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Frequent Closed Pattern Search By Row and Feature Enumeration

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Outline Problem Definition Related Work: Feature Enumeration Algorithms CARPENTER: Row Enumeration Algorithm COBBLER: Combined Enumeration Algorithm

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Problem Definition Frequent Closed Pattern: 1) frequent pattern: has support value higher than the threshold 2) closed pattern: there exists no superset which has the same support value Problem Definition: Given a dataset D which contains records consist of features, our problem is to discover all frequent closed patterns respect to a user defined support threshold.

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Related Work Searching Strategy: breadth-first & depth-first search Data Format: horizontal format & vertical format Data Compression Method: diffset, fp-tree, etc.

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Typical Algorithms APRIORI feature enumeration horizontal format breadth-first search CHARM feature enumeration vertical format depth-first search deffset technique CLOSET feature enumeration horizontal format depth-first search fp-tree technique

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CARPENTER CARPENTER stands for Closed Pattern Discovery by Transposing Tables that are Extremely Long Motivation Algorithm Prune Method Experiment

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Motivation Bioinformatic datasets typically contain large number of features with small number of rows. Running time of most of the previous algorithms will increase exponentially with the average length of the transactions. CARPENTER’s search space is much smaller than that of the previous algorithms on these kind of datasets and therefore has a better performance.

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Algorithm The main idea of CARPENTER is to mine the dataset row-wise. 2 steps: First, transpose the dataset Second, search in the row enumeration tree.

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Transpose Table Feature a, b, c, d. Row r1, r2, r3, r4. r1a b c r2b c d r3b c d r4d ar1 br1 r2 r3 c dr2 r3 r4 original table transposed table transpose b c dr4 project on (r2 r3) projected table

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Row Enumeration Tree According to the transposed table, we build the row enumeration tree which enumerates row ids with a pre- defined order. We do a depth first search in the row enumeration tree with out any prune strategies. { } r1 {abc} r2 {bcd} r3 {bcd} r4 {d} r1 r2 {bc} r1 r3 {bc} r1 r4 {} r1 r2 r3 {bc} r1 r2 r4 {} r1 r3 r4 { } r2 r3 {bcd} r2 r4 {d} r2 r3 r4 {d } r3 r4 {d } r1r2r3r4 { } minsup=2 bc: r1r2r3 bcd: r2r3 d: r2r3r4 ar1 br1 r2 r3 c dr2 r3 r4

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Prune Method 1 In the enumeration tree, the depth of a node is the corresponding support value. Prune a branch if there won’t be enough depth in that branch, which means the support of patterns found in the branch will not exceed the minimum support. r2 {bcd} r2 r3 {bcd} r2 r4 {d} depth= 1 sup =1 2 sub-nodes Max support value in branch “r2” will be 3, therefore prune this branch. minsup 4

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Prune Method 2 If r j has 100% support in projects table of r i, prune the branch of r j. br3 c dr3 r4 r2 {bcd}r2 r3 {bcd} r2 r4 {d} r3 has 100% support in the projected table of “r2”, therefore branch “r2 r3” will be pruned and whole branch is reconstructed. r2 r3 r4 {d} r2 r3 {bcd}r2 r3 r4 {d} b c dr4

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Prune Method 3 At any node in the enumeration tree, if the corresponding itemset of the node has been found before, we prune the branch rooted at this node. r2 {bcd}r2 r3 {bcd} r2 r4 {d} r3 {bcd}r3 r4 {d} Since itemset {bcd} has been found before, the branch rooted at “r3” will be pruned.

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Performance We compare 3 algorithms, CARPENTER, CHARM and CLOSET. Dataset (Lung Cancer) has 181 rows with features. We set 3 parameters, minsup, Length Ratio and Row Ratio.

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minsup Lung Cancer, 181 rows, length ratio 0.6,row ratio 1. Running time of CARNPENTER changes from 3 to 14 second

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Length Ratio Lung Cancer, 181 rows, sup 7 (4%), row ratio 1 Running time of CARPENTER changes from 3 to 33 seconds

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Row Ratio Lung Cancer, 181 rows, length ratio 0.6,sup 7 (4%) Running time of CARPENTER changes from 9 to 178 seconds

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Conclusion We propose an algorithm call CARPENTER for finding closed pattern on long biological datasets. CARPENTER perform row enumeration instead of column enumeration since the number of rows in such datasets are significantly smaller than the number of features. Performance studies show that CARPENTER is much more efficient in finding closed patterns compared to existing feature enumeration algorithms.

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COBBLER Motivation Algorithm Performance

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Motivation With the development of CARPENTER, existing algorithms can be separated into two parts. Feature enumeration: CHARM, CLOSET, etc. Row enumeration: CARPENTER We have two motivations to combine these two enumeration methods

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Motivation 1.We can see that these two enumeration methods have their own advantages on different type of data set. Given a dataset, the characteristic of its sub-dataset may change. 2.Given a dataset with both large number of rows and features, a single row enumeration algorithm or a single feature enumeration method can not handle the dataset. dataset sub- dataset more rows than features more features than rows project

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Algorithm There are two main points in the COBBLER algorithm How to build an enumeration tree for COBBLER. How to decide when the algorithm should switch from one enumeration to another. Therefore, we will introduce the idea of dynamic enumeration tree and switching condition

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Dynamic Enumeration Tree We call the new kind of enumeration tree used in COBBLER the dynamic enumeration tree. In dynamic enumeration tree, different sub-tree may use different enumeration method. We use the table as an example in later discussion r1a b c r2a c d r3b c r4d ar1 r2 br1 r3 cr1 r2 r3 dr2 r4 originaltransposed

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{ } a {r1r2} b {r1r3} c {r1r2r3} d {r2r4} ab {r1} ac {r1r2} ad {r2} abc {r1} abd { } acd { r2} bc {r1r3} bd { } bcd { } cd {r2 } { } r1 {abc} r2 {acd} r3 {bc} r4 {d} r1r2 {ac} r1r3 {bc} r1r4 { } r1r2r3 {c} r1r2r4 { } r1r3r4 { } r2r3 {c} r2r4 {d } r2r3r4 { } r3r4 { } Single Enumeration Tree r1r2r3r4 { } abcd { } Feature enumerationRow enumeration r1a b c r2a c d r3b c r4d

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Dynamic Enumeration Tree { } a {r1r2} b {r1r3} c {r1r2r3} d {r2r4} r1 {bc} r2 {cd} r1r2 {c} r1 {c} r3 { c} r1r3 { c} r2 {d } Feature enumeration to Row enumeration ab {r1} ac {r1r2} ad {r2} abc {r1} abd { } acd { r2} abcd { } a {r1r2} abc: {r1} ac: {r1r2} acd: {r2} r1bc r2cd br1 cr1 r2 dr2

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Dynamic Enumeration Tree { } r1 {abc} r2 {acd} r3 {bc} r4 {d} a {r2} b {r3} c {r2r3 } ab {} ac { r2} bc {r3 } a {r1} d {r4 } ac {r1 } b {r1 } Row enumeration to Feature Enumeration c {r1r3} ad { } acd { } cd { } c {r1r2 } bc {r1 } r1 {abc} r1r2 {ac} r1r3 {bc} r1r4 { } r1r2r3 {c} r1r2r4 { } r1r3r4 { } r1r2r3r4 { } ac: {r1r2} bc: {r1r3} c: {r1r2r3}

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Dynamic Enumeration Tree When we use different condition to decide the switching, the structure of the dynamic enumeration tree will change. No matter how it switches, the result set of closed pattern will be the same as the result of the single enumeration.

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Switching Condition The main idea of the switching condition is to estimate the processing time of the a enumeration sub-tree, i.e., row enumeration sub-tree or feature enumeration sub- tree. Define some characters.

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Switching Condition

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Suppose r=10, S(f1)=0.8, S(f2)=0.5, S(f3)=0.5, S(f4)=0.3 and minsup=2 Then the estimated deepest node under f1 is f1f2f3, since S(f1)*S(f2)*S(f3)*r=2 >minsup S(f1)*S(f2)*S(f3)*S(f4)*r=0.6 < minsup

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Experiments We compare 3 algorithms, COBBLER, CHARM and CLOSET+. One real-life dataset and one synthetic data. We set 3 parameters, minsup, Length Ratio and Row Ratio.

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minsup Synthetic dataReal-life data (thrombin)

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Length and Row ratio Synthetic data

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Discussion The combination of row and feature enumeration also makes some disadvantage The cost to calculate the switching condition and the cost of bad decision. The increased cost in pruning, maintain two set of pruning system.

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Discussion We may use other more complicated data structure in our algorithm to improve the performance, e.g., the vertical data format and diffset technique. And more efficient switching condition may improve the algorithm further.

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Conclusion The COBBLER algorithm gives better performance on dataset where the advantage of switching can be shown, e.g., complex dataset or dataset has both large number of rows and features. For simple characteristic data, a single enumeration algorithm may be better.

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Future Work Using other data structure and technique in the algorithm. Extend COBBLER to handle dataset that can not be fitted into memory.

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Thanks

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