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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 = {i1, i2, …, im}: 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 Lk: Set of frequent itemsets of size k (those with min support) Ck: Set of candidate itemset of size k (potentially frequent itemsets) L1 = {frequent items}; for (k = 1; Lk !=; k++) do begin Ck+1 = candidates generated from Lk; for each transaction t in database do increment the count of all candidates in Ck that are contained in t Lk+1 = candidates in Ck+1 with min_support end return k Lk;

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**How to Generate Candidates?**

Suppose the items in Lk-1 are listed in order Step 1: self-joining Lk-1 insert into Ck select p.item1, p.item2, …, p.itemk-1, q.itemk-1 from Lk-1 p, Lk-1 q where p.item1=q.item1, …, p.itemk-2=q.itemk-2, p.itemk-1 < q.itemk-1 Step 2: pruning forall itemsets c in Ck do forall (k-1)-subsets s of c do if (s is not in Lk-1) then delete c from Ck

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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 Ck 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: 104 frequent 1-itemset will generate 107 candidate 2-itemsets To discover a frequent pattern of size 100, e.g., {a1, a2, …, a100}, one needs to generate 2100 1030 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 f Pruning: (1) set infrequency**

Set enumeration tree of an ordered set f 1 2 3 4 Pruning: (1) set infrequency (2) Superset frequency 1,2 1,3 1,4 2,3 2,4 3,4 1,2,3 1,2,4 1,3,4 2,3,4 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 1,2,3,4 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 Cnew{ } for each g in C such that h(g) U t(g) is infrequent do F F U {Gen-sub-nodes(g, Cnew)} C Cnew 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 F1, the set of frequent 1-itemsets impose an ordering on items in F1 for each item i in F1 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 F1 (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**

TID Items 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} min_support = 0.5 {} f:4 c:1 b:1 p:1 c:3 a:3 m:2 p:2 m:1 Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 Steps: Scan DB once, find frequent 1-itemset (single item pattern) Order frequent items in frequency descending order 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**

Construct conditional pattern base for each node in the FP-tree Construct conditional FP-tree from each conditional pattern-base 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 {} f:4 c:1 b:1 p:1 c:3 a:3 m:2 p:2 m:1 Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 Conditional pattern bases item cond. pattern base c f:3 a fc:3 b fca:1, f:1, c:1 m fca:2, fcab:1 p fcam:2, cb:1

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**Properties of FP-tree for Conditional Pattern Base Construction**

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

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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 Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 f:4 c:1 All frequent patterns concerning m m, fm, cm, am, fcm, fam, cam, fcam {} f:3 c:3 a:3 m-conditional FP-tree c:3 b:1 b:1 a:3 p:1 m:2 b:1 p:2 m:1

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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 {(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-tree Conditional pattern-base Item

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**Step 3: Recursively mine the conditional FP-tree**

{} f:3 c:3 am-conditional FP-tree {} f:3 c:3 a:3 m-conditional FP-tree Cond. pattern base of “am”: (fc:3) {} 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 {} All frequent patterns concerning m m, fm, cm, am, fcm, fam, cam, fcam f:3 c:3 a:3 m-conditional FP-tree

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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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