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APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
APPLICATION OF CONDITIONAL PROBABILITY IN
PREDICTING INTERVAL PROBABILITY OF DATA
QUERYING
Rolly Intan
Faculty of Industrial Technology, Informatics Engineering Department,
Petra Christian University
Email: rintan@petra.ac.id
ABSTRACT: This paper discusses fuzzification of crisp domain into fuzzy classes
providing fuzzy domain. Relationship between two fuzzy domains, X i and X j , can
be represented by using a matrix,
wij . If X i has n elements of fuzzy data and X j
has m elements of fuzzy data then
wij is n × m matrix. Our primary goal in this
paper is to generate some formulas for predicting interval probability in the relation
to data querying, i.e., given John is 30 years old and he has MS degree, how about his
probability to get high salary.
Keywords: Fuzzy Conditional Probability Relation, Data Querying, Interval
Probability, Mass Assignment, Point Semantic Unification.
1. INTRODUCTION
In a system, we will find that every
component has relation one to each other.
For example, system CAREER has some
components such as education, age, and
salary in which we realize that all of them
has interrelationship as in general higher
education means higher salary, or older
someone will get higher salary, or in a
certain area, percentage of mid-40’s of
persons who has doctoral degree is 30
percent, etc.
In this paper, we process a certain
relational database by classifying every
domain into several value of data or
elements of the component, i.e., domain or
component age can be classified into
{about_20, about_25, …, about_60}. With
assumption that every classified data is a
fuzzy set, we must determine a membership function which represents degree of
element belonging to the fuzzy set, i.e.,
about_30={0.2/26, 04/27, 0.6/28, 0.8/29,
1/30, 0.8/31, 0.6/32, 0.4/33, 0.2/34}. Next,
we apply the all membership function into
the previous relational database to find
every membership value for every item
data. And then, by using conditional
probabilistic theory, we construct a model
to describe interrelationship among all
components of the system. Relationship
between two components, X 1 and X 2 , of
a system is expressed in a matrix w 12 . If
component X 1 has n elements, X 2 has m
elements then matrix w12 is n × m matrix,
where
a 12
ij ∈ w12
expresses weight or
degree dependency of
x2 j ∈ X 2
from
x1i ∈ X 1 , for 1 ≤ i ≤ n , 1 ≤ j ≤ m . Through
this model, we generate some formulas to
predict any value of component related to
a given query of input data i.e., given
John is 30 years old and he has MS
degree, how many his probability to get
high salary, and off course we have to
define high salary as a fuzzy data value.
Given input of data querying can be
precise as well as imprecise data (fuzzy
data), so first, before the data can be used
to make prediction, we must to find their
probabilistic matching related to element
of components of system by using Point
Semantic Unification Process as introduced in paper of Baldwin [1]. In this
case, Point Value Semantic Unification
can be considered as a conditional
probability between two fuzzy sets [3]. In
135
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
calculating prediction, we generate two
different formulas to provide upper and
lower bound probability of prediction.
Hence, result of prediction works into a
interval truth value [ a, b ] where a ≤ b as
proposed in [2].
Pr( f |g) = ∑ m ij .
2. BASIC CONCEPT
Definition 2.1 P( H | D) is defined as a
conditional probability for H given D.
Relation between conditional and unconditional probability satisfies the following
equation [5].
P( H ∩ D )
,
P( D)
(1)
where P( H ∩ D) is an unconditional
probability of compound events ’H and D
happen’. P(D) is unconditional probability
of event D.
2.2 Mass Assignment
Definition 2.2 Given f is a fuzzy set
defined
on
the
discrete
space
X = {x1 ,..., xn } , namely
n
f =∑
i =1
χi
xi
ment corresponding to the fuzzy set f is [6]
(2)
For example, given a fuzzy set
low_numbers = {1/1, 1/2, 0.5/3, 0.2/4}, the
mass assignment of the fuzzy set
low_numbers is
mlow_numbers = {1,2}:0.5, {1,2,3}:0.3,
{1,2,3,4}:0.2.
2.3 Point Semantic Unification
Definition 2.3 Let mf = {Li:li} and
mg={Ni:ni} be mass assignment associate
136
For example, let f = {1/a,0.7/b,0.2/c}
and g = {0.2/a,1/b,0.7/c,0.1/d} are
defined on X = {a,b,c,d,e }, so that
mf = {a}:0.3, {a,b}:0.5, {a,b,c}:0.2,
mg = {b}:0.3, {b,c}:0.5, {a,b,c}:0.1,
{a,b,c,d}:0.1.
From the following matrix,
0.3 {a}
0.5 {a,b}
0.2 {1,b,c}
0.3
{b}
0
0.15
0.06
0.5
{b,c}
0
0.125
0.1
0.1
{a,b,c}
0.01
0.0333
0.02
0.1
{a,b,c,d}
0.00075
0.025
0.015
the probability Pr(f | g)= 0.53905. It can
be proved that Point Semantic Unification
satisfies
Pr( f |g) + Pr(f |g ) = 1.
(5)
Thus, Point Semantic Unification is
considered as a conditional probability [3].
2.4 Interval Probability
Suppose f is a normal fuzzy set whose
elements
are
ordered
such
that:
χ1 = 1, χi ≤ χ j , if i ≤ j ; The mass assignmf = {{x1 , x2,...,xi } : χi − χi +1 } with xn+1 = 0
(4)
ij
2.1 Conditional Probability
P( H |D ) =
with the fuzzy set f and g, respectively.
From the matrix,
card(L i ∩ N j )
(3)
M = {m ij } =
⋅ li ⋅ n j.
card( M j )
The probability Pr(f | g) is given by [3]:
Definition 2.4 An interval probability
IP(E) can be interpreted as a scope of
probability of event E, P(E), i.e
IP(E)=[e1,e2] means e1 ≤ P( E) ≤e 2 , where
e1 and e2 are minimum and maximum
probability of E respectively[2].
For example, given two probabilities
P(A)=a and P(A)=b for event A and B,
where a , b ∈ [ 0,1].
Minimum probability of compound
event ’A and B happen’, P( A ∩ B ) min , is
the least intersection between A and B,
given by the following equation:
P( A ∩ B) min = max( 0, a + b − 1).
Maximum probability of compound
event ’A and B happen’, P( A ∩ B) max , is
the most intersection between A and B,
given by;
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
Thus interval probability of compound
event ’A and B happened’ is defined as
IP( A ∩ B) = [max(0, a + b − 1),min(a, b)].
(6)
Similarly, minimum and maximum
probability of compound event ’A or B
happens’, are max(a,b) and min(1,a+b)
respectively.
Thus, interval probability of compound event ’A or B happened’ is defined
as:
IP(A ∪ B) = [max(a, b),min(1, a + b)].
(7)
3. CONSTRUCTION MODEL OF SYSTEM
Definition 3.1 System is defined as
S(Er,X,Nm), where
Er: Number of entry data or number of
record or respondent of system.
X: Domain or components of system, if
there are n components then
X=(X1,…,X n).
Nm: Name of system.
For example, given CAREER DATABASE
in Table 1. By assuming that CAREER is
a system which has 24 entries and three
components, education, age, and salary,
therefore Er=24, X=(X1:education, X2:age,
X3: salary), Nm=CAREER. Now, we try to
find relation among education, age, and
salary.
Table 1. CAREER DATABASE
Name
Nm-1
Nm-2
Nm-3
Nm-4
Nm-5
Nm-6
Nm-7
Nm-8
Nm-9
Nm-10
Nm-11
Nm-12
Nm-13
Nm-14
Nm-15
Nm-16
Education
MS
SHS
PhD
JHS
ES
SHS
MS
SHS
MS
SHS
N
JHS
BA
SHS
BA
SHS
Age
35
24
44
45
35
37
39
27
45
56
60
33
54
47
41
21
Sallary
400,000
150,000
470,000
200,000
125,000
250,000
420,000
175,000
415,000
275,000
100,000
300,000
350,000
315,000
355,000
150,000
Nm-17
Nm-18
Nm-19
Nm-20
Nm-21
Nm-22
Nm-23
Nm-24
BA
PhD
ES
JHS
BA
SHS
BA
SHS
52
49
58
59
35
37
31
29
374,000
500,000
125,000
200,000
360,000
255,000
340,000
250,000
First, we classify all three domains or
components as follows.
education
age
salary
= (low_edu,mid_edu,hi_edu),
= (about_20,…,about_60),
= (low_slr,mid_slr,hi_slr).
where we assume that membership
functions of low_edu, mid_edu, and
high_edu:
µ (low_edu) = {1 / N ,0.8 / ES ,0.5 / JHS },
µ (mid_edu) = {0. 2 / ES,0. 5 / JHS ,0.9 / SHS ,0. 2 / BA},
µ (hi_edu) = {0. 1 / SHS , 0.8 / BA,1 / MS,1 / PhD}.
Membership function of age,
µ (about_ n) = {0 .2 /(n − 4),0.4 /(n − 3),0.6 /(n − 2),
0.8 /(n − 1),1 / n,0.8 /(n + 1),0.6 /(n + 2),0.4 /(n + 3),
0.2 /(n + 4)}.
Membership function of low_salary,
mid_salary, and high_salary :
µ (low_slr) = [1 / 0,1 /100000 ,0 / 150000 ],
µ (mid_slr) = [0 / 100000 ,1 /150000 ,1 / 250000 ,
0/300000 ],
µ (hi_slr) = [0 / 250000 ,1 / 300000 ].
By using all membership functions
above, we calculate and transform table 1
into table 2.
Table 2. CAREER FUZZY VALUE
Nama
Nm-1
Nm-2
Nm-3
Nm-4
Nm-5
Nm-6
Nm-7
Nm-8
Nm-9
Nm-10
Nm-11
Nm-12
Nm-13
Education
Age
Salary
LE ME HE 20 25 … 60 LS MS HS
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0.2 0.8 ... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0.5 0.5 0
0
0
... 0
0
1
0
0.8 0.2 0
0
0
... 0 0.5 0.5 0
0 0.9 0.1 0
0
... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0 0.6 ... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0
0
... 0.2 0 0.5 0.5
1
0
0
0
0
... 1
1
0
0
0 0.9 0.1 0
0
... 0
0
0
1
0 0.2 0.8 0
0
... 0
0
0
1
137
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
Nm-14 0 0.9 0.1 0
0
Nm-15 0 0.2 0.8 0
0
Nm-16 0 0.9 0.1 0.8 0.2
Nm-17 0 0.2 0.8 0
0
Nm-18 0
0
1
0
0
Nm-19 0.8 0.2 0
0
0
Nm-20 0 0.9 0.1 0
0
Nm-21 0 0.2 0.8 0
0
Nm-22 0 0.9 0.1 0
0
Nm-23 0 0.2 0.8 0
0
Nm-24 0 0.9 0.1 0 0.2
Σ 3.1 10.9 10 1 1.8
... 0
0
0
1
... 0
0
0
1
... 0
0
1
0
... 0
0
0
1
... 0
0
0
1
... 0.6 0 0.5 0.5
... 0.8 0
1
0
... 0
0
0
1
... 0
0 0.9 0.1
... 0
0
0
1
... 0
0
1
0
... 2.6 1.5 9.4 13.1
Definition 3.2 Xn is defined as compound
attribute to express component of the
system . Xn is a vector. If there are k
elements of Xn then Xn=(x n1,…,x nk), where
x ni is element i of compound attribute Xn
and for further, x ni is called attribute.
For example, if system CAREER has
three compound attributes and their
attributes as follows,
then x 11 =low_edu, x 25 =about_40, etc.
=1
(8)
as
shown
11
12
e4 = 0.5, e2 = 0.9, etc.
∑ P(x ni ) = 1
Given three compound attributes, X1,
X2 and X3. Relation among them can be
illustrated in Fig. 1. as follows.
w11
X1
w21
in
Table
∑e
1≤i≤Er
∑ N(x ni )
(9)
wmn
X 3 w33
a11nm
nm
a
= 21
M
nm
ak 1
a12nm
L
nm
a22
L
M
O
aknm2
L
a11mn
mn
a
= 21
M
mn
a j 1
a12mn
mn
a22
L
L
O
mn
j2
L
a
M
a1nmj
a 2nmj
,
M
a kjnm
a1mn
k
mn
a2 k
.
M
a mn
jk
(10)
1≤i≤k
For example, as shown in Table 2.,
N(x ni)=N(low_edu) = 3.1.
138
w32
w31
Definition 3.6 wnm is defined as weight
matrix, to express degree of dependency of
Xm from Xn. For a k-compound attribute
Xn and a j-compound attribute Xm, wnm
and wmn present two different matrices, as
follows.
2.,
If compound attribute Xn has k attributes
then,
Er =
w12 w13
w23
w22 X 2
Definition 3.4 N(x ni) is defined as sum of
entries value for attribute x ni. If there are
Er number of entries, then
N( x ni ) =
(12)
3.1 Relation Among Compound Attributes
wnm
ni
j
(11)
Figure 1. Relation Among Compound
Attributes, X1, X2 and X3.
Definition 3.3 e nij is defined as membership’s value of entry j for attribute x ni. If
compound attribute Xn has k attributes
then,
Example,
N( xni )
Er
If compound attribute Xn has k attributes
then,
= (low_edu,mid_edu,hi_edu),
= (about_20,…,about_60),
= (low_slr,mid_slr,hi_slr ).
X1 : education
X2 : age
X3 : salary
e nij
∑
1≤i≤k
P( xni ) =
as
1≤i≤k
Note: LE:low_edu, ME:mid_edu, HE =
hi_edu, 20:about_20, 25:about_25, …,
LS=low_slr, MS:mid_slr and HS:hi_slr.
∀j
Definition 3.5 P(x ni) is defined
probability of attribute x ni as follows.
Definition 3.7 Each element of matrix
wnm, entry aihnm
expresses numerical
probabilistic value of relation from xni ∈ X n
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
to xmh ∈ X n . aihnm can also be interpreted as
conditional probability as follows.
a nm
ih
P(x ni ∩ x mh )
= P(x ni|x mh ) =
P(x mh )
1
x1u
(13)
avu21
If there are Er number of entries, then
a
nm
ih
∑
=
1 ≤ j ≤ Er
∑
min (e nij , e mh
j )
1 ≤ j ≤Er
e mh
j
1
(14)
where P(xni ∩ xmh ) express probability of
entries which be inside x ni and x mh.
On the other hand, ahimn expresses
numerical probabilistic value of relation
from xmh ∈ X m to xni ∈ X n . ahimn can also be
interpreted as conditional probability as
follows.
a
mn
hi
P(x ni ∩ x mh )
= P(x mh|x ni ) =
P(x ni )
a mn
hi
min (enij , e mh
j )
1 ≤ j≤ Er
e ni
1 ≤ j ≤ Er j
P(x ni ∩ x ni )
.
P(x ni )
1 a
nn
a
1
wnn = 21
M
M
nn
ak1 aknn2
3.2 Relation
System
L
L
O
L
x3r
1
32
rv
Figure 2. Relation Among Attributes, x1u,
x2v, and x3r.
In order to understand the meaning of
this connection, we use relation of set.
P( x1u ∩ x3 r )
x1u
x2 v
x3 r
P(x1u ∩ x3r ) =
P(x1u ∩ x3r )
⋅ P(x1u ) = a 31
ru ⋅ P(x1u )
P(x1u )
=
P(x1u ∩ x3r )
⋅ P(x3r ) = aur13 ⋅ P(x3r ).
P(x3r )
P( x 2v ∩ x 3r )
x1u
x2 v
a
a
.
M
1
nn
1k
nn
2k
Among
the same area or quantity, are intersection between x 1u and x 3r. In the same
way, we can find two other relations,
32
21
and a12
a23
vr ⋅ P(x3r ) = arv ⋅ P(x2v )
uv⋅ P(x2v ) =avu ⋅ P(x1u ),
which are proved as follows.
(17)
If compound attribute Xn has k attributes,
then,
nn
12
a vr23
Both aru31 ⋅ P( x1u ) and a13
ur ⋅ P( x3r ) , point to
From equations (13), (14) and (15), (16),
we conclude that ahimn and aihnm are in
general different.
The above definition leads to the
conclusion that every attribute can be
used to determine itself perfectly.
∀X n , x ni ∈ X n ,
a13
ur
a
(16)
∑
auv12
(15)
If there are Er number of entries, then
∑
=
x2v
aru31
x3 r
P(x2 v ∩ x3r )
⋅ P(x2v ) = a rv32 ⋅ P(x2v )
P(x2v )
P(x2v ∩ x3r ) =
=
P(x2v ∩ x3r )
⋅ P(x3r ) = a vr23 ⋅ P(x3r ).
P(x3r )
x1u
Attributes
In
Given three attributes, x1u ∈ X1 ,
x2 v ∈ X 2 and x3r ∈ X 3 . Relation among
these three attributes can be seen in Fig.
2.
x2 v
P( x1u ∩ x2v )
x3 r
P(x1u ∩ x2v ) =
=
P(x1u ∩ x2v )
21
⋅ P(x1u ) = a vu
⋅ P(x1u )
P(x1u )
P(x1u ∩ x2v )
⋅ P(x2v ) = a 12
uv ⋅ P(x2v ).
P(x2 v )
139
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
From the relations above, we find the
following equation.
23
13
a12
a 21
uv .avr
vu.a ur
=
a32
a31
rv
ru
(19)
a12
arv32
uv
and
, respectively. Finally, again
avu21
avr23
we must convert all from x2 v into x3 r by
Proof:
12
21
auv
⋅ P(x2 v ) = avu
⋅ P(x1u ),
23
a 12
uv ⋅ (a vr ⋅
P(x3 r )
P(x3 r )
21
) = a vu
⋅ ( a13
)
ur ⋅
32
arv
a 31
ru
a12
uv ⋅
compared, they must be point to the same
area, in this case we use x2 v as base for
their comparison. Therefore, we must
convert them into x2 v by multiplying with
multiplying with
arv32
.
avr23
13
a vr23
21 a ur
=
a
⋅
.
vu
31
a 32
a ru
rv
12
12
min(a32
rv , auv ) = auv
Important characteristic of relation
among attributes is transitive relation, i.e.
21
23
32
given a12
uv , avu , avr , a rv and we would like to
x2 v
min{( 1 − a vu21 ) ⋅
x3r
x1u
find interval value of a13
ur , which satisfy
the two following equations.
a 23
vr
− 1)}. 32
.
a rv
(20)
a 23
vr
+
a 32
rv
(21)
a12
a 32
a 23
uv
23
rv
vr
min{(1 - a 21
).
,
(
1
−
a
).
}.
.
vu
vr
23
32
a 21
a
a
vu
vr
rv
Proof :
To find the upper bound of a 13
ur , first we
take the maximum area inside x2 v , result
of intersection between two intersection
areas which are intersection between x1u
and x2 v , expressed in a12
uv and intersection
between x3 r and x2 v , expressed in a 32
rv . The
maximum area that is result of
overlapping between the two intersection
areas, shown in Fig. 3., can be expressed
32
in min function applied to a12
uv and a rv .
The next, we plus with maximum
intersection between remain x1u and x2 v
which be outside of x2 v . Again, this area
can be expressed in min function applied
21
to (1 − avu
) and (1 − avr23 ) . Value of these
two area point to two different area, x1u
and x3 r . However, in order to be able to be
140
= (1 − a vr23 ) ⋅
between
Upper bound of a13
ur ,
32
12
a13
ur ≤ min{ a rv , a uv }.
a 32
rv }
a vr23
a rv32
a vr23
Figure 3. Maximum Area of Intersection
Lower bound of a13
ur ,
12
32
a13
ur ≥ max{0, ( a uv + a rv
(1 − a vr23 ) ⋅
a 12
uv
,
a 21
vu
x1u and x3 r inside x2 v .
To find the lower bound of a 13
ur , we
take the minimum area inside x2 v , result
of intersection between two intersection
areas which are intersection between x1u
and x2 v , expressed in a12
uv and intersection
between x3 r and x2 v , expressed in a 32
rv .
The minimum area which is result of as
much as possible avoid overlapping
between the two intersection areas, shown
in Fig. 4., can be expressed in max
32
function applied to a12
uv and a rv as shown
in (13). The next, we convert quantity of
the maximum area from x2 v into x3 r by
multiplying with
x1u
x2v
arv32
.
avr23
max(0, (arv32 + a12
uv − 1))
x3r
= (arv32 + a12
uv −1)
Figure 4. Minimum Area of Intersection
between x1u and x3 r inside x2 v .
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
mq i = {56,57,...,60} : 0. 2,{57 ,...60} : 0.2,
4. CALCULATING PREDICTION
After constructed the model of
system, it can be used to predict interval
probability (find lower and upper bound)
of any query data. In this section, we
generate formulas to calculate interval
probability of the query data. First, user
must give input related to data type of
compound attributes.
Definition 4.1 Q is define as set of
input data that be given by user to do
query for a certain data. If there are n
compound attributes then Q={q1,…,qn}
where qi is data input related to
compound attribute Xi.
For example, suppose CAREER
system has been constructed, given John
is old man and has MS degree as input for
age and education, respectively, then q1 =
old and q2=MS.
Definition 4.2 P(Xi,qi) is defined as
probabilistic matching of compound
attribute Xi toward given input data qi. If
there are k elements or attributes of
compound attribute Xi, then,
P(Xi,qi)=( pi1,…, pik),
Next, i.e. mass assignment for x i8=55
as one attribute of Xi is given by
m x = {51,..., 59 } : 0 .2 , {52 ,... 58} : 0 . 2,
i8
{53,...,57 } : 0.2, {54,55,56} : 0.2, {55} : 0.2.
Process to calculate Point Value
Semantic Unification of relation between
two fuzzy set, old and about_55 or
P(about_55,old) is shown in the following
table.
0.2
0.2
0.2
0.2
{56,…,60} {57,…,60} {58,59,60} {59,60}
0.2
{60}
0.2
{51,...,59}
0.032
0.03
0.026
0.02
0
0.2
{52,...,58}
0.024
0.02
0.013
0
0
0.016
0.01
0
0
0
0.008
0
0
0
0
0
0
0
0
0
0.2
{53,...,57}
0.2
{54,55,56}
0.2
{55}
(22)
From the table, we calculate
(23)
P (about _ 55, old ) = 0.032 + 0.03 + 0.026 + 0.02 +
0.024 + 0.02 + 0.013 + 0.016 +
0.01 + 0.008
= 0.199.
where
pij= P(x ij|qi),
{58,59,60} : 0.2,{59,60} : 0.2, {60} : 0.2.
expresses conditional probability for x ij
given qi. In this case Point Semantic
Unification Process [1,3] can be used to
calculate pij.
For example, given qi =old which is a
fuzzy set defined as qi ={0/55,1/60}.
Xi = age has 9 attributes as defined in
section 2, as follows.
Xi = {about_20,about_25,…,about_60}.
By using point semantic unification
process applying to membership function
of age which has been defined in section 2
and membership function of qi, we
calculate P(Xi,qi) as follows. First, we
calculate the mass assignment for qi. It is
equivalent to the basic probability
assignment of Dempster Shafer Theory
which we can write as
In the same way, we find P(about_
60,old) = 0.799, where P(about_20,old) =
P(about_25,old) = ... = P(about_50,old) =
0, because there is no intersection
between their members. Finally, we find,
P ( X i ,q i ) = P( age, old ) = (0 ,0,0 ,0,0 ,0,0, 0.199 , 0.799 ).
Definition 4.3 P( X i , q j )
is defined as
probability of attribute X i , influenced by
given input data q j . X i and q j have
different type of data, therefore to find
their probabilistic matching, first, we
must find P( X j , q j ) and then apply maxmultiply (*) operation between P( X j , q j )
and w ji as follows. If X i has k attributes
and X j has s attributes then,
141
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
P ( X i , q j ) = P ( X j , q j ) * w ji
(24)
a
ji
11
a
ji
12
... a
ji
1k
a 21ji a ji22 ... a 2kji
. . . .
= ( p j 1 ,..., p js ) *
.
.
. .
. .
input Q = {q1 , q2 , q3 }. .
(25)
.
.
max{p
j1
ji
1k
(26)
ji
sk
.a ,..., p js .a })
= ( P ( x i 1 , q j ),..., P ( x ik , q j )),
(27)
Definition 4.4
P( xir , Q) , which is
defined as probability of attribute xir
influenced by given set input data Q, is ∨
operation for all probabilities of relation
between xir and all members of Q. ∨
operation will be explained in the latter. If
there are n members of Q, {( q1 ,..., qn )} ,
then,
P( xir ,Q) = ∨ P( xir , q j ).
(28)
1≤ j ≤ n
Definition 4.5 P( X i ,Q) is defined as
probability of compound attribute X i
influenced by given set input data Q. If
there are n members of Q and k
attributes of X i , then
P(Xi,Q) = (
1≤ j ≤ n
P(xi1 , q j ),...,
1≤ j ≤n
P (x3 r , q3 ) = P ( x3r | q3 ) = p3 r and the second,
we call indirect predicted probability truth
of x3 r which is predicted from other
attributes value, P( x3 r , q1 ) ∨min P( x3 r , q2 ) .
where P( xir ,q j ) = max{ p j 1.a1jir ,..., p js .a srji }.
P(X i,Q) = (P(xi1 , Q),..., P(xik , Q)),
P ( x3r , Q )min = P ( x3r , q1 ) ∨ min P( x3r , q 2 ) ∨ min P( x3r , q3 ).
We separate formula above into two
parts. The first, we call direct predicted
probability
of
x3 r
which
is
a s1ji a s2ji ... a skji
= (max{ p j1 . a11ji ,..., p js .a sji1 },...,
shown in Fig. 2. We calculate minimum
probability truth of x3 r ∈ X 3 based on
(29)
P(xik , q j )). (30)
The next, we compare both of them by
applying max function as follows.
P( x3r , Q) min = max{ P( x3r , q1 ) ∨ min P( x3r ,q 2 ), p 3r )}. (32)
The problem now, is how to calculate
P( x3 r , qi ) ∨ min P( x3 r , q2 ) = δmin . i.e. X1 has s
attributes, X 2 has t attributes. Let’s say
that,
13
P ( x3 r ,q1 ) = max{ p11 .a113r ,..., p1s .a 13
sr } = p1u .aur ,
P ( x3 r ,q 2 ) = max{ p21. a123r ,..., p2 t .a tr23} = p 2v .avr23 .
We solve this problem by imaging
interrelationship among x1u , x2 v , and x3 r as
shown in Fig. 2, in the following three
conditions.
1. If | ( x1u ∩ x2 v ) |≤| ( x1u ∩ x3 r ) | and
| ( x1u ∩ x2 v ) |≤| ( x2 v ∩ x3 r ) | , then
( x1u ∩ x2 v ) will be put in x3 r .
23
a 12
u v a vr
) ⋅ p2 v
a rv32
( a vr −
23
4.1 Calculating minimum probability
truth of P( xir ,Q)
Now, we generate formula for calculating minimum probability of attribute
xir given Q = {q1 ,..., qn } , as input data.
Related to (27), we defined minimum
probability truth of P( xir , Q) as follows.
Pmin ( xir , Q) =
min
∨
P( xir , q j ).
(31)
x1u
a u12v a vr23
⋅ max(
a rv32
a
x3r
δ min = ( a 23
vr −
21
vu
a
31
a ru
13
ur
( a 13
ur −
⋅ max(
p 1 u , p 2 v ) or
p 1u , p 2 v )
a vu21 a 13
ur
) ⋅ p1u
a ru31
13
a 12uv a 23
a 21
vr
vu a ur
) ⋅ p 2 v + ( a 13
) ⋅ p1u +
ur −
a 32rv
a 31
ru
a 12uv a 23
vr
⋅ max( p 1 u , p 2 v ).
a 32
rv
or
1≤ j≤ n
To simplify the problem, let’s say that
system just has three compound attributes, X1 , X 2 , and X 3 and their relation
142
x2v
δ min = ( a 23
vr
13
a 12 a 23
a 21
vu a ur
− uv 32 vr ) ⋅ p 2 v + ( a 13
) ⋅ p1u +
ur −
31
a rv
a ru
a 21vu a 13ur
⋅ max( p 1 u , p 2 v )
a 31
ru
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
2. If | ( x3 r ∩ x2 v ) |≤| ( x1u ∩ x3 r ) | and
Related to (27), we defined maximum
probability truth of P( xir , Q) as follows.
| ( x3 r ∩ x2 v ) |≤| ( x2 v ∩ x1u ) | , then
( x2 v ∩ x3 r ) will be put in
( x1u ∩ x3 r ) .
x2 v
x1u
max
P max (x ir , Q)
avr23 ⋅ max( p1u , p2v )
13
23
δmin = a23
vr ⋅ max(p1u , p2 v ) + (aur − avr ) ⋅ p1u
3. If | ( x1u ∩ x3 r ) |≤| ( x1u ∩ x2 v ) | and
| ( x1u ∩ x3 r ) |≤| ( x2 v ∩ x3 r ) | , then
( x1u ∩ x3 r ) will be put in
( x2 v ∩ x3 r ) .
a 13
ur ⋅ max(
ir
(34)
,q j )
2v
We separate formula above into two
parts. The first, we call direct predicted
probability of x 3r which is P(x 3r|q3) =
P(x 3r|q3) = p3r and the second, we call
indirect predicted probability truth of x3r
which is predicted from other attributes
value, P (x 3r , q1 ) ∨ max P ( x3 r , q2 ) . The next, we
compare both of them by applying min
function as follows.
(35)
Pmax ( x3 r ,Q ) = min{1,( P( x3 r ,q1 ) ∨ max P( x3 r ,q 2 )) + p3 r )}
2v
p 1u , p
P(x
Pmax(x3r , Q ) = P ( x3 r , q1 ) ∨ max P (x3r , q2 ) ∨ max P (x3r , q3 )
x3 r
( a vr23 − a 13
ur ) ⋅ p
1 ≤ j ≤n
To simplify the problem, let’s say that
system just has three compound
attributes, X1, X2 and X3 and their
relation shown in Fig. 2.2. We calculate
maximum probability truth of x3 r ∈ X 3
based on input Q=(q1, q2, q3).
23
(a13
ur − avr ) ⋅ p1u
x2 v
= ∨
)
The problem now, is how to calculate
P(x 3r , q 2 ) = ä max .i.e. X 1 has s attributes,
x1u
X2 has t attributes. Let’s say that,
x3 r
13
13
P(x 3r , q1 ) = max{p11 ⋅ a1r
,..., p1s ⋅ asr
} = p1u ⋅ a 13
ur
13
13
ä min = (avr23 − a ur
) ⋅ p 2v + a ur
⋅ max(p1u , p 2v )
From the above conditions, we generate a formula that satisfy all conditions
as follows.
23
a 12
13
uv a vr
δ min = (a 23
−
min(
, a 23
vr
vr , a ur )) ⋅ p 2v +
a 32
rv
(a13
ur − min(
We solve this problem by imaging
interrelationship among x 1u, x 2v and x 3r as
shown in Fig. 2.2, in the following four
conditions.
31
21
1. If ( arv32 + a12
uv ≤ 1) and ( ar 1 + a vu ≤ 1)
23
and ( a13
ur + avr ≤ 1) then
23
a 12
13
uv a vr
, a 23
vr , a ur )).p 1u +
32
a rv
a 12 a 23
13
min( uv32 vr , a 23
vr , a ur ). max( p1u , p 2v ).
a rv
Finally, we find that
23
P(x 3r , q2 ) = max{p21 ⋅ a1r23 ,..., p1t ⋅ atr23 } = p2u ⋅ avr
(33)
Pmin ( x3 r , Q) = max{δmin , P( x3 r | q3 )}.
4.2 Calculating Maximum Probability
Truth of P( xir , Q)
Next, we generate formula for calculating maximum probability of attribute
x ir given Q=(q1,… qn), as input data.
x1u
x 2v
avr23 ⋅ p 2v
a13
ur ⋅ p1 u
x3 r
δmax = avr23 ⋅ p2 v + a13
ur ⋅ p1 u .
23
a vr
12
32
12
2. If ( a 32
rv + a uv > 1) and ( ( a rv + a uv − 1 ). 32 >
a rv
21
( a31
ru + a vu - 1).
13
ur
31
ru
a
a 23
12
vr
) and (( a32
>
rv + a uv - 1).
a
a 32
rv
23
( a13
ur + a vr - 1)), then
143
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
x1u
32
12
(a23
vr − (arv + auv −1) ⋅
x2 v x3 r
12
(a32
rv + auv − 1) ⋅
32
12
( a13
ur − ( arv + auv − 1) ⋅
(a
+a
23
13
( a 13
ur − ( a vr + a ur − 1)) ⋅ p1 u +
a23
vr
⋅ max( p1u , p2v )
a32
rv
a 23
vr
) ⋅ p 2v +
a32
rv
32
12
δ max = ( a 23
vr − max( 0 , ( a rv + a uv − 1 ) ⋅
a 13
ur
31
a ru
12
( a 32
rv + a uv - 1).
a
a
23
vr
32
rv
21
) and (( a 31
ru + a vu - 1).
a
a
13
ur
31
ru
>
21
( avr23 − (a31
ru + avu − 1) ⋅
21
vu
aur13
) ⋅ p2v
aru31
31
21
δ max = ( a13
ur − ( a ru + a vu − 1) ⋅
31
21
( a 23
vr − ( a ru + a vu − 1) ⋅
4.
a13
ur
) ⋅ p1u
a31
ru
a13
(a + a − 1) ⋅ ur
⋅ max( p1u , p2v )
a31
ru
31
ru
21
( a 31
ru + a vu − 1 ) ⋅
a 13
ur
) ⋅ p1 u +
a 31
ru
a 13
ur
) ⋅ p2 v +
a 31
ru
a13
ur
⋅ max( p 1 u , p 2 v )
a 31
ru
13
23
13
If ( a23
vr + a ur > 1 ) and (( a vr + a ur − 1).
a13
ur
>
a31
ru
13
ur
31
ru
21
( a31
ru + a vu - 1).
a
23
) and (( a13
ur + a vr - 1) >
a
12
( a32
rv + a uv - 1).
a23
vr
), then
a32
rv
x1u
x2 v
x3 r
13
( avr23 − ( a23
vr + aur −1)) ⋅ p2v
( avr23 + a 13
ur −1) ⋅ max(p1u , p2v )
23
13
( a 13
ur − ( avr + aur − 1)) ⋅ p1u
144
12
max( 0 , ( a 32
rv + a uv − 1 ) ⋅
a 23
21
vr
, ( a 31
ru + a vu − 1 ) ⋅
a 32
rv
a 13
13
ur
, ( a 23
vr + a ur − 1 )) ⋅ max( p 1 u , p 2v ).
a 31
ru
(36)
Finally, we find that
Pmax ( x3 r , Q) = max{1, δmax + P( x3 r | q3 )}.
31
21
(a13
ur − (aru + avu − 1) ⋅
x1u x
3r
13
13
32
12
, ( a 23
vr + a ur − 1 ))) ⋅ p 2v + ( a ur − max( 0 , ( a rv + a uv − 1 ) ⋅
23
23
( a13
ur + a vr - 1)), then
x 2v
a 23
21
vr
, ( a 31
ru + a vu − 1 ) ⋅
a 32
rv
a vr
a 13
21
13
ur
, ( a 31
, ( a 23
ru + a vu − 1 ) ⋅
vr + a ur − 1 ))) ⋅ p1 u +
a 32
a 31
rv
ru
a 23
− 1 ) ⋅ vr
⋅ max( p1 u , p2 v )
a 32
rv
a13
21
ur
> 1) and ((a31
ru + avu − 1). 31 >
aru
21
3. If (a31
ru + avu
13
( a 23
vr + a ur − 1) ⋅ max( p1 u , p 2 v ).
From the above conditions, we generate a
formula that satisfy all condition as
follows.
a 23
vr
) ⋅ p1 u +
a 32
rv
32
12
( a13
ur − ( a rv + a uv − 1) ⋅
12
uv
a
) ⋅ p2v
a
avr23
) ⋅ p1u
arv32
32
12
δ max = (a 23
vr − ( a rv + a uv − 1) ⋅
32
rv
23
13
δ max = ( a 23
vr − ( a vr + a ur − 1 )) ⋅ p 2 u +
23
vr
32
rv
5. CONCLUSION
This paper proposed a method based
on conditional probability relation to
approximately calculate interval probability of dependency of data for data
querying. Theoretically the formulation is
quite interesting. However, it seems to be
too complicated to calculate interaction of
three or more components. Practically the
formulas should be simplified, even
though the accuracy of prediction may be
decreased.
REFERENCES
1. Intan, R., Mukaidono, M., ‘Application
of Conditional Probability in Construc ting Fuzzy Functional Dependency (FFD)’, Proceedings of AFSS’00,
2000, pp.271-276.
2. Intan, R., Mukaidono, M., ‘A proposal
of Fuzzy Functional Dependency
based on Conditional Probability’,
Proceeding of FSS’00 (Fuzzy Systems
Symposium), 2000, pp. 199-202.
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
3. Intan, R., Mukaidono, M., ‘Fuzzy
Functional Dependency and Its
Application to Approximate Querying’, Proceedings of IDEAS’00, 2000,
pp.47-54.
4. Intan, R., Mukaidono, M., ‘Conditional Probability Relations in Fuzzy
Relational Database ’, Proceedings of
RSCTC’00, LNAI 2005, Springer &
Verlag, 2000, pp.251-260.
5. Baldwin J.F., ‘Knowledge from Data
using Fril and Fuzzy Methods’,Fuzzy
Logic,John Wiley & Sons Ltd 1996,
pp. 33–75.
6. Yukari Yamauchi, Masao Mukaidono ,
‘Interval and Paired Probabilities for
Treating Uncertain Events’, The
Institute of Electronics, Information
and Communication Engineers, Vol.
E82-D (May 1999), pp. 955–961.
7. Baldwin J.F., Martin T.P., and
Pilsworth B.W. FRIL-Fuzzy and
Evidential Reasoning in AI, Research
Studies Press and Willey, 1995.
8. Shafer G. A Mathematical Theory of
Evidence, Princeton Univ. Press.
1976.
9. Richard Jeffrey, ’Probabilistic Thinking’, Priceton University, 1995.
10. Baldwin J.F., Martin T.P., ’A Fuzzy
Data Browser in Fril’, Fuzzy Logic,
John Wiley & Sons Ltd. 1996, pp. 101123.
145
APPLICATION OF CONDITIONAL PROBABILITY IN
PREDICTING INTERVAL PROBABILITY OF DATA
QUERYING
Rolly Intan
Faculty of Industrial Technology, Informatics Engineering Department,
Petra Christian University
Email: rintan@petra.ac.id
ABSTRACT: This paper discusses fuzzification of crisp domain into fuzzy classes
providing fuzzy domain. Relationship between two fuzzy domains, X i and X j , can
be represented by using a matrix,
wij . If X i has n elements of fuzzy data and X j
has m elements of fuzzy data then
wij is n × m matrix. Our primary goal in this
paper is to generate some formulas for predicting interval probability in the relation
to data querying, i.e., given John is 30 years old and he has MS degree, how about his
probability to get high salary.
Keywords: Fuzzy Conditional Probability Relation, Data Querying, Interval
Probability, Mass Assignment, Point Semantic Unification.
1. INTRODUCTION
In a system, we will find that every
component has relation one to each other.
For example, system CAREER has some
components such as education, age, and
salary in which we realize that all of them
has interrelationship as in general higher
education means higher salary, or older
someone will get higher salary, or in a
certain area, percentage of mid-40’s of
persons who has doctoral degree is 30
percent, etc.
In this paper, we process a certain
relational database by classifying every
domain into several value of data or
elements of the component, i.e., domain or
component age can be classified into
{about_20, about_25, …, about_60}. With
assumption that every classified data is a
fuzzy set, we must determine a membership function which represents degree of
element belonging to the fuzzy set, i.e.,
about_30={0.2/26, 04/27, 0.6/28, 0.8/29,
1/30, 0.8/31, 0.6/32, 0.4/33, 0.2/34}. Next,
we apply the all membership function into
the previous relational database to find
every membership value for every item
data. And then, by using conditional
probabilistic theory, we construct a model
to describe interrelationship among all
components of the system. Relationship
between two components, X 1 and X 2 , of
a system is expressed in a matrix w 12 . If
component X 1 has n elements, X 2 has m
elements then matrix w12 is n × m matrix,
where
a 12
ij ∈ w12
expresses weight or
degree dependency of
x2 j ∈ X 2
from
x1i ∈ X 1 , for 1 ≤ i ≤ n , 1 ≤ j ≤ m . Through
this model, we generate some formulas to
predict any value of component related to
a given query of input data i.e., given
John is 30 years old and he has MS
degree, how many his probability to get
high salary, and off course we have to
define high salary as a fuzzy data value.
Given input of data querying can be
precise as well as imprecise data (fuzzy
data), so first, before the data can be used
to make prediction, we must to find their
probabilistic matching related to element
of components of system by using Point
Semantic Unification Process as introduced in paper of Baldwin [1]. In this
case, Point Value Semantic Unification
can be considered as a conditional
probability between two fuzzy sets [3]. In
135
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
calculating prediction, we generate two
different formulas to provide upper and
lower bound probability of prediction.
Hence, result of prediction works into a
interval truth value [ a, b ] where a ≤ b as
proposed in [2].
Pr( f |g) = ∑ m ij .
2. BASIC CONCEPT
Definition 2.1 P( H | D) is defined as a
conditional probability for H given D.
Relation between conditional and unconditional probability satisfies the following
equation [5].
P( H ∩ D )
,
P( D)
(1)
where P( H ∩ D) is an unconditional
probability of compound events ’H and D
happen’. P(D) is unconditional probability
of event D.
2.2 Mass Assignment
Definition 2.2 Given f is a fuzzy set
defined
on
the
discrete
space
X = {x1 ,..., xn } , namely
n
f =∑
i =1
χi
xi
ment corresponding to the fuzzy set f is [6]
(2)
For example, given a fuzzy set
low_numbers = {1/1, 1/2, 0.5/3, 0.2/4}, the
mass assignment of the fuzzy set
low_numbers is
mlow_numbers = {1,2}:0.5, {1,2,3}:0.3,
{1,2,3,4}:0.2.
2.3 Point Semantic Unification
Definition 2.3 Let mf = {Li:li} and
mg={Ni:ni} be mass assignment associate
136
For example, let f = {1/a,0.7/b,0.2/c}
and g = {0.2/a,1/b,0.7/c,0.1/d} are
defined on X = {a,b,c,d,e }, so that
mf = {a}:0.3, {a,b}:0.5, {a,b,c}:0.2,
mg = {b}:0.3, {b,c}:0.5, {a,b,c}:0.1,
{a,b,c,d}:0.1.
From the following matrix,
0.3 {a}
0.5 {a,b}
0.2 {1,b,c}
0.3
{b}
0
0.15
0.06
0.5
{b,c}
0
0.125
0.1
0.1
{a,b,c}
0.01
0.0333
0.02
0.1
{a,b,c,d}
0.00075
0.025
0.015
the probability Pr(f | g)= 0.53905. It can
be proved that Point Semantic Unification
satisfies
Pr( f |g) + Pr(f |g ) = 1.
(5)
Thus, Point Semantic Unification is
considered as a conditional probability [3].
2.4 Interval Probability
Suppose f is a normal fuzzy set whose
elements
are
ordered
such
that:
χ1 = 1, χi ≤ χ j , if i ≤ j ; The mass assignmf = {{x1 , x2,...,xi } : χi − χi +1 } with xn+1 = 0
(4)
ij
2.1 Conditional Probability
P( H |D ) =
with the fuzzy set f and g, respectively.
From the matrix,
card(L i ∩ N j )
(3)
M = {m ij } =
⋅ li ⋅ n j.
card( M j )
The probability Pr(f | g) is given by [3]:
Definition 2.4 An interval probability
IP(E) can be interpreted as a scope of
probability of event E, P(E), i.e
IP(E)=[e1,e2] means e1 ≤ P( E) ≤e 2 , where
e1 and e2 are minimum and maximum
probability of E respectively[2].
For example, given two probabilities
P(A)=a and P(A)=b for event A and B,
where a , b ∈ [ 0,1].
Minimum probability of compound
event ’A and B happen’, P( A ∩ B ) min , is
the least intersection between A and B,
given by the following equation:
P( A ∩ B) min = max( 0, a + b − 1).
Maximum probability of compound
event ’A and B happen’, P( A ∩ B) max , is
the most intersection between A and B,
given by;
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
Thus interval probability of compound
event ’A and B happened’ is defined as
IP( A ∩ B) = [max(0, a + b − 1),min(a, b)].
(6)
Similarly, minimum and maximum
probability of compound event ’A or B
happens’, are max(a,b) and min(1,a+b)
respectively.
Thus, interval probability of compound event ’A or B happened’ is defined
as:
IP(A ∪ B) = [max(a, b),min(1, a + b)].
(7)
3. CONSTRUCTION MODEL OF SYSTEM
Definition 3.1 System is defined as
S(Er,X,Nm), where
Er: Number of entry data or number of
record or respondent of system.
X: Domain or components of system, if
there are n components then
X=(X1,…,X n).
Nm: Name of system.
For example, given CAREER DATABASE
in Table 1. By assuming that CAREER is
a system which has 24 entries and three
components, education, age, and salary,
therefore Er=24, X=(X1:education, X2:age,
X3: salary), Nm=CAREER. Now, we try to
find relation among education, age, and
salary.
Table 1. CAREER DATABASE
Name
Nm-1
Nm-2
Nm-3
Nm-4
Nm-5
Nm-6
Nm-7
Nm-8
Nm-9
Nm-10
Nm-11
Nm-12
Nm-13
Nm-14
Nm-15
Nm-16
Education
MS
SHS
PhD
JHS
ES
SHS
MS
SHS
MS
SHS
N
JHS
BA
SHS
BA
SHS
Age
35
24
44
45
35
37
39
27
45
56
60
33
54
47
41
21
Sallary
400,000
150,000
470,000
200,000
125,000
250,000
420,000
175,000
415,000
275,000
100,000
300,000
350,000
315,000
355,000
150,000
Nm-17
Nm-18
Nm-19
Nm-20
Nm-21
Nm-22
Nm-23
Nm-24
BA
PhD
ES
JHS
BA
SHS
BA
SHS
52
49
58
59
35
37
31
29
374,000
500,000
125,000
200,000
360,000
255,000
340,000
250,000
First, we classify all three domains or
components as follows.
education
age
salary
= (low_edu,mid_edu,hi_edu),
= (about_20,…,about_60),
= (low_slr,mid_slr,hi_slr).
where we assume that membership
functions of low_edu, mid_edu, and
high_edu:
µ (low_edu) = {1 / N ,0.8 / ES ,0.5 / JHS },
µ (mid_edu) = {0. 2 / ES,0. 5 / JHS ,0.9 / SHS ,0. 2 / BA},
µ (hi_edu) = {0. 1 / SHS , 0.8 / BA,1 / MS,1 / PhD}.
Membership function of age,
µ (about_ n) = {0 .2 /(n − 4),0.4 /(n − 3),0.6 /(n − 2),
0.8 /(n − 1),1 / n,0.8 /(n + 1),0.6 /(n + 2),0.4 /(n + 3),
0.2 /(n + 4)}.
Membership function of low_salary,
mid_salary, and high_salary :
µ (low_slr) = [1 / 0,1 /100000 ,0 / 150000 ],
µ (mid_slr) = [0 / 100000 ,1 /150000 ,1 / 250000 ,
0/300000 ],
µ (hi_slr) = [0 / 250000 ,1 / 300000 ].
By using all membership functions
above, we calculate and transform table 1
into table 2.
Table 2. CAREER FUZZY VALUE
Nama
Nm-1
Nm-2
Nm-3
Nm-4
Nm-5
Nm-6
Nm-7
Nm-8
Nm-9
Nm-10
Nm-11
Nm-12
Nm-13
Education
Age
Salary
LE ME HE 20 25 … 60 LS MS HS
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0.2 0.8 ... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0.5 0.5 0
0
0
... 0
0
1
0
0.8 0.2 0
0
0
... 0 0.5 0.5 0
0 0.9 0.1 0
0
... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0 0.6 ... 0
0
1
0
0
0
1
0
0
... 0
0
0
1
0 0.9 0.1 0
0
... 0.2 0 0.5 0.5
1
0
0
0
0
... 1
1
0
0
0 0.9 0.1 0
0
... 0
0
0
1
0 0.2 0.8 0
0
... 0
0
0
1
137
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
Nm-14 0 0.9 0.1 0
0
Nm-15 0 0.2 0.8 0
0
Nm-16 0 0.9 0.1 0.8 0.2
Nm-17 0 0.2 0.8 0
0
Nm-18 0
0
1
0
0
Nm-19 0.8 0.2 0
0
0
Nm-20 0 0.9 0.1 0
0
Nm-21 0 0.2 0.8 0
0
Nm-22 0 0.9 0.1 0
0
Nm-23 0 0.2 0.8 0
0
Nm-24 0 0.9 0.1 0 0.2
Σ 3.1 10.9 10 1 1.8
... 0
0
0
1
... 0
0
0
1
... 0
0
1
0
... 0
0
0
1
... 0
0
0
1
... 0.6 0 0.5 0.5
... 0.8 0
1
0
... 0
0
0
1
... 0
0 0.9 0.1
... 0
0
0
1
... 0
0
1
0
... 2.6 1.5 9.4 13.1
Definition 3.2 Xn is defined as compound
attribute to express component of the
system . Xn is a vector. If there are k
elements of Xn then Xn=(x n1,…,x nk), where
x ni is element i of compound attribute Xn
and for further, x ni is called attribute.
For example, if system CAREER has
three compound attributes and their
attributes as follows,
then x 11 =low_edu, x 25 =about_40, etc.
=1
(8)
as
shown
11
12
e4 = 0.5, e2 = 0.9, etc.
∑ P(x ni ) = 1
Given three compound attributes, X1,
X2 and X3. Relation among them can be
illustrated in Fig. 1. as follows.
w11
X1
w21
in
Table
∑e
1≤i≤Er
∑ N(x ni )
(9)
wmn
X 3 w33
a11nm
nm
a
= 21
M
nm
ak 1
a12nm
L
nm
a22
L
M
O
aknm2
L
a11mn
mn
a
= 21
M
mn
a j 1
a12mn
mn
a22
L
L
O
mn
j2
L
a
M
a1nmj
a 2nmj
,
M
a kjnm
a1mn
k
mn
a2 k
.
M
a mn
jk
(10)
1≤i≤k
For example, as shown in Table 2.,
N(x ni)=N(low_edu) = 3.1.
138
w32
w31
Definition 3.6 wnm is defined as weight
matrix, to express degree of dependency of
Xm from Xn. For a k-compound attribute
Xn and a j-compound attribute Xm, wnm
and wmn present two different matrices, as
follows.
2.,
If compound attribute Xn has k attributes
then,
Er =
w12 w13
w23
w22 X 2
Definition 3.4 N(x ni) is defined as sum of
entries value for attribute x ni. If there are
Er number of entries, then
N( x ni ) =
(12)
3.1 Relation Among Compound Attributes
wnm
ni
j
(11)
Figure 1. Relation Among Compound
Attributes, X1, X2 and X3.
Definition 3.3 e nij is defined as membership’s value of entry j for attribute x ni. If
compound attribute Xn has k attributes
then,
Example,
N( xni )
Er
If compound attribute Xn has k attributes
then,
= (low_edu,mid_edu,hi_edu),
= (about_20,…,about_60),
= (low_slr,mid_slr,hi_slr ).
X1 : education
X2 : age
X3 : salary
e nij
∑
1≤i≤k
P( xni ) =
as
1≤i≤k
Note: LE:low_edu, ME:mid_edu, HE =
hi_edu, 20:about_20, 25:about_25, …,
LS=low_slr, MS:mid_slr and HS:hi_slr.
∀j
Definition 3.5 P(x ni) is defined
probability of attribute x ni as follows.
Definition 3.7 Each element of matrix
wnm, entry aihnm
expresses numerical
probabilistic value of relation from xni ∈ X n
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
to xmh ∈ X n . aihnm can also be interpreted as
conditional probability as follows.
a nm
ih
P(x ni ∩ x mh )
= P(x ni|x mh ) =
P(x mh )
1
x1u
(13)
avu21
If there are Er number of entries, then
a
nm
ih
∑
=
1 ≤ j ≤ Er
∑
min (e nij , e mh
j )
1 ≤ j ≤Er
e mh
j
1
(14)
where P(xni ∩ xmh ) express probability of
entries which be inside x ni and x mh.
On the other hand, ahimn expresses
numerical probabilistic value of relation
from xmh ∈ X m to xni ∈ X n . ahimn can also be
interpreted as conditional probability as
follows.
a
mn
hi
P(x ni ∩ x mh )
= P(x mh|x ni ) =
P(x ni )
a mn
hi
min (enij , e mh
j )
1 ≤ j≤ Er
e ni
1 ≤ j ≤ Er j
P(x ni ∩ x ni )
.
P(x ni )
1 a
nn
a
1
wnn = 21
M
M
nn
ak1 aknn2
3.2 Relation
System
L
L
O
L
x3r
1
32
rv
Figure 2. Relation Among Attributes, x1u,
x2v, and x3r.
In order to understand the meaning of
this connection, we use relation of set.
P( x1u ∩ x3 r )
x1u
x2 v
x3 r
P(x1u ∩ x3r ) =
P(x1u ∩ x3r )
⋅ P(x1u ) = a 31
ru ⋅ P(x1u )
P(x1u )
=
P(x1u ∩ x3r )
⋅ P(x3r ) = aur13 ⋅ P(x3r ).
P(x3r )
P( x 2v ∩ x 3r )
x1u
x2 v
a
a
.
M
1
nn
1k
nn
2k
Among
the same area or quantity, are intersection between x 1u and x 3r. In the same
way, we can find two other relations,
32
21
and a12
a23
vr ⋅ P(x3r ) = arv ⋅ P(x2v )
uv⋅ P(x2v ) =avu ⋅ P(x1u ),
which are proved as follows.
(17)
If compound attribute Xn has k attributes,
then,
nn
12
a vr23
Both aru31 ⋅ P( x1u ) and a13
ur ⋅ P( x3r ) , point to
From equations (13), (14) and (15), (16),
we conclude that ahimn and aihnm are in
general different.
The above definition leads to the
conclusion that every attribute can be
used to determine itself perfectly.
∀X n , x ni ∈ X n ,
a13
ur
a
(16)
∑
auv12
(15)
If there are Er number of entries, then
∑
=
x2v
aru31
x3 r
P(x2 v ∩ x3r )
⋅ P(x2v ) = a rv32 ⋅ P(x2v )
P(x2v )
P(x2v ∩ x3r ) =
=
P(x2v ∩ x3r )
⋅ P(x3r ) = a vr23 ⋅ P(x3r ).
P(x3r )
x1u
Attributes
In
Given three attributes, x1u ∈ X1 ,
x2 v ∈ X 2 and x3r ∈ X 3 . Relation among
these three attributes can be seen in Fig.
2.
x2 v
P( x1u ∩ x2v )
x3 r
P(x1u ∩ x2v ) =
=
P(x1u ∩ x2v )
21
⋅ P(x1u ) = a vu
⋅ P(x1u )
P(x1u )
P(x1u ∩ x2v )
⋅ P(x2v ) = a 12
uv ⋅ P(x2v ).
P(x2 v )
139
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
From the relations above, we find the
following equation.
23
13
a12
a 21
uv .avr
vu.a ur
=
a32
a31
rv
ru
(19)
a12
arv32
uv
and
, respectively. Finally, again
avu21
avr23
we must convert all from x2 v into x3 r by
Proof:
12
21
auv
⋅ P(x2 v ) = avu
⋅ P(x1u ),
23
a 12
uv ⋅ (a vr ⋅
P(x3 r )
P(x3 r )
21
) = a vu
⋅ ( a13
)
ur ⋅
32
arv
a 31
ru
a12
uv ⋅
compared, they must be point to the same
area, in this case we use x2 v as base for
their comparison. Therefore, we must
convert them into x2 v by multiplying with
multiplying with
arv32
.
avr23
13
a vr23
21 a ur
=
a
⋅
.
vu
31
a 32
a ru
rv
12
12
min(a32
rv , auv ) = auv
Important characteristic of relation
among attributes is transitive relation, i.e.
21
23
32
given a12
uv , avu , avr , a rv and we would like to
x2 v
min{( 1 − a vu21 ) ⋅
x3r
x1u
find interval value of a13
ur , which satisfy
the two following equations.
a 23
vr
− 1)}. 32
.
a rv
(20)
a 23
vr
+
a 32
rv
(21)
a12
a 32
a 23
uv
23
rv
vr
min{(1 - a 21
).
,
(
1
−
a
).
}.
.
vu
vr
23
32
a 21
a
a
vu
vr
rv
Proof :
To find the upper bound of a 13
ur , first we
take the maximum area inside x2 v , result
of intersection between two intersection
areas which are intersection between x1u
and x2 v , expressed in a12
uv and intersection
between x3 r and x2 v , expressed in a 32
rv . The
maximum area that is result of
overlapping between the two intersection
areas, shown in Fig. 3., can be expressed
32
in min function applied to a12
uv and a rv .
The next, we plus with maximum
intersection between remain x1u and x2 v
which be outside of x2 v . Again, this area
can be expressed in min function applied
21
to (1 − avu
) and (1 − avr23 ) . Value of these
two area point to two different area, x1u
and x3 r . However, in order to be able to be
140
= (1 − a vr23 ) ⋅
between
Upper bound of a13
ur ,
32
12
a13
ur ≤ min{ a rv , a uv }.
a 32
rv }
a vr23
a rv32
a vr23
Figure 3. Maximum Area of Intersection
Lower bound of a13
ur ,
12
32
a13
ur ≥ max{0, ( a uv + a rv
(1 − a vr23 ) ⋅
a 12
uv
,
a 21
vu
x1u and x3 r inside x2 v .
To find the lower bound of a 13
ur , we
take the minimum area inside x2 v , result
of intersection between two intersection
areas which are intersection between x1u
and x2 v , expressed in a12
uv and intersection
between x3 r and x2 v , expressed in a 32
rv .
The minimum area which is result of as
much as possible avoid overlapping
between the two intersection areas, shown
in Fig. 4., can be expressed in max
32
function applied to a12
uv and a rv as shown
in (13). The next, we convert quantity of
the maximum area from x2 v into x3 r by
multiplying with
x1u
x2v
arv32
.
avr23
max(0, (arv32 + a12
uv − 1))
x3r
= (arv32 + a12
uv −1)
Figure 4. Minimum Area of Intersection
between x1u and x3 r inside x2 v .
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
mq i = {56,57,...,60} : 0. 2,{57 ,...60} : 0.2,
4. CALCULATING PREDICTION
After constructed the model of
system, it can be used to predict interval
probability (find lower and upper bound)
of any query data. In this section, we
generate formulas to calculate interval
probability of the query data. First, user
must give input related to data type of
compound attributes.
Definition 4.1 Q is define as set of
input data that be given by user to do
query for a certain data. If there are n
compound attributes then Q={q1,…,qn}
where qi is data input related to
compound attribute Xi.
For example, suppose CAREER
system has been constructed, given John
is old man and has MS degree as input for
age and education, respectively, then q1 =
old and q2=MS.
Definition 4.2 P(Xi,qi) is defined as
probabilistic matching of compound
attribute Xi toward given input data qi. If
there are k elements or attributes of
compound attribute Xi, then,
P(Xi,qi)=( pi1,…, pik),
Next, i.e. mass assignment for x i8=55
as one attribute of Xi is given by
m x = {51,..., 59 } : 0 .2 , {52 ,... 58} : 0 . 2,
i8
{53,...,57 } : 0.2, {54,55,56} : 0.2, {55} : 0.2.
Process to calculate Point Value
Semantic Unification of relation between
two fuzzy set, old and about_55 or
P(about_55,old) is shown in the following
table.
0.2
0.2
0.2
0.2
{56,…,60} {57,…,60} {58,59,60} {59,60}
0.2
{60}
0.2
{51,...,59}
0.032
0.03
0.026
0.02
0
0.2
{52,...,58}
0.024
0.02
0.013
0
0
0.016
0.01
0
0
0
0.008
0
0
0
0
0
0
0
0
0
0.2
{53,...,57}
0.2
{54,55,56}
0.2
{55}
(22)
From the table, we calculate
(23)
P (about _ 55, old ) = 0.032 + 0.03 + 0.026 + 0.02 +
0.024 + 0.02 + 0.013 + 0.016 +
0.01 + 0.008
= 0.199.
where
pij= P(x ij|qi),
{58,59,60} : 0.2,{59,60} : 0.2, {60} : 0.2.
expresses conditional probability for x ij
given qi. In this case Point Semantic
Unification Process [1,3] can be used to
calculate pij.
For example, given qi =old which is a
fuzzy set defined as qi ={0/55,1/60}.
Xi = age has 9 attributes as defined in
section 2, as follows.
Xi = {about_20,about_25,…,about_60}.
By using point semantic unification
process applying to membership function
of age which has been defined in section 2
and membership function of qi, we
calculate P(Xi,qi) as follows. First, we
calculate the mass assignment for qi. It is
equivalent to the basic probability
assignment of Dempster Shafer Theory
which we can write as
In the same way, we find P(about_
60,old) = 0.799, where P(about_20,old) =
P(about_25,old) = ... = P(about_50,old) =
0, because there is no intersection
between their members. Finally, we find,
P ( X i ,q i ) = P( age, old ) = (0 ,0,0 ,0,0 ,0,0, 0.199 , 0.799 ).
Definition 4.3 P( X i , q j )
is defined as
probability of attribute X i , influenced by
given input data q j . X i and q j have
different type of data, therefore to find
their probabilistic matching, first, we
must find P( X j , q j ) and then apply maxmultiply (*) operation between P( X j , q j )
and w ji as follows. If X i has k attributes
and X j has s attributes then,
141
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
P ( X i , q j ) = P ( X j , q j ) * w ji
(24)
a
ji
11
a
ji
12
... a
ji
1k
a 21ji a ji22 ... a 2kji
. . . .
= ( p j 1 ,..., p js ) *
.
.
. .
. .
input Q = {q1 , q2 , q3 }. .
(25)
.
.
max{p
j1
ji
1k
(26)
ji
sk
.a ,..., p js .a })
= ( P ( x i 1 , q j ),..., P ( x ik , q j )),
(27)
Definition 4.4
P( xir , Q) , which is
defined as probability of attribute xir
influenced by given set input data Q, is ∨
operation for all probabilities of relation
between xir and all members of Q. ∨
operation will be explained in the latter. If
there are n members of Q, {( q1 ,..., qn )} ,
then,
P( xir ,Q) = ∨ P( xir , q j ).
(28)
1≤ j ≤ n
Definition 4.5 P( X i ,Q) is defined as
probability of compound attribute X i
influenced by given set input data Q. If
there are n members of Q and k
attributes of X i , then
P(Xi,Q) = (
1≤ j ≤ n
P(xi1 , q j ),...,
1≤ j ≤n
P (x3 r , q3 ) = P ( x3r | q3 ) = p3 r and the second,
we call indirect predicted probability truth
of x3 r which is predicted from other
attributes value, P( x3 r , q1 ) ∨min P( x3 r , q2 ) .
where P( xir ,q j ) = max{ p j 1.a1jir ,..., p js .a srji }.
P(X i,Q) = (P(xi1 , Q),..., P(xik , Q)),
P ( x3r , Q )min = P ( x3r , q1 ) ∨ min P( x3r , q 2 ) ∨ min P( x3r , q3 ).
We separate formula above into two
parts. The first, we call direct predicted
probability
of
x3 r
which
is
a s1ji a s2ji ... a skji
= (max{ p j1 . a11ji ,..., p js .a sji1 },...,
shown in Fig. 2. We calculate minimum
probability truth of x3 r ∈ X 3 based on
(29)
P(xik , q j )). (30)
The next, we compare both of them by
applying max function as follows.
P( x3r , Q) min = max{ P( x3r , q1 ) ∨ min P( x3r ,q 2 ), p 3r )}. (32)
The problem now, is how to calculate
P( x3 r , qi ) ∨ min P( x3 r , q2 ) = δmin . i.e. X1 has s
attributes, X 2 has t attributes. Let’s say
that,
13
P ( x3 r ,q1 ) = max{ p11 .a113r ,..., p1s .a 13
sr } = p1u .aur ,
P ( x3 r ,q 2 ) = max{ p21. a123r ,..., p2 t .a tr23} = p 2v .avr23 .
We solve this problem by imaging
interrelationship among x1u , x2 v , and x3 r as
shown in Fig. 2, in the following three
conditions.
1. If | ( x1u ∩ x2 v ) |≤| ( x1u ∩ x3 r ) | and
| ( x1u ∩ x2 v ) |≤| ( x2 v ∩ x3 r ) | , then
( x1u ∩ x2 v ) will be put in x3 r .
23
a 12
u v a vr
) ⋅ p2 v
a rv32
( a vr −
23
4.1 Calculating minimum probability
truth of P( xir ,Q)
Now, we generate formula for calculating minimum probability of attribute
xir given Q = {q1 ,..., qn } , as input data.
Related to (27), we defined minimum
probability truth of P( xir , Q) as follows.
Pmin ( xir , Q) =
min
∨
P( xir , q j ).
(31)
x1u
a u12v a vr23
⋅ max(
a rv32
a
x3r
δ min = ( a 23
vr −
21
vu
a
31
a ru
13
ur
( a 13
ur −
⋅ max(
p 1 u , p 2 v ) or
p 1u , p 2 v )
a vu21 a 13
ur
) ⋅ p1u
a ru31
13
a 12uv a 23
a 21
vr
vu a ur
) ⋅ p 2 v + ( a 13
) ⋅ p1u +
ur −
a 32rv
a 31
ru
a 12uv a 23
vr
⋅ max( p 1 u , p 2 v ).
a 32
rv
or
1≤ j≤ n
To simplify the problem, let’s say that
system just has three compound attributes, X1 , X 2 , and X 3 and their relation
142
x2v
δ min = ( a 23
vr
13
a 12 a 23
a 21
vu a ur
− uv 32 vr ) ⋅ p 2 v + ( a 13
) ⋅ p1u +
ur −
31
a rv
a ru
a 21vu a 13ur
⋅ max( p 1 u , p 2 v )
a 31
ru
APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
2. If | ( x3 r ∩ x2 v ) |≤| ( x1u ∩ x3 r ) | and
Related to (27), we defined maximum
probability truth of P( xir , Q) as follows.
| ( x3 r ∩ x2 v ) |≤| ( x2 v ∩ x1u ) | , then
( x2 v ∩ x3 r ) will be put in
( x1u ∩ x3 r ) .
x2 v
x1u
max
P max (x ir , Q)
avr23 ⋅ max( p1u , p2v )
13
23
δmin = a23
vr ⋅ max(p1u , p2 v ) + (aur − avr ) ⋅ p1u
3. If | ( x1u ∩ x3 r ) |≤| ( x1u ∩ x2 v ) | and
| ( x1u ∩ x3 r ) |≤| ( x2 v ∩ x3 r ) | , then
( x1u ∩ x3 r ) will be put in
( x2 v ∩ x3 r ) .
a 13
ur ⋅ max(
ir
(34)
,q j )
2v
We separate formula above into two
parts. The first, we call direct predicted
probability of x 3r which is P(x 3r|q3) =
P(x 3r|q3) = p3r and the second, we call
indirect predicted probability truth of x3r
which is predicted from other attributes
value, P (x 3r , q1 ) ∨ max P ( x3 r , q2 ) . The next, we
compare both of them by applying min
function as follows.
(35)
Pmax ( x3 r ,Q ) = min{1,( P( x3 r ,q1 ) ∨ max P( x3 r ,q 2 )) + p3 r )}
2v
p 1u , p
P(x
Pmax(x3r , Q ) = P ( x3 r , q1 ) ∨ max P (x3r , q2 ) ∨ max P (x3r , q3 )
x3 r
( a vr23 − a 13
ur ) ⋅ p
1 ≤ j ≤n
To simplify the problem, let’s say that
system just has three compound
attributes, X1, X2 and X3 and their
relation shown in Fig. 2.2. We calculate
maximum probability truth of x3 r ∈ X 3
based on input Q=(q1, q2, q3).
23
(a13
ur − avr ) ⋅ p1u
x2 v
= ∨
)
The problem now, is how to calculate
P(x 3r , q 2 ) = ä max .i.e. X 1 has s attributes,
x1u
X2 has t attributes. Let’s say that,
x3 r
13
13
P(x 3r , q1 ) = max{p11 ⋅ a1r
,..., p1s ⋅ asr
} = p1u ⋅ a 13
ur
13
13
ä min = (avr23 − a ur
) ⋅ p 2v + a ur
⋅ max(p1u , p 2v )
From the above conditions, we generate a formula that satisfy all conditions
as follows.
23
a 12
13
uv a vr
δ min = (a 23
−
min(
, a 23
vr
vr , a ur )) ⋅ p 2v +
a 32
rv
(a13
ur − min(
We solve this problem by imaging
interrelationship among x 1u, x 2v and x 3r as
shown in Fig. 2.2, in the following four
conditions.
31
21
1. If ( arv32 + a12
uv ≤ 1) and ( ar 1 + a vu ≤ 1)
23
and ( a13
ur + avr ≤ 1) then
23
a 12
13
uv a vr
, a 23
vr , a ur )).p 1u +
32
a rv
a 12 a 23
13
min( uv32 vr , a 23
vr , a ur ). max( p1u , p 2v ).
a rv
Finally, we find that
23
P(x 3r , q2 ) = max{p21 ⋅ a1r23 ,..., p1t ⋅ atr23 } = p2u ⋅ avr
(33)
Pmin ( x3 r , Q) = max{δmin , P( x3 r | q3 )}.
4.2 Calculating Maximum Probability
Truth of P( xir , Q)
Next, we generate formula for calculating maximum probability of attribute
x ir given Q=(q1,… qn), as input data.
x1u
x 2v
avr23 ⋅ p 2v
a13
ur ⋅ p1 u
x3 r
δmax = avr23 ⋅ p2 v + a13
ur ⋅ p1 u .
23
a vr
12
32
12
2. If ( a 32
rv + a uv > 1) and ( ( a rv + a uv − 1 ). 32 >
a rv
21
( a31
ru + a vu - 1).
13
ur
31
ru
a
a 23
12
vr
) and (( a32
>
rv + a uv - 1).
a
a 32
rv
23
( a13
ur + a vr - 1)), then
143
JURNAL INFORMATIKA Vol. 5, No. 2, Nopember 2004: 135 - 146
x1u
32
12
(a23
vr − (arv + auv −1) ⋅
x2 v x3 r
12
(a32
rv + auv − 1) ⋅
32
12
( a13
ur − ( arv + auv − 1) ⋅
(a
+a
23
13
( a 13
ur − ( a vr + a ur − 1)) ⋅ p1 u +
a23
vr
⋅ max( p1u , p2v )
a32
rv
a 23
vr
) ⋅ p 2v +
a32
rv
32
12
δ max = ( a 23
vr − max( 0 , ( a rv + a uv − 1 ) ⋅
a 13
ur
31
a ru
12
( a 32
rv + a uv - 1).
a
a
23
vr
32
rv
21
) and (( a 31
ru + a vu - 1).
a
a
13
ur
31
ru
>
21
( avr23 − (a31
ru + avu − 1) ⋅
21
vu
aur13
) ⋅ p2v
aru31
31
21
δ max = ( a13
ur − ( a ru + a vu − 1) ⋅
31
21
( a 23
vr − ( a ru + a vu − 1) ⋅
4.
a13
ur
) ⋅ p1u
a31
ru
a13
(a + a − 1) ⋅ ur
⋅ max( p1u , p2v )
a31
ru
31
ru
21
( a 31
ru + a vu − 1 ) ⋅
a 13
ur
) ⋅ p1 u +
a 31
ru
a 13
ur
) ⋅ p2 v +
a 31
ru
a13
ur
⋅ max( p 1 u , p 2 v )
a 31
ru
13
23
13
If ( a23
vr + a ur > 1 ) and (( a vr + a ur − 1).
a13
ur
>
a31
ru
13
ur
31
ru
21
( a31
ru + a vu - 1).
a
23
) and (( a13
ur + a vr - 1) >
a
12
( a32
rv + a uv - 1).
a23
vr
), then
a32
rv
x1u
x2 v
x3 r
13
( avr23 − ( a23
vr + aur −1)) ⋅ p2v
( avr23 + a 13
ur −1) ⋅ max(p1u , p2v )
23
13
( a 13
ur − ( avr + aur − 1)) ⋅ p1u
144
12
max( 0 , ( a 32
rv + a uv − 1 ) ⋅
a 23
21
vr
, ( a 31
ru + a vu − 1 ) ⋅
a 32
rv
a 13
13
ur
, ( a 23
vr + a ur − 1 )) ⋅ max( p 1 u , p 2v ).
a 31
ru
(36)
Finally, we find that
Pmax ( x3 r , Q) = max{1, δmax + P( x3 r | q3 )}.
31
21
(a13
ur − (aru + avu − 1) ⋅
x1u x
3r
13
13
32
12
, ( a 23
vr + a ur − 1 ))) ⋅ p 2v + ( a ur − max( 0 , ( a rv + a uv − 1 ) ⋅
23
23
( a13
ur + a vr - 1)), then
x 2v
a 23
21
vr
, ( a 31
ru + a vu − 1 ) ⋅
a 32
rv
a vr
a 13
21
13
ur
, ( a 31
, ( a 23
ru + a vu − 1 ) ⋅
vr + a ur − 1 ))) ⋅ p1 u +
a 32
a 31
rv
ru
a 23
− 1 ) ⋅ vr
⋅ max( p1 u , p2 v )
a 32
rv
a13
21
ur
> 1) and ((a31
ru + avu − 1). 31 >
aru
21
3. If (a31
ru + avu
13
( a 23
vr + a ur − 1) ⋅ max( p1 u , p 2 v ).
From the above conditions, we generate a
formula that satisfy all condition as
follows.
a 23
vr
) ⋅ p1 u +
a 32
rv
32
12
( a13
ur − ( a rv + a uv − 1) ⋅
12
uv
a
) ⋅ p2v
a
avr23
) ⋅ p1u
arv32
32
12
δ max = (a 23
vr − ( a rv + a uv − 1) ⋅
32
rv
23
13
δ max = ( a 23
vr − ( a vr + a ur − 1 )) ⋅ p 2 u +
23
vr
32
rv
5. CONCLUSION
This paper proposed a method based
on conditional probability relation to
approximately calculate interval probability of dependency of data for data
querying. Theoretically the formulation is
quite interesting. However, it seems to be
too complicated to calculate interaction of
three or more components. Practically the
formulas should be simplified, even
though the accuracy of prediction may be
decreased.
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APPLICATION OF CONDITIONAL PROBABILITY IN PREDICTING INTERVAL PROBABILITY OF DATA QUERYING (Rolly Intan)
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