1. Mining Long Sequential Patterns in a Noisy Environment Jiong Yang, Wei Wang, Philip S. Yu, and Jiawei Han SIGMOD 2002 Presented by: Eddie Date: 2002/12/23

2. Outline Introduction A Model of Obscure Patterns A Border Collapsing Approach Experimental Results Some thoughts

3. Introduction In a noisy environment, an observed sequence may not accurately reflect the underlying behavior. For example, in a protein sequence, the amino acid N is likely to mutate to D with little impact to the biological function of the protein. The substitution may prevent an occurrence of a pattern from being recognized and in turn slashes the support of that pattern. It would be desirable if the occurrence of D in the observation can be related to a possible mutation from N in an appropriate manner.

4. Introduction In order to accommodate the above circumstance, it is necessary to allow some flexibility in pattern matching. A so-called compatibility matrix is introduced to enable a clear representation of the likelihood of symbol substitutions. –The following matrix shows an example of the compatibility matrix between 5 symbols d 1, d 2, d 3, d 4, and d 5. An entry C(d i,d j ) > 0 indicates that d i might be (mis)represented as d j in the observation; while C(d i,d j ) = 0 implies that the symbol d i cannot be represented as d j despite the presence of noise.

5. Introduction It provides a meaningful measure to reveal the substance and the assessment of each entry has great impact to the final result. In practice, this matrix can either given by a domain expert or learned from a training data set. d1d2d3d4d5 d10.90.1000 d20.050.80.050.10 d30.0500.70.150.1 d400.1 0.750.05 d5000.1500.85 observed value true value

6. A Model of Obscure Patterns The authors proposed a new metric, namely match, to characterize the significance of the pattern in a symbol sequences. In particular, they utilize the conditional probability to assess the match of the patter. Def. Given a pattern P and a subseuence of l symbols s, the match of P in s is defined as the conditional probability Prob(P|s). (or M(P, s)) M(P, s) = П 1 ≦ i ≦ l C(d i,d i ’) For example, P 1 =d 1 d 2 in a subsequence s=d 1 d 3 M(P 1, s) = C(d 1,d 1 )  C(d 2,d 3 ) = 0.9  0.05=0.045 Where the pattern P 2 =d 1 d 5 does not match s because M(P 2, s) = C(d 1,d 1 )  C(d 2,d 5 ) = 0.9  0=0

7. A Model of Obscure Patterns Def. For a symbol sequence S and a pattern P, the match of P in S is defined as the maximal match of P in every distinct subsequence in S. That is, M(P,S) = max s  S M(P, s) Therefore, the dynamic programming method is used to compute the match. –E.g.: The match of P = d 1 d 2 in the sequence d 1 d 3 d 2 d 3 d 4 d 1 is 0.72. d1d3d2d3d4d1 0.9 d20.0450.72

8. A Model of Obscure Patterns Def. Given a pattern P and a database D of N sequences, the match of P in D is the average match of P in every sequence in D, i.e. M(P,D) = Similar to the traditional support model, all patterns that meet the min_match threshold are then referred to as frequent patterns.

9. A Model of Obscure Patterns A direct implication of the Apriori property is that, given a min-match threshold, the set of frequent patterns occupy a “contiguous portion” in the pattern lattice, and can be described using the notion of border.

10. A Border Collapsing Approach The Chernoff bound is used to estimate the range of the match of a pattern from a sample of the data with a high statistical confidence (e.g., 99.99%). Chernoff bound: Let X be a random variable whose spread is R(e.g. 1), suppose that we have n independent observations of X, and the mean is μ. The Chernoff Bound states that with probability 1-δ, the true mean of X is at least μ-ε, where ε = E.g. μ is the mean of 10000 samples of random variables, the true value of the random variable is at least μ-0.0215 with 99.99% confidence.

11. A Border Collapsing Approach Claim (Chernoff bound estimation) Given a set of sample data and a threshold min-match, a pattern is frequent with probability 1-δ if μ match >min_match+ε and is infrequent with probability 1-δ if μ match < min_match-ε, other patterns remain undecided and are referred as ambiguous patterns. –For example, assume min_match=0.02, ε=0.003, μ match (d 1 d 2 )=0.025, μ match (d 3 d 4 )=0.015, μ match (d 5 d 6 )=0.021, then we can know that d 1 d 2 is frequent, d 3 d 4 is infrequent, and d 5 d 6 is ambiguous.

12. A Border Collapsing Approach Claim (Restricted spread) The restricted spread R for the match of a pattern d 1 d 2 …d l is R = min 1 ≦ i ≦ l match[d i ] where match[d i ] is the match of the symbol d i in the entire sequence database. –For example, if the match of d1d2 are 0.1 and 0.05 in the database respectively, then the match of d1d2 has to be between 0 and 0.05 in the database. Thus, we can use R = 0.05 and reduce the value of ε by 95%.

13. A Border Collapsing Approach Hence, the following three-fold algorithm is developed for mining the obscure patterns: –Finding the match of individual symbols and taking a random sample of sequences –Identify the borders that embrace the set of ambiguous patterns(i.e., whose match is between min_match–ε and min_match+ε) –Locate the border of frequent patterns via border collapsing

14. A Border Collapsing Approach

15. Experimental Results Accuracy and completeness of the two models While the accuracy and the completeness are very high, the quality of the resutls by the support model degrades significantly. For example, when  = 0.6, the accuracy and completeness of the support model are 61% and 33%, respectively.

16. Experimental Results Accuracy and completeness of the two models With a given degree of noise (e.g.,  = 0.1), the accuracy and completeness of the support and match models with different pattern lengths are shown as above.

17. Some thoughts They devised a border-collapsing algorithm. It is very helpful to speed up the reduction of candidates. I learned a novel technique about database sampling. This paper mentioned me about the noise problem. I think that I should take more consideration about it on my thesis.