1. False Positive or False Negative: Mining Frequent Itemsets from High Speed Transactional Data Streams
Jeffrey Xu Yu , Zhihong Chong（崇志宏） , Hongjun Lu , Aoying Zhou
VLDB 2004

2. Introduction Mining data stream: Data items arrive continuously
One scan of data
Limited memory
Bounded error

3. Introduction
In this paper, develop algorithm of effectively mining frequent itemset with bound of memory consumption
Use false-negative

4. False Positive
Most existing algorithm of mining frequent itemset are false-positive oriented
Control memory consumption by error parameter ε
Allow item’s support below min support s but above s –ε as frequent
Approximate frequency counts over data streams (VLDB 02)

5. False Positive Memory bound : O ( ．log (εN))
Dilemma of false-positive approach
ε smaller, less # of false-positive item included
Memory consumption increase reciprocally in terms of ε
In Apriori, k-th frequent itemset generate (k+1)-th candidate itemset

6. False Positive & False Negative
All itemsets will output
All itemsets will output
S + ε
Some will output s
Some will output
S - ε
False Positive
False Negative

7. False Negative Error control and pruning
ε : prune data, control error bound,
changeable
ε decrease and approach to zero when # of observation increase
ε reciprocal of n
s : minimum support
n : # of observation

8. False Negative Memory control
δ : reliability, instead ε control memory consumption
Memory consumption related to ln(1/ δ)
In this approach not allow 1-itemsets with support below s as frequent

9. Comparison: False Positive & False Negative
Recall and Precision
A : true frequent itemsets
B : obtained frequent itemsets
Recall =
Precision =
|A∩B|
|A|
|A∩B|
|B|

10. Comparison: False Positive
δ=0.1
S(%)
True Size
Mined Size
Recall
Precision
0.08
21,361
126,307
1.00
0.17
0.10
12,252
68,275
0.18
0.20
2,359
23,154
0.16

11. Comparison: False Negative
s+ε: minimum support
S (%)
True Size
Mined Size
Recall
Precision
0.08
21,361
18,351
0.86
1.00
0.10
12,252
10,411
0.85
0.20
2,359
1,739
0.74

12. Chernoff Bound
Chernoff Bound give certain probabilistic guarantee on estimation of statistics about underlying data
Pr{ T ≥ еE[T]} ≤ е-E[T]
For example : Pick a lottery number
0000,0001, …,9999.
1,000,000 people buy $1 ticket
E[#winners] = 100
Pr{T≧273} ≦ e-100

13. Chernoff Bound Bernolli trails (coin flips): for any γ> 0
Pr[oi=1]=p, Pr[oi=0]=1-p
r : # of heads in n coin flips
np: expectation of r
for any γ> 0

14. Chernoff Bound Let r as r/n, min support s as p Replace sγ with ε
Right of equation be δ
Pr{|RunningSupport – TrueSupport|≥εn } ≤δ

15. Frequent or Infrequent
A pattern X is potential infrequent if
count(X) / n < s –εn in terms of n
A pattern X is potential frequent if it
is not potential infrequent in terms of n

16. FDPM-1(s, δ)

17. FDPM-1(s, δ) Delete infrequent items A D B C Memory is full
Count D B C 1 2 1 1
Memory is full
Compute new εn C D A
Source

18. FDPM-1(s, δ) Algorithm ensure :
item whose true frequency exceeds sN are output with probability of at least 1-δ
No item whose true frequency is less than sN are output
Probability of the estimated support that equal true support no less than 1-δ

19. Memory Bound Sup(X) ≥ ( s – εn) n |P| ≤ 1/( s – εn), when s – εn>0
|P| = n = =
n = 1
S – εn

20. FDPM-2(s, δ)

21. Mining Frequent Itemsets from a Data Stream
{B}
{AB} …
Count
{C}
{F}
{EF}
{D}
{E} 5 F 13 10 9 8 6 3 6 4 3
Mining Frequent Itemsets
{A,B}
……..
{E,F,G}
Memory is full, compute new εn
Delete infrequent itemsets n1
Item
Set
{A}
{B}
{AB}
{E}
{F}
{EF}
Count 4 5 6 …… P
Source

22. Conclusion False negative Limited memory
Error bound with some probability