Mining Regular Patterns in Data Streams Using Vertical Format

G. Vijay Kumar, M. Sreedevi & NVS Pavan Kumar
International Journal of Computer Science and Security (IJCSS), Volume (6) : Issue (2) : 2012 142
Mining Regular Patterns in Data Streams Using Vertical Format
G. Vijay Kumar gvijay_73@yahoo.co.in
School of Computing
K L University
Guntur – 522 502, India
M. Sreedevi msreedevi_27@yahoo.co.in
School of Computing
K L University
NVS Pavan Kumar nvspavankumar@gmail.com
School of Computing
K L University
Abstract
The increasing prominence of data streams has been lead to the study of online mining in order
to capture interesting trends, patterns and exceptions. Recently, temporal regularity in occurrence
behavior of a pattern was treated as an emerging area in several online applications like network
traffic, sensor networks, e-business and stock market analysis etc. A pattern is said to be regular
in a data stream, if its occurrence behavior is not more than the user given regularity threshold.
Although there has been some efforts done in finding regular patterns over stream data, no such
method has been developed yet by using vertical data format. Therefore, in this paper we
develop a new method called VDSRP-method to generate the complete set of regular patterns
over a data stream at a user given regularity threshold. Our experimental results show that highly
efficiency in terms of execution and memory consumption.
Keywords: Data Streams, Temporal Regularity, Regular Pattern, Vertical Data.
1. INTRODUCTION
Unlike mining static databases, data stream mining [1, 2, 3] creates many new challenges. It is
unrealistic to keep the entire stream in the main memory or even in a secondary storage device
because data stream is a continuous, massive (e.g., terabytes in volume), unbounded, timely
ordered series of data elements generates at a rapid rate. Incredible volumes of data streams are
often generated by communication networks, Internet traffic, real-time surveillance systems, online
transactions in the financial market, remote sensors, scientific and engineering experiments and
other dynamic environments. Discovering knowledge in data streams is an important research
area in data mining and knowledge discovery process.
Mining Frequent patterns [4, 5] from static databases has been broadly studied in Stream data
mining [1, 2, 3]. Apriori algorithm [5] is a classical algorithm proposed by R. Agarwal and R.
Srikanth in 1993 for mining frequent item sets for Boolean association rules. The algorithm uses
prior knowledge and employs an iterative approach known as a level–wise search to generate
frequent item sets. First it generates with 1-item sets, recursively generates 2-item set and then
frequent 3-item set and continues until all the frequent item sets are generated. Later Han et. al
[4] proposed the frequent pattern tree (FP-tree) and FP-growth algorithm to mine frequent
patterns without candidate generation. The Apriori and FP-growth algorithms find outs the
occurrence frequencies of a pattern i.e., support. Several algorithms have been proposed so far
to mine frequent patterns in a transaction databases as well as in data streams. However, the
significance of a pattern may not always depend upon the occurrence frequency of a pattern (i.e.,

support). The significance of a pattern may also depend upon other occurrence characteristics
such as temporal regularity of a pattern. For example, to improve web site design the web site
administrator may be interested in regularly visited web page sequence rather than heavily hit
web pages only for a specific period of time. Also, in a retail market some products may have
regular demand than other products. To know how regularly a product has been sold is essential
rather than the occurrence frequency of a product. Therefore, finding patterns at regular intervals
also plays an important role in data mining.
Recently, Tanbeer et. al [6] introduced a new problem of discovering Regular Patterns that follow
a temporal regularity in their occurrence behavior. With the help of user given maximum regularity
measure at which pattern occurs in a database is called a regular pattern. They also extended the
same problem in Data Streams. They proposed a tree based data-structure, called RPS-tree [7]
that captures user-given regularity threshold and mines regular patterns in a data stream with the
help of FP-growth algorithm and conditional pattern bases and corresponding conditional trees.
Therefore, in this paper, we propose a new method called Vertical Data Stream Regular Patterns
method (VDSRP - method in short), using the same Data Stream which is in [7] to mine regular
patterns using vertical data format. By using Vertical Data Format [8, 9, 10, 11], it will be able to
judge the non-regular item sets before generating candidate item sets. The main idea of our new
method is to develop a simple, but yet powerful, that captures the data stream content in a
window by using sliding-window technique to find regular items. The experimental results show
the effectiveness of VDSRP-method in finding regular patterns in a Data Stream.
The rest of the paper is organized as follows. Section 2 summarizes the existing tree structure to
mine regular patterns. Section 3 introduces the problem definition of regular pattern mining. The
method of VDSRP to find regular patterns using vertical data format are given in section 4. Section
5, our experimental results are shown. Finally, we conclude the paper in section 6.
2. RELATED WORK
In data mining, one of the most important techniques is Association rule mining. It was first
introduced by Agarwal et al. [5]. It extracts frequent patterns, correlations, associations among
sets of items in databases. The main drawback with the classical Apriori algorithm is that it needs
repeated scans to generate candidate set. After that Frequent pattern tree and FP-growth
algorithm [4] is introduced by Han et al. to mine frequent patterns without candidate generation.
Periodic patterns [12], [13] and Cyclic patterns [14] are also closely related with Regular patterns.
Periodic pattern mining in time-series data focuses on the cyclic behavior of patterns either in
whole or some part of time-series. Although periodic pattern mining is closely related with our
work, it cannot be applied directly to mine regular patterns from a data stream because it process
with either time-series or sequential data.
Tanbeer et al. [7] have proposed a tree based data-structure, called RPS-tree that captures user-
given regularity threshold and mines regular patterns in a data stream with the help of FP-growth
[4] algorithm and conditional pattern bases and corresponding conditional trees. First, they
constructed RPS-tree consists of one root node referred to as “null” and a set of item-prefix sub-
trees called children of the root. Each node in an RPS-tree represents an itemset in the path from
the root up to that node. The RPS-tree maintains the occurrence information of all transactions in
the current window with the tree structure. Also RPS-tree maintains two types of nodes called
ordinary nodes and tail nodes. Nodes of both types explicitly maintain parent, children and node
traversal pointers. In addition each tail node maintains a tid-list and a tail-node pointer. The tail-
node pointer points to either the next tail node in the tree if any, or “null”. Then they construct an
item header table called RPS-table consists of each distinct item in the current window with
relative regularity and a pointer pointing to the first node in the RPS-tree that carries the item.
RPS-table of a RPS-tree consists of three fields, they are item name (i), regularity of i (r), and a
pointer to the RPS-tree for i (p). Similar to FP-growth mining, they mine the RPS-tree of
decreasing size to generate regular patterns by creating conditional pattern-bases and
corresponding conditional trees.
.

3. PROBLEM DEFINITION
Let I = {i1, i2, . . . , in} be the set of items. A set X = {ij, . . . , ik} ⊆ I, where j ≤ k and j, k ∈ [1, n] is
called a pattern (or an itemset). A transaction t = (tid, Y) is a couple where tid is a transaction-id
and Y is a patter or an itemset. If X ⊆ Y, which means that t contains X or X occurs in t. Let size(t)
be the size of t, i.e., the number of items in Y.
3.1 Definition 1 (Data Stream)
A data stream DS can be defined as infinite sequence of transactions, i.e., DS = [t1, t2, …, tm, …],
i ∈ [1, m] where ti is the i-th
arrived transaction. A window W can be referred to as a set of all
transactions between the i-th
and j-th
arrival of transactions, where j > i and the size of W is W= j
– i, i.e., the number of transactions between i-th
and j-th
arrival of transactions. Let each slide of
window introduce and expire slide_size, 1 ≤ slide_size ≥ W, transactions into and from the
current window. If X occurs in tj, j ∈ [1, W], such transactions-id is denoted as tj
X
, j ∈ [1, W].
Therefore Tw
X
= { tj
X
, . . . tk
X
}, j, k ∈ [1, W] and j ≤ k is the set of all transaction-ids where X
occurs in the current window W.
3.2 Definition 2 (A period of X in W)
Let tX
j+1 and tj
X
, j ∈ [1, ( W-1)], be two consecutive transaction-ids in Tw
X
. the number of
transactions between t
X
j+1 and tj
X
is defined as a period of X, say p
X
(i.e., p
X
= t
X
j+1 - tj
X
, j ∈ [1, (
W-1)]). For the simplicity of period computation, a “null” transaction with no item is considered
at the beginning of W, i.e., tf = 0(null), where tf represents the tid of the first transaction to be
considered. Similarly, tl, the tid of the last transaction to be considered, is the tid of the W
-th
transaction in the window, i.e., tl = t W. For instance, the stream data in Table 1, consider the
window is composed of eight transactions (i.e., tid = 1 to tid = 8 make the first window, say W1).
Then set of transactions in W1 where pattern (b, c) appears in (2, 3, 5). Therefore, the periods for
(b, c) are {(2 – tf) = 2, (3 – 2) = 1, (5 – 3) = 2 and (tl – 5) = 3}, where tf = 0 and tl = 8.
The occurrence periods of X in W defined as above will be the precise information about the
occurrence behaviour of a pattern. A pattern will not be a regular in W, if it appears after large
period at any stage. The largest occurrence period of a pattern can provide the upper limit of its
periodic occurrence characteristic. Hence, the measure of the characteristic of a pattern of being
regular in a W (i.e., the regularity of a pattern in W) can be defined as follows.
3.3 Definition 3 (Regularity of a pattern X in W)
Let in a TwX, PwX be the set of all periods of X in W i.e., PwX = { p1X , . . . , psX}, where s is
the total number of periods of X in W. Then, the regularity of X in W can be denoted as regw(X) =
Max(p1X, ..., psX}. For example, in DS of Table 1 regw(b, c) = 3, since Pw1{b, c} = Max(2, 1, 2,
3) = 3. Therefore a pattern is called a regular pattern in W if its regularity in W must not more than
a user given maximum regularity threshold called max_reg λ, with 1 ≤ λ ≤ W. The regularity
threshold is given as the percentage of window size.
Therefore the regular patterns in W satisfy the downward closure property [6]. i.e., if a pattern is
found to be regular, then all of its non-empty subsets will be regular. Accordingly, if a pattern is
not regular, then none of its supersets can be regular. Given DS, W, and max_reg, finding the
complete set of regular patterns in W, Rw that have regularity of not greater than the max_reg
value is the problem of mining regular patterns in data stream.
4. MINING REGULAR PATTERNS
In contrast with traditional data sets, the continuous flow (in and out) of stream data in a computer
system updates with varying rates. So, we had taken sliding window technique and vertical data
format to mine regular patterns from the data stream.

Let Figure 1. be the data stream which contains transaction-id i.e., tid, and itemset i.e., transaction
with respect to tid. Now consider the window size may be 8 i.e., the size of the window |W| = 8. Let
the first window W1 handles the transactions of data stream from tid-1 to tid-8, now convert this W1
into vertical data format i.e., (itemset : tid). Then find out the periods for each itemset which are
given in the data stream to get the regular patterns which are less than or equal to the user given
regularity threshold. After finding W1 regular patterns generate second window W2 and repeat the
same procedure to find out latest regular patterns from the data stream.
tid transaction
1. a, c, e, f
2. b, c, f
3. b, c, f
4. c, d, e
5. a, b, c, e
6. c, d, e
7. a, c, d, e
8. c, d, e, f
9. a, c
Window size | W | = 8
FIGURE 1: A data stream DS
Our proposed method is given below to mine regular patterns from the data stream with the help of
sliding-window technique and vertical data format. Both the Apriori algorithm and FP-growth
algorithm mine frequent patterns in Horizontal data format (i.e., {TID : itemset}), where TID is a
transaction-id and itemset is the set of items in transaction TID. But the data can also be present in
{item : TID-set} format where item is an item name and TID-set is the set of transactions containing
the item. This is known as Vertical data format. We are going to mine regular patterns from the
given data stream using vertical format.
VDSRP – Method
Input : DS, λ = 3
Output: Complete set of regular patterns
Procedure:
1. For each window W of size 8 in DS 12. Delete inext
2. Convert Tw into VTw 13. Else
3. For each item i in X where X ⊆ VTw 14. Result ← Result ∪ (i, next)
4. If(Find R(i) > λ) //FindR returns max_reg 15. Do “and” operation till all regular
5. Delete i itemsets found
6. Else 16. }
7. { 17. Update W(i, j)
8. Result ← Result ∪ (i)
9. For each item inext in X
10. {
11. If(Find R (inext) > λ) }
By using the example data stream in Figure 1. We convert the first window into vertical database
format and then we calculated period of X as explained in problem definition.
Streamflow
window1
window2

Itemset Tid-set P
X
R
a 1, 5, 7 1,4,2 4
b 2, 3, 5 2,1,2,3 2
c 1,2,3,4,5,6,7,8 1,1,1,1,1,1,1,1 1
d 4,6,7,8 4,2,1,1 4
e 1,4,5,6,7,8 1,3,1,1,1,1 3
f 1,2,3,8 1,1,1,5 5
TABLE 1: VDSRP-table with PX
and R
Finding periods is a simple procedure in our VDSRP-method. Create individual array to every
itemset. Find all the periods of each itemset by subtracting i.e., pX
= tX
j+1 - tj
X
. Among number of
periods that we get from the window we consider the maximum period from the tid-set of each
itemset. For example in Table 1, the periods of {a} are (1, 4, 2). Its regularity is 4 because it is the
maximum regularity for {a} and compare with the user given regularity threshold i.e., λ = 3. So {a}
is not a regular itemset because it is greater than the user given threshold.
Itemset Tid-set PX
R
(b, c) 2,3,5 2,1,1,3 3
(b, e) 5 5,3 5
(c, e) 1,4,5,6,7,8 1,3,1,1,1,1 3
(b, c, e) 5 5,3 5
TABLE 2: VDRSP-table with PX
and R
After getting 1-item set we go for 2-item set as shown in Table 2. The regular patterns in W1 satisfy
the down-ward closure property [5] i.e., if a pattern that found regular then all of its non-empty
subsets will be regular, if a pattern which is not regular then none of its supersets can be regular.
So we consider only the itemsets which are found regular to generate k+1 itemsets. From Table 2
we can say that itemsets (b, c), (c, e) are 2-item regular itemsets. We mine with the same
procedure until no regular item set generated in the window.
Itemset Tid-set P
X
R
a 4,6,8 4,2,2 4
b 1,2,4 1,1,2,4 4
c 1,2,3,4,5,6,7,8 1,1,1,1,1,1,1,1, 1
d 3,5,6,7 3,2,1,1,1 3
e 3,4,5,6,7 3,1,1,1,1,1 3
f 1,2,7 1,1,5,1 5
TABLE 3: VDRSP-table with PX
and R from W2
Generally, the regularity of patterns may change with the sliding of window i.e., with the deletion of
old transaction and the insertion of new transaction. For example, in Table 1 and Table 2 the
regular patterns {b} and {b, c} in W1 become irregular patterns in W2 because their regularity is
greater than max_reg. Again, the irregular patterns {d} and {c, d, e} in W1 become regular in W2.
Therefore to reflect the correct regularity of each item in the current window, we perform the
refreshing operation on VDSRP-table to get new window. The process continues to W2, W3 and
soon to find out the latest regular patterns from the data streams.

Itemset Tid-set P
X
R
(c, d) 3,5,6,7 3,2,1,1,1 3
(c, e) 3,4,5,6,7 3,1,1,1,1,1 3
(d, e) 3,5,6,7 3,2,1,1,1 3
(c, d, e) 3,5,6,7 3,2,1,1,1 3
TABLE 4: VDRSP-table with P
X
and R from W2
5. EXPERIMENT RESULTS
In this section we are going to present our results. All the programs are written in VC++ 6.0 and
executed in Windows XP on a 2.66 GHz machine with 2GB of main memory. We used our
VDSRP-method over several synthetic and real datasets which are frequently used to find out
frequent pattern mining experiments. With our proposed method we present our experiment
results by comparing with the existing RPS-tree. The detailed characteristics of the datasets are
available in Table 5 which are obtained from [15].
Dataset #Trans #Items MaxTL AvgTL Type
Kosarak 9,90,000 41,270 673 8.10 Real
Mushroom 8,124 119 23 23 Real
T1014D100K 1,00,759 870 30 10.10 Synthetic
TABLE 5: Database Characteristics
The above table shows some statistical information about the datasets. We consider the
slide_size = 1 for all the experiments. We report the results on Kosarak dataset which contain
9,90K transactions, 41,270 items and 8.10 average transaction length. We also report on
T1014D100K dataset which contains 1,00,759 transactions, 870 items, 10.10 average transaction
length and also on mushroom dataset which contains 8,124 transactions, 119 items and 23 is the
average transaction length.
5.1 Memory Efficiency
The memory requirements for our VDSRP-method on different datasets with different window
sizes are shown in Table 6. For example, in kosorak dataset when window size is 100K, the
memory required on an average of 3.57MB and when window size is 500K, it requires on an
average of 16.83 MB. Hence from table 6 it can be observed that VDSRP-method is memory
efficient on different real and synthetic datasets.
Kosorak W1 (100K) W2 (300K) W3 (500K) W4 (700K) W5 (900K)
3.57 MB 10.71 MB 16.83 MB 24.96 MB 32.1 MB
Mushroom W1 (1 K) W2 (3 K) W3 (5 K) W4 (7 K) W5 (8 K)
0.7 MB 0.14 MB 0.31 MB 0.48 MB 0.56 MB
T1014D100K W1 (20 K) W2 (40 K) W3 (60 K) W4 (80 K) W5 (100 K)
0.82 MB 1.74 MB 2.57 MB 3.31 MB 4.12 MB
TABLE 6: Memory Requirement for different window sizes
5.2 Runtime Efficiency
From figures 2(a) and 2(b) we can see that our proposed method runs faster than RPS-tree under
various regularity thresholds and with different window sizes respectively. We conducted
experiments on kosarak dataset with window size 500K on different max_reg(%) values. In figure
2(a) y-axis shows different regularity threshold values and x-axis shows the average total time
taken to convert the data into vertical format and mining time as well. Our proposed method is
taking on an average 38 seconds time when max_reg is 0.04%. If the max_reg value increases,
the execution time also increases to mine regular patterns from the window.

FIGURE 2 (a) On Kosarak (|W|)= 500K
In figure 2(b) the graph shows x-axis with different window sizes and y-axis with the average time
taken in seconds to mine regular patterns at max_reg is 0.06%. Our proposed method takes less
average time when compare with RPS-tree on different window sizes. For example, when window
size is 300K the average time taken is only 45 seconds.
FIGURE 2 (b) On Kosarak (max_reg = 0.06%)
6. CONCLUSION
In this paper we presented a VDSRP method which is much better than the existing RPS-tree
algorithm because it uses sliding-window technique and the advantages of Vertical Database
Format. This method is very simple to use with simple operations like arrays, unions, intersection,
deletion etc. to find out regular patterns over data streams. Our experiment results outperforms in
both execution and memory consumption.
7. REFERENCES
[1] S.K. Tanbeer, C.F. Ahmed, B.-S. Jeong, Y.-K. Lee “Sliding Window-based Frequent Pattern
Mining over Data Streams. Information Sciences”, 179, 2006, pp. 3843-3865.
Time(sec)
max_reg(%)
Time(sec)
window size (K)

[2] C.K.-S. Leung, , Q.I. Khan “DSTree: A Tree Structure for the mining of Frequent Sets from
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[3] H.-F. Li, S.-Y. Lee “Mining Frequent Itemsets over Data Streams Using Efficient Window
Sliding Techniques.” Expert Systems with Applications 36, 2009, pp. 1466-1477.
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[12]M.G. Elfeky, W.G. Aref, A.K. Elmagarmid “Periodicity detection in time series databases.”
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[14]B. Ozden, S. Ramaswamy, A. Silberschatz. “Cyclic Association Rules.” In.: 14th
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[15]Frequent Itemset Mining Dataset Repository http://fimi.cs.helsinki.fi/data/ and UCI machine
learning repository (University or California).

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