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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 2, February 2011
Mining Maximal Dense Intervals from
Temporal Interval Data
F. A. Mazarbhuiya1 M.A.Khaleel1 A. K. Mahanta2 H. K. Baruah2
1 2
Dept. of Computer Science Department of Computer Science
1 2
College of Computer Science Gauhati University, India
1 2
King Khalid University, Abha Saudi Arabia Email: anjanagu@yahoo.co.in, hemanta_bh@yahoo.com
1
Email:{fokrul_2005, khaleel_dm}@yahoo.com
Abstract- Some real life data are associated with duration of algorithm to mine maximal dense fuzzy intervals. In such cases,
events instead of point events. The most common example of such we define the amount of contribution (also called vote) of a
data is data of cellular industry where each transaction is transaction t associated with time interval [t1, t2] for a given
associated with a time interval. Mining maximal fuzzy intervals fuzzy interval A as the ratio of the area bounded by the
from such data allows the user to group the transactions with
membership function A(x) (associated with the fuzzy interval)
similar behavior together. Earlier works were devoted to mining
frequent as well as maximal frequent non-fuzzy intervals. We and the real line included within the interval [t1, t2] to the total
propose here a method of mining maximal dense fuzzy intervals area covered by A(x) and the real line. If the total average of the
where density of an interval quite similar to the frequency of an votes of all the transactions in a fuzzy interval A exceeds a pre-
interval. defined threshold, then the fuzzy interval is called a dense fuzzy
interval. Similarly a dense fuzzy interval will be maximal if no
Keywords- Frequent intervals, Maximal frequent intervals, Density dense fuzzy interval contains it. The well-known A-priori
of a fuzzy interval, Minimum density, Contribution (vote) of a algorithm cannot be used here directly as the downward and
transaction on a fuzzy interval, join of two fuzzy intervals. upward closure property of frequent sets does not hold in this
case (it is proved with an example). We propose a variation of
the A-priori algorithm that works in this situation and gives us
I INTRODUCTION
the maximal dense fuzzy intervals.
Among the various types of data mining applications, analysis
of transactional data has been considered important. One
important extension of this mining problem is to include a II. RELATED WORKS
temporal dimension. Most of the earlier works done in this area
do not take into account the time factor. By taking into account One of the very useful extensions of conventional data mining
the time aspect, more interesting patterns that are time dependent is temporal data mining. In recent times it has been able to attract
can be extracted. Recently data mining in temporal data sets has a lot of researcher to work in this area. Considering the time
arisen as an important data mining problem [[2], [10]]. dimension in the conventional data mining problem, more
interesting patterns can be extracted that are time dependent.
Many real life problems are associated with duration events
There are mainly two broad directions of temporal data mining
instead of point events. In this paper we are considering such
[7]. One concerns the discovery of causal relationships among
datasets i.e. dataset having time intervals. Such datasets are
temporally oriented events. Ordered events from sequences and
called as temporal interval datasets. A record in such data
the cause of an event always occur before it. The other concerns
typically consists of the starting time and ending time (or the
the discovery of similar patterns within the same time sequence
length of the transaction) in addition to other fields. In [5] an
or among different time sequences. The underlying problem is to
algorithm for mining maximal frequent intervals from such data
find frequent sequential pattern in the temporal databases.
sets has been given
Wong et al [9] introduced the fuzzy concept into the
In practice however most of the time people make statements association rule mining to deal with quantitative attributes.
using vague terms like the early morning, late evening etc Quantitative attributes are normally handled by partitioning the
instead of mentioning strict time intervals. There is no strict attribute domains and then combining adjacent partitions [8].
boundary for separating early morning from morning. To Although this method can solve problems introduced by finite
represent such vague terms, fuzzy sets are required. In this paper domain, it causes the sharp boundary problem. To soften the
we discuss the problem of mining dense intervals using a fuzzy affect of soft boundaries, fuzzy sets are used. Here each
concept. The objective of this paper is three fold. First we quantitative attribute is associated with several fuzzy sets. A
propose the definition of density of a fuzzy interval over a fuzzy association rule looks like if X is A then Y is B, where X
transactional (where each transaction is associated with a time and Y are attributes and A and B are fuzzy sets which describe X
duration) dataset. Secondly, we propose to define a join and Y respectively. Prade et al [6] defined support and
operation on the fuzzy intervals and lastly we propose an confidence of a fuzzy association rule.
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In [2], Rossi and Ale extended the well-known A-priori A(x) for all x ∈[a, b] is known as left reference function and A(x)
algorithm for mining association rules to temporal data and for x ∈ [c, d] is known as the right reference function. The left
described a technique to find interesting patterns on the data that reference function is non-decreasing and the right reference
are time bounded. function is non-increasing [see e.g. [4]]. The area of a fuzzy
In [5], the problem of mining maximal frequent intervals is interval is defined as the area bounded by the membership
discussed. They define a maximal frequent interval as an interval function of the fuzzy interval and the real line.
that is frequent which means that it is present in sufficient
number of transactions and no other frequent interval contains it. B. Contribution (vote) of a transaction to a fuzzy interval
Using a pre-fix traversal algorithm, the maximal frequent We define vote of a transaction t associated with the time
intervals have been found and it was also found experimentally interval [t/, t//] for the fuzzy interval A= [a, b, c, d] as follows:
that pre-order traversal algorithm outperforms the A-priori based t //
algorithm.
Our approach is different from the above approaches. We are votet A =
∫t/
A( x)dx
d
taking into account the fact that the intervals of time are of fuzzy
nature. By calculating density of the fuzzy intervals in a ∫a
A( x)dx
particular transactional dataset where transactions are associated where A(x) is the membership function associated with the fuzzy
with time intervals (non-fuzzy) as mentioned in the next section, interval.
we first compute the dense fuzzy time intervals by using some t //
user defined minimum density value and then apply a join Here ∫t/
A( x)dx is the portion of the area bounded by A(x) and
operation to join neighboring intervals to find maximal dense d
fuzzy intervals. The fuzzy intervals and their membership
functions are provided by domain experts.
the real line included in the time interval [t/, t//]. ∫a
A( x)dx is
the total area bounded by A(x) and the real line.
III PROBLEM DEFINITION Obviously votet A lies in [0,1] and if A⊆[t/, t//], then votet A =
/ //
1 and if A∩[t , t ] =Φ, then votet A =0.
A. Some basic definitions related to fuzziness
Let E be the universe of discourse. A fuzzy set A in E is C. Density of a fuzzy time interval in a data set
characterized by a membership function A(x) lying in [0,1]. A(x) The density of a fuzzy interval over a given temporal interval
for x ∈E represents the grade of membership of x in A. Thus a dataset D is computed by summing up the votes of all the
fuzzy set A is defined as transactions of D for the corresponding fuzzy time interval and
A={(x, A(x)), x ∈ E } dividing it by the total number of transactions in D. Each record
A Fuzzy set A is said to be normal if A(x) =1 for at least one x contributes a vote, which falls in [0, 1].
∈ E. density D A = ∑ votet A / | D |
An α-cut of a fuzzy set is an ordinary set of elements with t∈D
membership grade greater than or equal to a threshold α, 0≤α≤1. A fuzzy interval is dense if its density is more than a user
Thus an α-cut Aα of a fuzzy set A is characterized by specified threshold called min_density.
Aα={x ∈E; A(x) ≥ α} [see e.g. [3]]
A fuzzy set is said to be convex if all its α-cuts are convex
sets. D. Join of two fuzzy intervals
The fuzzy intervals are given by the user as input. Two fuzzy
A fuzzy number is a convex normalized fuzzy set A defined
intervals A and B are called neighbors or adjacent to each other
on the real line R such that
if supp(A ∩ B) ≠Φ where supp(A ∩ B) ={x; (A ∩ B)(x) > 0 }[see
1. there exists an x0 ∈ R such that A(x0) =1, and e.g.[4]]. We assume that the input fuzzy intervals are such that if
2. A(x) is piecewise continuous. the intervals are arranged in the ascending order according to
Thus a fuzzy number can be thought of as containing the real their starting time then each fuzzy interval has a unique left
numbers within some interval to varying degrees. neighbor and a unique right neighbor. Let A = [a1, b1, c1, d1] and
Fuzzy intervals are special fuzzy numbers satisfying the B = [a2, b2, c2, d2] be two adjacent fuzzy intervals. Without loss
following. of generality we can assume that a1 < a2. Also we assume that for
1. there exists an interval [a, b] ⊂ R such that A(x0) =1 for any two adjacent fuzzy intervals such as A and B above c1 = a2
all x0∈ [a, b], and and d1 = b2 and for c1 ≤ x ≤ d1 A(x) = 1 – B(x). Our assumption is
2. A(x) is piecewise continuous. natural since otherwise some points will be given more emphasis
and some less emphasis. We define the join of A and B denoted
A fuzzy interval can be thought of as a fuzzy number with a flat by A∧ B is defined as
region. A fuzzy interval A is denoted by A = [a, b, c, d] with a <
A∧ B = [a1, b1, c2, d2]
b < c < d where A(a) = A(d) = 0 and A(x) = 1 for all x ∈[b, c].
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0, x ≤ 4 and x ≥ 9
Where (A∧ B)(x) = A(x), a1 ≤ x ≤ b1 B(x) = (x – 4)/2, 4≤ x ≤ 6
A(x) + B(x)=1,b1 ≤ x ≤ c2 1, 6≤x≤7
B(x) for c2 ≤ x ≤ d2 (9-x)/2, 7≤ x ≤ 9
To explain the joining operation we again consider two fuzzy 3
intervals [a1,b1,c1,d1] and [a2,b2,c2,d2] whose membership ∫ A( x)dx =1/3
votet1 A = 1
functions are shown in the figure1. Here c1 = a2 and b2 = d1. Any 6
point in between c1and d1 will have a membership value of A(x)
corresponding to A and corresponding to B it will have a
∫ A( x)dx1
6
membership value of B(x) = 1 – A(x) so that A(x) + B(x) = 1.
Thus our joined fuzzy interval will be [a1, b1, c2, d2] (shown in vote A =
∫ A( x)dx = 1
1
t2 6
fig.2).
B C F G ∫ A( x)dx1
6
a1 b1 c1=a2 d1=b2 c2 d2 vote A =
∫ A( x)dx =2/3
3
t3 6
A E D H
∫ A( x)dx1
Fig 1: Join of two fuzzy intervals 6
B G vote A =
∫ A( x)dx = 2.75/3
2
t4 6
a1 b1 c2 d2 ∫ A( x)dx1
A H 7
Fig 2: Joined interval
vote A =
∫ A( x)dx =.25/3
5
t5 6
A dense fuzzy interval is maximal if no super set of it is dense. ∫ A( x)dx1
However a subset of it may not be dense because the downward 7
and upward closure property for dense sets may not hold in this
case. vote A =
∫ A( x)dx = 0
6
t6 6
E. Theorem
∫ A( x)dx 1
2
The join of two fuzzy intervals is not dense if both of the fuzzy
intervals are not dense and dense if at least one of the fuzzy vote A =
∫ A( x)dx =.25/3
1
t7 6
intervals is dense.
∫ A( x)dx1
7
Proof. To prove the above result we consider a data set D with 8
transactions. The time-intervals associated with the transactions
vote A =
∫ A( x)dx = .25/3
5
are shown below. t8 6
∫ A( x)dx1
Transac Therefore,
tion id t1 t2 t3 t4 t5 t6 t7 t8
votet1 A+ votet 2 A+ votet 3 A+ votet 4 A+ votet 5 A+ votet 6 A+ votet 7 A+ votet 8 A
Time- Density ( A) = 8
interval [1,3] [1,6] [3,6] [2,6] [5,7] [6,7] [1,2] [5,7]
[ti , tj] =3.1666666/8
Table1: Transaction datasets = 0.395833325
Similarly
Consider the fuzzy intervals A = [1, 3, 4, 6] and B = [4, 6, 7, 9] 3
where the membership functions of A and B are respectively
votet1 B=
∫ B( x)dx =0
1
9
0, x ≤ 1 and x ≥ 6 ∫ B( x)dx
4
A(x) = (x – 1)/2, 1≤ x ≤ 3 6
1, 3≤x≤4
votet2 B=
∫ B( x)dx = 1/3
1
(6-x)/2, 4≤ x ≤ 6 9
∫ B( x)dx
4
and
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6 7
∫ B( x)dx = 1/3 ∫ ( A B)( x)dx =2/6
^
vote B =
t3
3
9
votet5 ( A B) =
^ 5
9
∫ B( x)dx ∫ ( A B)( x)dx
^
4 1
6 7
∫ B( x)dx = 1/3 ∫ ( A B)( x)dx = 1/6
^
vote B =
t4
2
9
votet6 ( A B) =
^ 6
9
∫ B( x)dx ∫ ( A B)( x)dx
^
4 1
7 2
∫ B( x)dx =1.75 ∫ ( A B)( x)dx =.25/6
^
vote B =
t5
5
9
votet7 ( A B) =
^ 1
9
∫ B( x)dx ∫ ( A B)( x)dx
^
4 1
7 7
∫ B( x)dx = 1/3 ∫ ( A B)( x)dx = 2/6
^
vote B =
t6
6
9
votet8 ( A B) =
^ 5
9
∫ B( x)dx ∫ ( A B)( x)dx
^
4 1
2
vote B =
∫ B( x)dx =01 Therefore,
t7 9
∫ B( x)dx 4 Density ( A ^ B ) =
votet1 A+votet 2 A+ votet 3 A+ votet 4 A+ votet 5 A+ votet 6 A+ votet 7 A+ votet 8 A
8
7
vote B =
∫ B( x)dx = 1.75/3
5
Therefore
^
Density ( A B ) = 2.83333/8
t8 9 = 0.35416625
∫ B( x)dx 4
So if we take min_dense = 0.35 then we see that A is dense but B
Therefore, is not dense whereas (A^B) is dense. This establishes that the
downward as well as upward closure property is not satisfied for
votet1 B + votet 2 B + votet 3 B + votet 4 B + votet 5 B + votet 6 B + votet 7 B + votet 8 B dense fuzzy intervals.
Density ( B ) = 8
= 2.5/8 = 0.3125 IV. PROPOSED ALGORITH
^
Now, ( A B ) = [1, 3, 7, 9] The algorithm is a level wise algorithm similar to the A-priori
algorithm used for frequent item set mining [1]. Input to the
0, x ≤ 1 and x ≥ 9 algorithm is a temporal interval data set say D, n fuzzy intervals
^
( A B ) (x)= (x–1)/2, 1≤ x ≤ 3 (called basic fuzzy intervals here) satisfying both the
assumptions made in definition of join of fuzzy intervals defined
1, 3≤x≤7
on the time period covered by the dataset and with a value of
(9-x)/2, 7≤ x ≤ 9
min_density (minimum density value). The algorithm first finds
3
the dense basic fuzzy intervals by going through the dataset once
∫ ( A B)( x)dx =1/6
^
and using the definition C given in section III. They are dense
votet1 ( A B) =
^ 1
9
fuzzy intervals at level 1 we denote this set of dense intervals by
∫ ( A B)( x)dx L1. Next each dense fuzzy interval at level 1 is joined with its left
^
1 neighbour and right neighbour both of which are basic intervals
6 (may not be dense) using the join operation defined definition D
∫ ( A B)( x)dx = 4/6
^
in section III. They are the candidates C2 at level 2. Using the
votet2 ( A B) =
^ 1
9 same technique, going through the data set once more the dense
∫ ( A B)( x)dx
^
1 fuzzy intervals at level 2 say L2 are obtained. These are kept and
6 the others removed. If any of the intervals obtained by joining a
∫ ( A B)( x)dx = 3/6
^
dense interval say A with its neighbours turn out to be dense then
votet3 ( A B) =
^ 3
9 A is removed from the list of dense intervals maintained at the
∫ ( A B)( x)dx
^
previous level. This level wise extraction goes on till a particular
1
6
level becomes empty. Then the intervals kept at each level are
∫ ( A B)( x)dx =2.75/6
^ the maximal dense fuzzy intervals. It is mentioned here that at
votet4 ( A B) =
^ 2
9
any level the dense intervals are joined with their neighbors from
∫ ( A B)( x)dx the basic fuzzy intervals only. This is done because two new
^
1 fuzzy intervals obtained by joining basic intervals although
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neighbors may not satisfy our second assumption (Definition D) Thus the set of first level dense fuzzy number is
for being conformable for the join operation. When two intervals L1= {D, E}
A and B are joined where A is the left neighbor of B, then the left Candidates for the second pass are
neighbor of A becomes the left neighbor of A^B and the right C2 = {C^D, D^E, E^F}
neighbor of B becomes the right neighbor of A ^B. where each members of C2 are formed by joining the members of
L1 with their left right neighbor of C1 using the definition of join
and C^D = [3, 4, 5, 6], D^E = [4, 5, 6, 7]. E^F = [5, 6, 7, 8]
• Algorithm 1 After the second pass, we get Density(C^D) = 0.4375,
Input C1 = { Ai ; i = 1, 2,…n} /* set of fuzzy intervals */ Density(D^E) = 0.5, Density(E^F) = 0.34375.
Set Density[i]=0;for i = 1,2,…,n /* Density[i] stores the Thus the second level dense sets are
Density of Ai */ L2 = {C^D, D^E}
for each transaction t in D Joining with their left and right neighbors from the basic fuzzy
{ numbers we obtain the candidates for the third pass as
Compute votet(Ai) for i = 1, 2, ….n C3 = {B^C^D, C^D^E, D^E^F}
Density[i] += votet(Ai) After third pass, we get Density(B^C^D) = 0.458333333,
} Density(C^D^E) = 0.458333333, Density (D^E^F) =
for(i = 1, 2,….,n) do 0.3958333333.
{ Thus the third level dense sets are
if( ( Density[i])/D ≥ min_density ) L3= {B^C^D, C^D^E}
Add Ai to L1 Similarly candidates for the fourth pass as
} C4 = {A^B^C^D, B^C^D^E, C^D^E^F}
k=1 After the fourth pass, we get Density(A^B^C^D) = 0.40625,
L1= [Dense fuzzy intervals at level 1] Density(B^C^D^E) = 0.0.4375, Density(C^D^E^F) = 0.390625.
for (k = 2 ; Lk ≠ φ ; k++) Thus the fourth level dense sets are
{ L4 = {A^B^C^D, C^D^E^F}
do Candidates for the fifth pass as
{ C5 = {A^B^C^D^E, B^C^D^E^F}
Ck = candidate-gen (Lk-1) After the fifth pass, we get Density(A^B^C^D^E) = 0.425,
Compute Lk by going through the transactions Density(B^C^D^E^F) = 0.3875.
in the dataset Thus the fifth level frequent sets are
k=k+1 L5 = {A^B^C^D^E}
} Candidates for the sixth pass are
} C6 = {A^B^C^D^E^F}
After the sixth pass Density(A^B^C^D^E^F) = 0.385416666,
which is less than min_ density.
Candidate-gen(Lk-1, Ck) Thus the sixth level is empty which is empty. So the algorithm
{ terminates giving the following maximal dense sets A^B^C^D^E.
for all A∈ Lk-1
form A^L and A^R where L and R are the left
and right neighbours of A respetively in case CONCLUSIONS
these exists.
/* For the extreme intervals both the In this paper, we have introduced the concept of fuzziness in
neighbours may not exist */ mining maximal dense intervals. In our datasets each transaction
Ck = Ck ∪ {A^L, A^R} has associated with it a time interval of the form [start_time,
} end_time]. It is a level-wise method of generating dense fuzzy
intervals. At the bottom level we have the basic dense fuzzy
To illustrate the above algorithm we again consider the example intervals. In subsequent levels the already obtained dense fuzzy
given in the section-III. For the sake of convenience, consider the intervals are expanded by joining them with their neignbours
basic fuzzy interval as fuzzy number with triangular membership from the basic fuzzy intervals and their density counted by going
function, which will be the input intervals for the first level i.e. through the dataset to check whether they are frequent or not.
C1 = {A, B, C, D, E, F}, where A = [1, 2, 3], B = [2, 3, 4], C = [3, The process continues till no candidate is generated or some
4, 5], D = [4, 5, 6], E = [5, 6, 7] and F = [6, 7, 8] and min_density level is empty. The algorithm finally gives only the maximal
= 0.4. dense fuzzy intervals. This algorithm although looks like A-
After the first pass we have, Density(A) = 0.375, Density(B) = priori algorithm, has a slight variation in the sense that it has to
0.375, Density(C) = 0.375, Density(D) = 0.5, Density(E) = 0.5, take into account the fact that the downward and upward closure
Density(F) = 0.1875. properties of dense interval do not hold here.
106 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 2, February 2011
Mohammed Abdul Khaleel received B.Sc.
degree in Mathematics from Osmania University,
India and M.C.A degree from Osmania
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Technical report, Department information management, Science, Gauhati University. She has a good number of
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[7] Roddick, J. F., Spillopoulou, M. (1999), A Biblography of and M.Sc. degree in Statistics from Gauhati University, India.
Temporal, Spatial and Spatio-Temporal Data Mining After that he received Ph. D. in Mathematics from IIT
Research, ACM SIGKDD. Kharagpur, India. He worked as a Lecturer in Mathematics in
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the 1996 ACM SIGMOD Conference on management of he is working as a Professor in the Department of Statistics,
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AUTHOR’S PROFILE
Fokrul Alom Mazarbhuiya received B.Sc.
degree in Mathematics from Assam University,
India and M.Sc. degree in Mathematics from
Aligarh Muslim University, India. After this he
obtained the Ph.D. degree in Computer Science
from Gauhati University, India. Since 2008 he
has been serving as an Assistant Professor in College of
Computer Science, King Khalid University, Abha, kingdom of
Saudi Arabia. His research interest includes Data Mining,
Information security, Fuzzy Mathematics and Fuzzy logic.
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ISSN 1947-5500
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