1. A Fast Algorithm for Subspace Clustering by Pattern Similarity
Haixun Wang, Fang Chu, Wei Fan, Philip S Yu, and Jian Pei
In the 16th International Conference on Scientific and Statistical Database Management, 2004
2019/1/3
報告人:吳建良

2. Abstract Clustering by pattern similarity SeqClus: a novel model
Objects exhibit a coherent pattern of rise and fall in subspaces
SeqClus: a novel model
Intuitive for capturing subspace pattern similarity
Reduce computation complexity dramatically
Discovery pattern similarity embedded in data sequence

3. Subspace pattern similarity
(b) A Shifting pattern in subspace {b, c, h, j, e}
(a) Raw data: 3 objects, 10 columns
(c) A Scaling pattern in subspace {f, d, a, g, i}

4. Applications Analysis of Large Scientific Datasets
Gene expression data
Example: VPS8, EFB1, CYS3 changes coherently under {CH1I, CH1D, CH2B}

5. Applications (cont.) Example 1. Counting Example 2. Clustering
How many genes whose expression level in sample CH1I is about 100±5 units higher than that in CH2B, 280±5 units higher than that in CH1D, and 75±5 units higher than that in CH2I?
Find clusters of genes that exhibit coherent subspace patterns, given the following constraints: i) the subspace pattern has dimensionality higher than minCols; and ii) the number of objects in the cluster is larger than minRows

6. Applications (cont.) Discovery of Sequential Patterns
Network event logs
Focus on two attributes: Event and Timestamp

7. Applications (cont.) Example 3. Sequential Pattern
Mining subspace pattern
Events as condition on X axis
Timestamp as Y axis
Sliding Window
Event CiscoDCDLinkUp is followed by MLMStatusUp that is followed, in turn, by CiscoDCDLinkUp, under the constraint that the interval between the first two events is about 20±2 seconds, and the interval between the 1st and 3rd events is about 40±2 seconds
Event1
Event2
Event3 …
Timestamp t1 t2 t3 t4

8. Object Definition
Use sequences to represent objects in a tabular dataset D
A={c1, c2,…cn}: the set of columns
: total order
Object x:
Where is the value of x in column ci
Example:
EFB1: <(CH1I, 318), (CH1B, 280), (CH1D, 37)…>

9. Sequence-based pattern similarity
Distance function:
Given two objects x and y, a subspace S, an arbitrary dimension ck
Example:
(1) ck
max x y c1 c3 c2 c4 1 4 2 3 5
distk,S(x, y)=3

10. Sequence-based pattern similarity
Property1
Proof:

11. Pattern Definition Pattern p:
a tuple (T,δ), T is an ordered sequence of (column, value) pair
Object x exhibits pattern p in subspace S={c1,…,ck}
(2)

12. Pattern Definition (cont.)
Example:
Pattern p: <(c1,0), (c2,2), (c3,1), 2>
Object x: <(c1,2), (c2,3), (c3,1)>
x’: <(c1,0), (c2,1), (c3,-1)>
High density pattern
The number of objects that satisfy Eq(2) reach a user-defined density threshold c1 c3 c2 1 2 3 4 -1 -2 +δ -δ

13. Construct a Counting tree
A compact summary structure of density patterns, like suffix trie
Create a counting tree
For each object x, insert its relevant subsequences (length≧) into the tree. At the end up node t, increase the t’s count by 1
With depth-first traversal, label each tree node t as a triple: (ID1, ID2, Count)

14. Construct a Counting tree (cont.)
Relevant subsequence
The relevant subsequence of an object x in an n-dimension space are:
(ID1, ID2, Count)
ID1: unique identification of node t
ID2: is the largest ID1 of t’s descendent nodes
Count:
If t is a leaf node, Count is the number of objects end up at t
Otherwise, it is the sum of the counts of its child nodes

15. Example: =2 x c1 c3 c2 c4 z y x1 c1 c3 c2 c4 x3 x2 y1 c1 c3 c2 c4 y3
Relevant subsequences x c1 c3 c2 c4 4 3 2 z 1 y x1 c1 c3 c2 c4 -1 -4 -2 x3 x2 -3 y1 c1 c3 c2 c4 1 -2 y3 y2 -3 -1 z1 c1 c3 c2 c4 1 2 z3 -2 z2 -1 c1 c2 c3 c4
[1, , ]
[2, , ]
[3, , ]
[4, ,1]
[1,9, ]
[2,4, ]
[3,4, ]
[4,4,1]
[ , ,1]
[1,9,3]
[2,4,1]
[3,4,1]
[4,4,1] x -1 -4 -2 1 y -2
x, y -3 -1
[ , ,1]
[5,9,2]
[6,7,1]
[7,7,1]
[5, , ]
[6, , ]
[7, ,1]
[5,9, ]
[6,7, ]
[7,7,1] z 2
[8,9, ]
[9,9,1]
[8, , ]
[9, ,1]
[8,9,1]
[9,9,1]
[ , ,1]
[10, , ]
[11, , ]
[12, ,2]
[10,14, ]
[11,12, ]
[12,12,2]
[ , ,2]
[ , ,1]
[10,14,3]
[11,12,2]
[12,12,2] z 1 -1
[13,14,1]
[14,14,1]
[ , ,1]
[13, , ]
[14, ,1]
[13,14, ]
[14,14,1]

16. Counting list: count pattern occurrences during the depth-first traversal
Link head
List of node labels
(c1,c1,0)
[1,9,3]
[1,9,3]
[2,4,1]
[3,4,1]
[4,4,1]
(c1,c2,-1)
[2,4,1] x -1 -4 -2
[5,9,2]
[6,7,1]
[7,7,1]
(c1,c3,-4)
[3,4,1] 1 y
(c1,c4,-2)
[4,4,1] -2
[8,9,1]
[9,9,1] z
(c1,c2,1)
[5,9,2] 2
[10,14,3]
[11,12,2]
[12,12,2]
(c1,c3,-2)
[6,7,1]
x, y -3 -1
(c1,c4,0)
[7,7, 1 ]
[9,9,1+1]
[13,14,1]
[14,14,1] 1 z
(c1,c3,2)
[8,9,1] -1
(c2,c2,0)
[10,14,3]
(c2,c3,-3)
[11,12,2]
(c2,c4,-1)
[12,12,2]
(c2,c3,1)
[13,14,1]
[14,14,3]

17. Counting Pattern Occurrence-1 1 2
[1,12,3]
[2,5,1]
[3,5,1]
[4,5,1]
[6,12,2]
[7,9,1]
[8,9,1]
[10,12,1]
[11,12,1] c5
[5,5,1]
[9,9,1]
[12,12,1]
Link head
List of node labels
(c1,c1,0)
[1,12,3]
(c1,c2,-1)
[2,5,1]
(c1,c3,1)
[3,5,1]
[7,9,2]
(c1,c4,1)
[4,5,1]
[8,9,2]
[11,12,3]
(c1,c5,2)
[5,5,1]
[9,9,2]
[12,12,3]
(c1,c2,1)
[6,12,2]
<(c1,0), (c3,1), (c4,1)>出現次數=2
(c1,c3,2)
[10,12, 1]
(c1,c1,0)
[1,12,3]
(c1,c3,1)
[3,5,1]
[7,9,2]
計算規則:
If IDv is the first element of the list, then there are cntw objects
Otherwise, there are cntw-cntu objects
(c1,c4,1)
[4,5,1]
[8,9,2]
[11,12,3]

18. Clustering Construct Cluster Tree
node: is the triple (item, count, range-list)
Count the occurrences of all 2-column pattern
If it is frequent (count≧minRows), insert it under root node of cluster tree
For each node p on current level, join p with its eligible nodes to derive nodes on the next level

19. Clustering (cont.) A node q is node p’s eligible nodes Join operation
q is on the same level as p;
if p denotes a-b=v, and q denotes c-d=v’,
then
Join operation
p: (a-b=v, count, range-list)
New node: (c-b=v’, count’, range-list’)
q: (c-d=v’, count’, range-list’)

20. Clustering (cont.) minRows=2 (c2-c1=-1, 1, [2,4]) (c2-c1=1, 2, [5,9])X
(c2-c1=1, 2, [5,9])
(c4-c1=0, 2, [7,7], [9,9])
join
(c3-c1=-4, 1, [3,4]) X
(c3-c1=-2, 1, [6,7])
從root到leaf代表了freq. pattern
<(c1,0),(c2,1),(c4,0)>
<(c1,0),(c4,0)>
<(c2,0),(c3,-3),(c4,-1)>
<(c2,0),(c4,-1)> X
(c3-c1=2, 1, [8,9]) X
Root
(c4-c1=-2, 1, [4,4]) X
(c4-c1=0, 2, [7,7], [9,9])
(c3-c2=-3, 2, [11,12])
(c4-c2=-1, 3, [12,12], [14,14])
join
(c3-c2=1, 1, [13,14]) X
(c4-c2=-1, 3, [12,12], [14,14])

21. Experiment Synthetic data Tabular form Sequential form
Value range: 0~300
Embed clusters: δ=0, 2, 4, 6,…
Sequential form
(id, timestamp)
Generated by probabilistic distribution

22. Experiment (cont.)
Scalability

23. Experiment (cont.) Gene expression data Event management data
Yeast micro-array
2884 genes, 17 conditions,
Expression level range: 0~600, discretized into 40 bins
Event management data
NETVIEW
EventType: 241
10 days’ worth of event logs

24. Experiment (cont.)
Yeast micro-array
NETVIEW