be second factor.
( ) ( ) ( ) ( )1122221111 ,,...,,,, ++ =→====== kkkkkk xXyYxXyYxXyYxXIf
We assume that there are k antecedent ( ) ( ) ( )( )kk YXYXYX ,,...,,,, 2211
and one consequent 1+kX

75
M-Factor and k-th Order Fuzzy Time Series Model
( ) ( )
( ) ( )
11 11 12 12 1 1 21 21 22 22 2 2
1 1 2 2 1, 1 1, 1
, ,..., , , ,..., ,...,
, ,...,
1,2,..., , 1,2,...,
k k k k
m m m m mk mk m k m k
X x X x X x X x X x X x
X x X x X x X x
for i m j k
+ + + +
= = = = = =
= = = =
= =
If
then
In the similar way, we can define m-factor (i=1,2,…,m) and kth
order (k=1,2,…,k) fuzzy time series as

76
• We can define relationship among present and future state of a
time series with the help of fuzzy sets.
Fuzzy Time Series
kA , respectively, where UAA kj ∈, ,then kj AA → represented the
Assume the fuzzified data of the thi ( )thi 1+ day areand jA and
andjAfuzzy logical relationship between kA

77
Fuzzy Time Series
• Let ( ) ( ),...2,1,0...,,tY =t be the universe of discourse and ( ) RtY ⊆
Assume that ( ) ,...2,1, =itfi is defined in the universe of discourse
( )tY and ( )tF is a collection of ( ) ( ),...2,1,0...,,tf i =i , then ( )tF
is called a fuzzy time series of ( ) ,...2,1,tY =i
• Using fuzzy relation, we define, ( ) ( ) ( )1,1 −−= ttRtFtF  , where ( ), 1R t t −
is a fuzzy relation and “  ” is the max–min composition operator,
then ( )tF is caused by ( )1−tF ( )tF ( )1−tF, where and
are fuzzy sets.

78
( )tF ( )tF
( ) ( ) ( )1 , 2 ,...,F t F t F t n− − −
Let be a fuzzy time series. If is caused by
then the fuzzy logical relationship is represented by
( ) ( ) ( ) ( )tFtFtFntF →−−− 1,2,...,
is called the one-factor nth order fuzzy time series forecasting
model.
Univariate Vector Fuzzy Logic Inferencing
New Forecasting Methods Based on M-Factors High-Order Fuzzy
Time Series

79
Bivariate Vector Fuzzy Logic Inferencing
Time Series
Let ( )tFbe a fuzzy time series. If( )tF is caused by
then this fuzzy logical relationship is represented by
( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 2 1 2 1 21 , 1 , 2 , 2 ,..., ,F t F t F t F t F t n F t n− − − − − −
( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )tFtFtFtFtFntFntF →−−−−−− 1,1,2,2,...,, 212121
and
is called the two-factors nth order fuzzy time series forecasting
model, where ( )tF1
( )tF2 are called the main factor and the
Secondary factor FTS respectively.

80
• In the similar way, we can define m-factor nth-order fuzzy logical
relationship as
Mutivariate Vector Fuzzy Logic Inferencing
( ) ( ) ( )( ) ( ) ( ) ( )( )
( ) ( ) ( )( ) ( )tFtFtFtF
tFtFtFntFntFntF
m
mm
→−−−
−−−−−−
1,...,1,1
,2,...,2,2,...,,...,,
21
2121
Here ( )1F t is called the main factor and ( ) ( ) ( )2 3, ,..., mF t F t F t
secondary factor FTS.
are called
Time Series

81
New Forecasting Methods Based on M-Factors High-
Order Fuzzy Time Series
Steps:
Step 1) Define the universe of discourse, U of the main factor
[ ]min 1 max 2,U D D D D= − −
where
minD and maxD are the minimum and the maximum values of the main
1D , 2D
proper positive real numbers to divide the universe of discourse into
n-equal length intervals 1 2,, ..., lu u u
factor of the known historical data, respectively, and are two

82
Some Observations
• Here we can implement any of the fuzzy membership function to
define the FTS in above equations.
• Comparative study by using different membership functions is
also possible. However, we have used triangular membership
function due to low computational cost.
• Using fuzzy composition rules, we establish a fuzzy inference
system for FTS forecasting with higher accuracy
• Using fuzzy composition rules, we establish a fuzzy inference
system for FTS forecasting with higher accuracy
• The accuracy of forecast can be improved by considering higher
number of factors and higher dependence on history.

Some work in Actuarial Science ,
Health Management
• E- Health Management etc.Management
• Mobile Health Management(Internet of
Things)
• Temporal Database and Fuzzy Logic
• Software Engineering (SCM)
• Medical Image Analysis using Soft
Computing Techniques
83

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