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Mining Association Rules Part II

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Data Mining Overview Data Mining Data warehouses and OLAP (On Line Analytical Processing.) Association Rules Mining Clustering: Hierarchical and Partitional approaches Classification: Decision Trees and Bayesian classifiers Sequential Patterns Mining Advanced topics: outlier detection, web mining

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Problem Statement I = { i 1, i 2, …, i m }: a set of literals, called items Transaction T : a set of items s.t. T I Database D : a set of transactions A transaction contains X, a set of items in I, if X T An association rule is an implication of the form X Y, where X,Y I The rule X Y holds in the transaction set D with confidence c if c% of transactions in D that contain X also contain Y The rule X Y has support s in the transaction set D if s% of transactions in D contain X Y Find all rules that have support and confidence greater than user- specified min support and min confidence

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Problem Decomposition 1. Find all sets of items that have minimum support (frequent itemsets) 2. Use the frequent itemsets to generate the desired rules

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Mining Frequent Itemsets: the Key Step Find the frequent itemsets: the sets of items that have minimum support A subset of a frequent itemset must also be a frequent itemset i.e., if {AB} is a frequent itemset, both {A} and {B} should be a frequent itemset Iteratively find frequent itemsets with cardinality from 1 to k (k-itemset) Use the frequent itemsets to generate association rules.

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The Apriori Algorithm L k : Set of frequent itemsets of size k (those with min support) C k : Set of candidate itemset of size k (potentially frequent itemsets) L 1 = {frequent items}; for (k = 1; L k != ; k++) do begin C k+1 = candidates generated from L k ; for each transaction t in database do increment the count of all candidates in C k+1 that are contained in t L k+1 = candidates in C k+1 with min_support end return k L k ;

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How to Generate Candidates? Suppose the items in L k-1 are listed in order Step 1: self-joining L k-1 insert into C k select p.item 1, p.item 2, …, p.item k-1, q.item k-1 from L k-1 p, L k-1 q where p.item 1 =q.item 1, …, p.item k-2 =q.item k-2, p.item k-1 < q.item k- 1 Step 2: pruning forall itemsets c in C k do forall (k-1)-subsets s of c do if (s is not in L k-1 ) then delete c from C k

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How to Count Supports of Candidates? Why counting supports of candidates a problem? The total number of candidates can be very huge One transaction may contain many candidates Method: Candidate itemsets are stored in a hash-tree Leaf node of hash-tree contains a list of itemsets and counts Interior node contains a hash table Subset function: finds all the candidates contained in a transaction

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Hash-tree:search Given a transaction T and a set C k find all of its members contained in T Assume an ordering on the items Start from the root, use every item in T to go to the next node If you are at an interior node and you just used item i, then use each item that comes after i in T If you are at a leaf node check the itemsets

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Is Apriori Fast Enough? — Performance Bottlenecks The core of the Apriori algorithm: Use frequent (k – 1)-itemsets to generate candidate frequent k-itemsets Use database scan and pattern matching to collect counts for the candidate itemsets The bottleneck of Apriori: candidate generation Huge candidate sets: 10 4 frequent 1-itemset will generate 10 7 candidate 2-itemsets To discover a frequent pattern of size 100, e.g., {a 1, a 2, …, a 100 }, one needs to generate candidates. Multiple scans of database: Needs (n +1 ) scans, n is the length of the longest pattern

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Max-Miner Max-miner finds long patterns efficiently: the maximal frequent patterns Instead of checking all subsets of a long pattern try to detect long patterns early Scales linearly to the size of the patterns

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Max-Miner: the idea ,21,31,42,32,4 3,4 1,2,3 1,2,41,3,4 1,2,3,4 2,3,4 Set enumeration tree of an ordered set Pruning: (1) set infrequency (2) Superset frequency Each node is a candidate group g h(g) is the head: the itemset of the node t(g) tail: an ordered set that contains all items that can appear in the subnodes Example: h({1}) = {1} and t({1}) = {2,3,4}

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Max-miner pruning When we count the support of a candidate group g, we compute also the support for h(g), h(g) t(g) and h(g) {i} for each i in t(g) If h(g) t(g) is frequent, then stop expanding the node g and report the union as frequent itemset If h(g) {i} is infrequent, then remove i from all subnodes (just remove i from any tail of a group after g) Expand the node g by one and do the same

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The algorithm Max-Miner Set candidate groups C {} Set of Itemsets F {Gen-Initial-Groups(T,C)} while C not empty do scan T to count the support of all candidate groups in C for each g in C s.t. h(g) U t(g) is frequent do F F U {h(g) U t(g)} Set candidate groups C new { } for each g in C such that h(g) U t(g) is infrequent do F F U {Gen-sub-nodes(g, C new )} C C new remove from F any itemset with a proper superset in F remove from C any group g s.t. h(g) U t(g) has a superset in F return F

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The algorithm (2) Gen-Initial-Groups(T, C) scan T to obtain F 1, the set of frequent 1-itemsets impose an ordering on items in F 1 for each item i in F 1 other than the greatest itemset do let g be a new candidate with h(g) = {i} and t(g) = {j | j follows i in the ordering} C C U {g} return the itemset F 1 (an the C of course) Gen-sub-nodes(g, C) /* generation of new itemsets at the next level*/ remove any item i from t(g) if h(g) U {i} is infrequent reorder the items in t(g) for each i in t(g) other than the greatest do let g’ be a new candidate with h(g’) = h(g) U {i} and t(g’) = {j | j in t(g) and j is after i in t(g)} C C U {g’} return h(g) U {m} where m is the greatest item in t(g) or h(g) if t(g) is empty

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Item Ordering By re-ordering items we try to increase the effectiveness of frequency-pruning Very frequent items have higher probability to be contained in long patterns Put these item at the end of the ordering, so they appear in many tails

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Mining Frequent Patterns Without Candidate Generation Compress a large database into a compact, Frequent-Pattern tree (FP-tree) structure highly condensed, but complete for frequent pattern mining avoid costly database scans Develop an efficient, FP-tree-based frequent pattern mining method A divide-and-conquer methodology: decompose mining tasks into smaller ones Avoid candidate generation: sub-database test only!

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Construct FP-tree from a Transaction DB {} f:4c:1 b:1 p:1 b:1c:3 a:3 b:1m:2 p:2m:1 Header Table Item frequency head f4 c4 a3 b3 m3 p3 min_support = 0.5 TIDItems bought (ordered) frequent items 100{f, a, c, d, g, i, m, p}{f, c, a, m, p} 200{a, b, c, f, l, m, o}{f, c, a, b, m} 300 {b, f, h, j, o}{f, b} 400 {b, c, k, s, p}{c, b, p} 500 {a, f, c, e, l, p, m, n}{f, c, a, m, p} Steps: 1.Scan DB once, find frequent 1-itemset (single item pattern) 2.Order frequent items in frequency descending order 3.Scan DB again, construct FP-tree

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Benefits of the FP-tree Structure Completeness: never breaks a long pattern of any transaction preserves complete information for frequent pattern mining Compactness reduce irrelevant information—infrequent items are gone frequency descending ordering: more frequent items are more likely to be shared never be larger than the original database (if not count node-links and counts) Example: For Connect-4 DB, compression ratio could be over 100

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Mining Frequent Patterns Using FP-tree General idea (divide-and-conquer) Recursively grow frequent pattern path using the FP- tree Method For each item, construct its conditional pattern-base, and then its conditional FP-tree Repeat the process on each newly created conditional FP-tree Until the resulting FP-tree is empty, or it contains only one path (single path will generate all the combinations of its sub-paths, each of which is a frequent pattern)

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Major Steps to Mine FP-tree 1)Construct conditional pattern base for each node in the FP-tree 2)Construct conditional FP-tree from each conditional pattern-base 3)Recursively mine conditional FP-trees and grow frequent patterns obtained so far If the conditional FP-tree contains a single path, simply enumerate all the patterns

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Step 1: From FP-tree to Conditional Pattern Base Starting at the frequent header table in the FP-tree Traverse the FP-tree by following the link of each frequent item Accumulate all of transformed prefix paths of that item to form a conditional pattern base Conditional pattern bases itemcond. pattern base cf:3 afc:3 bfca:1, f:1, c:1 mfca:2, fcab:1 pfcam:2, cb:1 {} f:4c:1 b:1 p:1 b:1c:3 a:3 b:1m:2 p:2m:1 Header Table Item frequency head f4 c4 a3 b3 m3 p3

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Properties of FP-tree for Conditional Pattern Base Construction Node-link property For any frequent item a i, all the possible frequent patterns that contain a i can be obtained by following a i 's node-links, starting from a i 's head in the FP-tree header Prefix path property To calculate the frequent patterns for a node a i in a path P, only the prefix sub-path of a i in P need to be accumulated, and its frequency count should carry the same count as node a i.

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Step 2: Construct Conditional FP-tree For each pattern-base Accumulate the count for each item in the base Construct the FP-tree for the frequent items of the pattern base m-conditional pattern base: fca:2, fcab:1 {} f:3 c:3 a:3 m-conditional FP-tree All frequent patterns concerning m m, fm, cm, am, fcm, fam, cam, fcam {} f:4c:1 b:1 p:1 b:1c:3 a:3 b:1m:2 p:2m:1 Header Table Item frequency head f4 c4 a3 b3 m3 p3

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Mining Frequent Patterns by Creating Conditional Pattern-Bases Empty f {(f:3)}|c{(f:3)}c {(f:3, c:3)}|a{(fc:3)}a Empty{(fca:1), (f:1), (c:1)}b {(f:3, c:3, a:3)}|m{(fca:2), (fcab:1)}m {(c:3)}|p{(fcam:2), (cb:1)}p Conditional FP-treeConditional pattern-base Item

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Step 3: Recursively mine the conditional FP-tree {} f:3 c:3 a:3 m-conditional FP-tree Cond. pattern base of “am”: (fc:3) {} f:3 c:3 am-conditional FP-tree Cond. pattern base of “cm”: (f:3) {} f:3 cm-conditional FP-tree Cond. pattern base of “cam”: (f:3) {} f:3 cam-conditional FP-tree

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Single FP-tree Path Generation Suppose an FP-tree T has a single path P The complete set of frequent pattern of T can be generated by enumeration of all the combinations of the sub-paths of P {} f:3 c:3 a:3 m-conditional FP-tree All frequent patterns concerning m m, fm, cm, am, fcm, fam, cam, fcam

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Principles of Frequent Pattern Growth Pattern growth property Let be a frequent itemset in DB, B be 's conditional pattern base, and be an itemset in B. Then is a frequent itemset in DB iff is frequent in B. “abcdef ” is a frequent pattern, if and only if “abcde ” is a frequent pattern, and “f ” is frequent in the set of transactions containing “abcde ”

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Why Is Frequent Pattern Growth Fast? Our performance study shows FP-growth is an order of magnitude faster than Apriori, and is also faster than tree-projection Reasoning No candidate generation, no candidate test Use compact data structure Eliminate repeated database scan Basic operation is counting and FP-tree building

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FP-growth vs. Apriori: Scalability With the Support Threshold Data set T25I20D10K

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FP-growth vs. Tree-Projection: Scalability with Support Threshold Data set T25I20D100K

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Presentation of Association Rules (Table Form )

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Visualization of Association Rule Using Plane Graph

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Visualization of Association Rule Using Rule Graph

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Iceberg Queries Icerberg query: Compute aggregates over one or a set of attributes only for those whose aggregate values is above certain threshold Example: select P.custID, P.itemID, sum(P.qty) from purchase P group by P.custID, P.itemID having sum(P.qty) >= 10 Compute iceberg queries efficiently by Apriori: First compute lower dimensions Then compute higher dimensions only when all the lower ones are above the threshold

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