Frequent Pattern Mining S L I D E S B Y : S H R E E J A S W A L

Transcription

1 Frequent Pattern Mining S L I D E S B Y : S H R E E J A S W A L

2 Topics to be covered Market Basket Analysis, Frequent Itemsets, Closed Itemsets, and Association Rules; Frequent Pattern Mining, Efficient and Scalable Frequent Itemset Mining Methods, The Apriori Algorithm for finding Frequent Itemsets Using Candidate Generation, Generating Association Rules from Frequent Itemsets, Improving the Efficiency of Apriori, A pattern growth approach for mining Frequent Itemsets; Mining Frequent itemsets using vertical data formats; Mining closed and maximal patterns; Introduction to Mining Multilevel Association Rules and Multidimensional Association Rules; From Association Mining to Correlation Analysis, Pattern Evaluation Measures; Introduction to Constraint-Based Association Mining. 2

3 Which Chapter from which Text Book? Chapter 6: Mining Frequent Patterns, Associations and Correlations: Basic Concepts and Methods from Han, Kamber, "Data Mining Concepts and Techniques", Morgan Kaufmann 3nd Edition Chapter 7: Advanced Pattern Mining from Han, Kamber, "Data Mining Concepts and Techniques", Morgan Kaufmann 3nd Edition 3

4 What Is Frequent Pattern Analysis? Frequent pattern: a pattern (a set of items, subsequences, substructures, etc.) that occurs frequently in a data set First proposed by Agrawal, Imielinski, and Swami [AIS93] in the context of frequent itemsets and association rule mining Motivation: Finding inherent regularities in data What products were often purchased together? Beer and diapers?! What are the subsequent purchases after buying a PC? What kinds of DNA are sensitive to this new drug? Can we automatically classify web documents? Applications Basket data analysis, cross-marketing, catalog design, sale campaign analysis, Web log (click stream) analysis, and DNA sequence analysis. 4

5 Why Is Freq. Pattern Mining Important? Discloses an intrinsic and important property of data sets Forms the foundation for many essential data mining tasks Association, correlation, and causality analysis Sequential, structural (e.g., sub-graph) patterns Pattern analysis in spatiotemporal, multimedia, timeseries, and stream data Classification: associative classification Cluster analysis: frequent pattern-based clustering Data warehousing: iceberg cube and cube-gradient Semantic data compression: fascicles Broad applications 5

6 Association Rule Mining Given a set of transactions, find rules that will predict the occurrence of an item based on the occurrences of other items in the transaction 7 Market-Basket transactions TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke Example of Association Rules {Diaper} {Beer}, {Milk, Bread} {Eggs,Coke}, {Beer, Bread} {Milk}, Implication means co-occurrence, not causality!

7 Definition: Frequent Itemset Itemset A collection of one or more items Example: {Milk, Bread, Diaper} k-itemset An itemset that contains k items Support count ( ) Frequency of occurrence of an itemset E.g. ({Milk, Bread,Diaper}) = 2 Support Fraction of transactions that contain an itemset E.g. s({milk, Bread, Diaper}) = 2/5 Frequent Itemset An itemset whose support is greater than or equal to a minsup threshold TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke

8 Association Rule Definition: Association Rule An implication expression of the form X Y, where X and Y are itemsets Example: {Milk, Diaper} {Beer} Rule Evaluation Metrics Support (s) Fraction of transactions that contain both X and Y i.e P(X U Y) Confidence (c) Measures how often items in Y appear in transactions that contain X i.e. P(X Y) 9 TID Example: Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke { Milk, Diaper} Beer (Milk, Diaper, Beer) s T (Milk,Diaper, Beer) c (Milk, Diaper)

9 Association Rule Mining Task Given a set of transactions T, the goal of association rule mining is to find all rules having support minsup threshold 10 confidence minconf threshold Brute-force approach: List all possible association rules Compute the support and confidence for each rule Prune rules that fail the minsup and minconf thresholds Computationally prohibitive!

10 Mining Association Rules 11 TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke Observations: Example of Rules: {Milk,Diaper} {Beer} (s=0.4, c=0.67) {Milk,Beer} {Diaper} (s=0.4, c=1.0) {Diaper,Beer} {Milk} (s=0.4, c=0.67) {Beer} {Milk,Diaper} (s=0.4, c=0.67) {Diaper} {Milk,Beer} (s=0.4, c=0.5) {Milk} {Diaper,Beer} (s=0.4, c=0.5) All the above rules are binary partitions of the same itemset: {Milk, Diaper, Beer} Rules originating from the same itemset have identical support but can have different confidence Thus, we may decouple the support and confidence requirements

11 Two-step approach: Mining Association Rules 1. Frequent Itemset Generation Generate all itemsets whose support minsup 2. Rule Generation Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset Frequent itemset generation is still computationally expensive 12

12 Frequent Itemset Generation null 13 A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE ABCDE Given d items, there are 2 d possible candidate itemsets

13 Closed Patterns and Max-Patterns A long pattern contains a combinatorial number of subpatterns, e.g., {a 1,, a 100 } contains ( 1001 ) + ( 1002 ) + + ( ) = = 1.27*10 30 sub-patterns! 14 Solution: Mine closed patterns and max-patterns instead An itemset X is closed if X is frequent and there exists no super-pattern Y כ X, with the same support as X (proposed by Pasquier, et ICDT 99) An itemset X is a max-pattern if X is frequent and there exists no frequent super-pattern Y כ X (proposed by SIGMOD 98) Closed pattern is a lossless compression of freq. patterns Reducing the # of patterns and rules

14 Closed Patterns and Max-Patterns Exercise. DB = {<a 1,, a 100 >, < a 1,, a 50 >} Min_sup = 1. What is the set of closed itemset? <a 1,, a 100 >: 1 < a 1,, a 50 >: 2 What is the set of max-pattern? <a 1,, a 100 >: 1 15

15 Scalable Methods for Mining Frequent Patterns The downward closure property of frequent patterns Any subset of a frequent itemset must be frequent If {beer, diaper, nuts} is frequent, so is {beer, diaper} i.e., every transaction having {beer, diaper, nuts} also contains {beer, diaper} Scalable mining methods: Three major approaches Apriori (Agrawal & Srikant@VLDB 94) Freq. pattern growth (FPgrowth Han, Pei & 00) Vertical data format approach (Charm Zaki & 02) 16

16 Apriori: A Candidate Generation-and-Test Approach Apriori pruning principle: If there is any itemset which is 17 infrequent, its superset should not be generated/tested! (Agrawal & 94, Mannila, et KDD 94) That is, If an itemset is frequent, then all of its subsets must also be frequent Apriori principle holds due to the following property of the support measure: X, Y : ( X Y) s( X) s( Y) Support of an itemset never exceeds the support of its subsets This is known as the anti-monotone property of support

17 Apriori: A Candidate Generation-and-Test Approach Method: Initially, scan DB once to get frequent 1-itemset Generate length (k+1) candidate itemsets from length k frequent itemsets Test the candidates against DB Terminate when no frequent or candidate set can be generated 18

18 Illustrating Apriori Principle Item Count Bread 4 Coke 2 Milk 4 Beer 3 Diaper 4 Eggs 1 Minimum Support = 3 Items (1-itemsets) 19 Itemset Count {Bread,Milk} 3 {Bread,Beer} 2 {Bread,Diaper} 3 {Milk,Beer} 2 {Milk,Diaper} 3 {Beer,Diaper} 3 Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs) Triplets (3-itemsets) If every subset is considered, 6 C C C 3 = 41 With support-based pruning, = 13 Itemset Count {Bread,Milk,Diaper} 3

19 Illustrating Apriori Principle null A B C D E AB AC AD AE BC BD BE CD CE DE Found to be Infrequent ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE Pruned supersets ABCDE 20

20 The Apriori Algorithm An Example Sup Itemset sup Database TDB min = 2 {A} 2 L Tid Items C 1 1 {B} 3 10 A, C, D 20 B, C, E 30 A, B, C, E 40 B, E 1 st scan C 2 C 2 {A, B} 1 L 2 Itemset sup 2 nd scan {A, C} 2 {B, C} 2 {B, E} 3 {C, E} 2 Itemset 21 {C} 3 {D} 1 {E} 3 sup {A, C} 2 {A, E} 1 {B, C} 2 {B, E} 3 {C, E} 2 Itemset sup {A} 2 {B} 3 {C} 3 {E} 3 Itemset {A, B} {A, C} {A, E} {B, C} {B, E} {C, E} C Itemset 3 3 rd scan L 3 {B, C, E} Itemset sup {B, C, E} 2

21 The Apriori Algorithm Pseudo-code: C k : Candidate itemset of size k L k : frequent itemset of size k 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 L k+1 22 increment the count of all candidates in C k+1 that are contained in t = candidates in C k+1 with min_support end return L 1 = k L k ;

22 Important Details of Apriori How to generate candidates? Step 1: self-joining L k Step 2: pruning How to count supports of candidates? Example of Candidate-generation L 3 ={abc, abd, acd, ace, bcd} Self-joining: L 3 *L 3 abcd from abc and abd acde from acd and ace Pruning: 23 acde is removed because ade is not in L 3 C 4 ={abcd}

23 What s Next? 24 Having found all of the frequent items. This completes our Apriori Algorithm. What s Next? These frequent itemsets will be used to generate strong association rules( where strong association rules satisfy both minimum support & minimum confidence).

24 Generating Association Rules from Frequent Itemsets Procedure: 25 For each frequent itemset l, generate all nonempty subsets of l. For every nonempty subset s of l, output the rule s ->(l-s) if support_count(l)/support_count(s) >= min_conf where min_conf is minimum confidence threshold. Lets take l = {I1,I2,I5}. Its all nonempty subsets are {I1,I2}, {I1,I5}, {I2,I5}, {I1}, {I2}, {I5}.

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27 Methods to Improve Apriori s Efficiency Hash-based technique can be used to reduce the size of the candidate k-itemsets, Ck, for k>1. For example when scanning each transaction in the database to generate the frequent 1-itemsetes, L1, from the candidate 1-itemsets in C1, we can generate all of the 2- itemsets for each transaction, hash them into a different buckets of a hash table structure and increase the corresponding bucket counts: a. H(x,y)=((order of x)*10+(order of y)) mod 7 e.g. H(I1,I4)=((1*10)+4))mod7=0 b. A 2-itemset whose corresponding bucket count in the hash table is below the threshold cannot be frequent and thus should be removed from the candidate set.

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29 Methods to Improve Apriori s Efficiency contd Transaction reduction a transaction that does not contain any frequent k itemsets cannot contain any frequent k+1 itemsets. Therefore, such a transaction can be marked or removed from further consideration because subsequent scans of the database for j-itemsets, where j>k, will not require it.

30 Methods to Improve Apriori s Efficiency contd Partitioning (partitioning the data to find candidate itemsets): A partitioning technique can be used that requires just two database scans to mine the frequent itemsets. It consists of two phases. In phase I, the algorithm subdivides the transactions of D into n nonoverlapping partitions. If the minimum support threshold for transactions in D is min_sup, then the minimum support count for a partition is min_sup * the number of transactions in that partition. For each partition, all frequent itemsets within the partition are found. These are referred to as local frequent itemsets. A local frequent itemset may or may not be frequent with respect to the entire database, D. Any itemset that is potentially frequent with respect to D must occur as a frequent itemset in at least one of the partitions. Therefore all local frequent itemsets are candidate itemsets with respect to D. The collection of frequent itemsets from all partitions forms the global candidate itemsets with respect to D. In phase II, a second scan of D is conducted in which the actual support of each candidate is assessed in order to determine the global frequent itemsets

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32 Methods to Improve Apriori s Efficiency contd Sampling (mining on a subset of a given data): The basic idea of the sampling approach is to pick a random sample S of the given data D, and then search for frequent itemsets in S instead of D. In this way, we trade off some degree of accuracy against efficiency. The sample size of S is such that the search for frequent itemsets in S can be done in main memory, and so only one scan of the transactions in S in required overall. Because we are searching for frequent itemsets in S rather than in D, it is possible that we will miss some of the global frequent itemsets. To lessen this possibility, we use a lower support threshold than minimum support to find the frequent itemsets local to S.

33 Methods to Improve Apriori s Efficiency contd Dynamic itemset counting (adding candidate itemsets at different points during a scan): A dynamic itemset counting technique was proposed in which the database is partitioned into blocks marked by start points. In this variation new candidate itemsets can be added at any start point, which determines new candidate itemsets only immediately before each complete database scan. The resulting algorithm requires fewer database scan than Apriori.

34 Mining Frequent Patterns Without Candidate Generation 36 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!

35 FP-Growth Method: Construction of FP-Tree First, create the root of the tree, labeled with null. Scan the database D a second time. (First time we scanned it to create 1-itemset and then L). The items in each transaction are processed in L order (i.e. descending order). A branch is created for each transaction with items having their support count separated by colon. Whenever the same node is encountered in another transaction, we just increment the support count of the common node or Prefix. To facilitate tree traversal, an item header table is built so that each item points to its occurrences in the tree via a chain of node-links. Now, The problem of mining frequent patterns in database is transformed to that of mining the FP-Tree. 37

36 FP-Growth Method: Construction of FP-Tree 38

37 Mining the FP-Tree by Creating Conditional (sub) pattern bases Steps: 1.Start from each frequent length-1 pattern(as an initial suffix pattern). 2.Construct its conditional pattern base which consists of the set of prefix paths in the FP-Tree co-occurring with suffix pattern. 3.Then, Construct its conditional FP-Tree& perform mining on such a tree. 4.The pattern grow this achieved by concatenation of the suffix pattern with the frequent patterns generated from a conditional FP-Tree. 5.The union of all frequent patterns (generated by step 4) gives the required frequent itemset. 39

38 FP-Tree Example Continued 40

39 FP-Tree Example Continued 41 Out of these, Only I1 & I2 is selected in the conditional FP- Tree because I3 is not satisfying the minimum support count. For I1, support count in conditional pattern base = = 2 For I2, support count in conditional pattern base = = 2 For I3, support count in conditional pattern base = 1 Thus support count for I3 is less than required min_sup which is2 here. Now, We have conditional FP-Tree with us. All frequent pattern corresponding to suffix I5 are generated by considering all possible combinations of I5 and conditional FP-Tree. The same procedure is applied to suffixes I4, I3 and I1. Note:I2 is not taken into consideration for suffix because it doesn t have any prefix at all.

40 Why Frequent Pattern Growth Fast? Performance study shows 43 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

41 Example 1 44 Transaction T1 T2 T3 T4 T5 Items a, c, d, f, g, i, m, p a, b, c, f, l, m, o b, f, h, j, o b, c, k, n, p a, c, e, f, l, m, n, p Min support = 60%

42 Step 1 45 Find support of every item. Item Support Item Support a 3 i 1 b 3 j 1 c 4 k 1 d 1 l 2 e 1 m 3 f 4 n 2 g 1 o 2 h 1 p 3

43 Step 2 Choose items that are above min support and arrange them in descending order. 46 item support f 4 c 4 a 3 b 3 m 3 p 3

44 Step 3 47 Rewrite transactions with only those items that have more than min support, in descending order Transactio n Items Frequent items T1 a, c, d, f, g, i, m, p f, c, a, m, p T2 a, b, c, f, l, m, o f, c, a, b, m T3 b, f, h, j, o f, b T4 b, c, k, n, p c, b, p T5 a, c, e, f, l, m, n, p f, c, a, m, p

45 Step 4 48 Introduce a Root Node Root = NULL Give the first transaction to the root. Transaction T1: f, c, a, m, p NULL Root f : 1 c : 1 a : 1 m : 1 p : 1

46 Transaction T2: f, c, a, b, m 49 Root f : 2 c : 2 a : 2 m : 1 b : 1 p : 1 m : 1

47 Transaction T3 : f, b 50 Root f : 3 c : 2 b : 1 a : 2 m : 1 b : 1 p : 1 m : 1

48 Transaction T4: c, b, p 51 Root f : 3 c : 1 c : 2 b : 1 a : 2 b : 1 m : 1 b : 1 p : 1 p : 1 m : 1

49 Transaction T5: f, c, a, m, p 52 Root f : 4 c : 1 c : 3 b : 1 a : 3 b : 1 m : 2 b : 1 p : 1 p : 2 m : 1

50 Mining the FP Tree 53 Frequent items Condition pattern Conditional tree p fcam:2, cb: 1 c:3 {c,p:3} Frequent Patterns m fca: 2, fcab: 1 f:3, c:3, a:3 {f,m:3},{c,m:3},{a,m:3},{a,c,f,m:3} b fca:1, f:1, c:1 Empty {b:3} a fc:3 f:3, c:3 {f,a:3},{c,a:3},{f,c,a:3} c f:3 f:3 {f,c:3} f Empty Empty {f:4}

51 Example 2 54 Transaction Id t1 t2 t3 t4 t5 Items Purchased b, e a, b, c, e b, c, e a, c, d a Given: min support = 40%

52 Step 1/ 2: Step: 3 55 Items Support a 3 b 3 c 3 d 1 e 3 Transactions t1 t2 t3 t4 t5 Items in desc order b, e b, e, c, a b, e, c a, c a

53 Step: 4 Root 56 b : 3 a : 2 e: 3 c:1 c:2 a : 1 Step: 5 Mining the FP Tree Item Conditional pattern conditional tree a bec: 1 empty b empty empty c be: 2, a: 1 b:2, e:2 e b:3 b:3

54 Mining frequent itemsets using vertical data format 57 The Apriori and the FP-growth methods mine frequent patterns from a set of transactions in TIDitemset format, where TID is a transaction id and itemset is the set of items bought in transaction TID. This data format is known as horizontal data format. Alternatively data can also be presented in item- TID_set format, where item is an item name and TID_set is the set of transaction identifiers containing the item. This format is known as vertical data format.

55 Procedure 58 First we transform the horizontally formatted data to the vertical format by scanning the data set once. The support count of an itemset is simply the length of the TID_set of the itemset. Starting with k=1 the frequent k-itemsets can be used to construct the candidate (k+1)-itemsets. This process repeats with k incremented by 1 each time until no frequent itemsets or no candidate itemsets can be found.

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60 Advantages Advantages of this algorithm: Better than Apriori in the generation of candidate (k+1)-itemset from frequent k itemsets There is no need to scan the database to find the support (k+1) itemsets (for k>=1). This is because the TID_set of each k-itemset carries the complete information required for counting such support. The disadvantage of this algorithm consist in the TID_set being to long, taking substantial memory space as well as computation time for intersecting the long sets. 63

61 Multilevel AR It is difficult to find interesting patterns at a too primitive level high support = too few rules low support = too many rules, most uninteresting Approach: reason at suitable level of abstraction 64 A common form of background knowledge is that an attribute may be generalized or specialized according to a hierarchy of concepts Dimensions and levels can be efficiently encoded in transactions Multilevel Association Rules: rules which combine associations with hierarchy of concepts

62 Hierarchy of concepts 65 Department FoodStuff Sector Frozen Refrigerated Fresh Bakery Etc... Family Vegetable Fruit Dairy Etc... Product Banana Apple Orange Etc...

63 Multilevel AR 66 Fresh [support = 20%] Dairy [support = 6%] Fruit [support = 1%] Vegetable [support = 7%] Fresh Bakery [20%, 60%] Dairy Bread [6%, 50%] Fruit Bread [1%, 50%] is not valid

64 Support and Confidence of Multilevel Association Rules Generalizing/specializing values of attributes affects support and confidence 67 from specialized to general: support of rules increases (new rules may become valid) from general to specialized: support of rules decreases (rules may become not valid, their support falls under the threshold)

65 Mining Multilevel AR 68 Hierarchical attributes: age, salary Association Rule: (age, young) (salary, 40k) age salary young middle-aged old low medium high k 40k 50k 60k 70k 80k 100k Candidate Association Rules: (age, 18 ) (salary, 40k), (age, young) (salary, low), (age, 18 ) (salary, low)

66 Mining Multilevel AR Calculate frequent itemsets at each concept level, until no more frequent itemsets can be found For each level use Apriori A top_down, progressive deepening approach: First find high-level strong rules: fresh bakery [20%, 60%]. Then find their lower-level weaker rules: fruit bread [6%, 50%]. Variations at mining multiple-level association rules. Level-crossed association rules: fruit wheat bread 69

67 Mining Multiple-Level Association Rules 70 Items often form hierarchies Flexible support settings Items at the lower level are expected to have lower support Exploration of shared multi-level mining (Agrawal & 95, Han & 95) uniform support Level 1 min_sup = 5% Milk [support = 10%] reduced support Level 1 min_sup = 5% Level 2 min_sup = 5% 2% Milk [support = 6%] Skim Milk [support = 4%] Level 2 min_sup = 3%

68 Multi-level Association: Uniform Support vs. Reduced Support Uniform Support: the same minimum support for all levels 71 One minimum support threshold. No need to examine itemsets containing any item whose ancestors do not have minimum support. If support threshold too high miss low level associations. too low generate too many high level associations. Reduced Support: reduced minimum support at lower levels - different strategies possible.

69 Multi-level Association: Redundancy Filtering 72 Some rules may be redundant due to ancestor relationships between items. Example milk wheat bread [support = 8%, confidence = 70%] 2% milk wheat bread [support = 2%, confidence = 72%] We say the first rule is an ancestor of the second rule. A rule is redundant if its support is close to the expected value, based on the rule s ancestor.

70 Mining Multi-Dimensional Association Single-dimensional rules: 73 buys(x, milk ) buys(x, bread ) Multi-dimensional rules: 2 dimensions or predicates Inter-dimension assoc. rules (no repeated predicates) age(x, ) occupation(x, student ) buys(x, coke ) hybrid-dimension assoc. rules (repeated predicates) age(x, ) buys(x, popcorn ) buys(x, coke ) Categorical(nominal) Attributes: finite number of possible values, no ordering among values data cube approach Quantitative Attributes: numeric, implicit ordering among values discretization, clustering, and gradient approaches

71 Interestingness Measure: Correlations (Lift) play basketball eat cereal [40%, 66.7%] is misleading The overall % of students eating cereal is 75% > 66.7%. play basketball not eat cereal [20%, 33.3%] is more accurate, although with lower support and confidence Measure of dependent/correlated events: lift If value less than 1 then negatively correlated and if value greater than 1 then positively correlated If value equal to 1 then they are independent lift P( A B) P( A) P( B) Basketball Not basketball Sum (row) Cereal Not cereal Sum(col.) / / 5000 lift( B, C) 0.89 lift( B, C) / 5000*3750 / / 5000*1250 / 5000 Jaswal Slides by Shree 74

72 Constraint-based (Query-Directed) Mining Finding all the patterns in a database autonomously? unrealistic! The patterns could be too many but not focused! Data mining should be an interactive process 77 User directs what to be mined using a data mining query language (or a graphical user interface) Constraint-based mining User flexibility: provides constraints on what to be mined System optimization: explores such constraints for efficient mining constraint-based mining

73 Constraints in Data Mining 78 Knowledge type constraint: classification, association, etc. Data constraint using SQL-like queries find product pairs sold together in stores in Mumbai in Dec. 16 Dimension/level constraint in relevance to region, price, brand, customer category Rule (or pattern) constraint small sales (price < $10) triggers big sales (sum > $200) Interestingness constraint strong rules: min_support 3%, min_confidence 60%

74 Constrained Mining vs. Constraint-Based Search Constrained mining vs. constraint-based search/reasoning Both are aimed at reducing search space Finding all patterns satisfying constraints vs. finding some (or one) answer in constraint-based search in AI Constraint-pushing vs. heuristic search 79 It is an interesting research problem on how to integrate them Constrained mining vs. query processing in DBMS Database query processing requires to find all Constrained pattern mining shares a similar philosophy as pushing selections deeply in query processing