Advanced Topics in Data Mining: Association Rules

Advanced Topics in Data Mining: Association Rules

What Is Association Mining?
Association Rule Mining Finding frequent patterns, associations, correlations, or causal structures among item sets in transaction databases, relational databases, and other information repositories Applications Market basket analysis (marketing strategy: items to put on sale at reduced prices), cross-marketing, catalog design, shelf space layout design, etc Examples Rule form: Body ® Head [Support, Confidence]. buys(x, “Computer”) ® buys(x, “Software”) [2%, 60%] major(x, “CS”) ^ takes(x, “DB”) ® grade(x, “A”) [1%, 75%]

Market Basket Analysis
Typically, association rules are considered interesting if they satisfy both a minimum support threshold and a minimum confidence threshold.

Rule Measures: Support and Confidence
Let minimum support 50%, and minimum confidence 50%, we have A  C [50%, 66.6%] C  A [50%, 100%]

Support & Confidence

Association Rule: Basic Concepts
Given (1) database of transactions, (2) each transaction is a list of items (purchased by a customer in a visit) Find all rules that correlate the presence of one set of items with that of another set of items Find all the rules A  B with minimum confidence and support support, s, P(A  B) confidence, c, P(B|A)

Association Rule Mining: A Road Map
Boolean vs. quantitative associations (Based on the types of values handled in the rule set) buys(x, “SQLServer”) ^ buys(x, “DM Book”)  buys(x, “DBMiner”) [0.2%, 60%] age(x, “30..39”) ^ income(x, “42..48K”)  buys(x, “PC”) [1%, 75%] Single dimension vs. multiple dimensional associations Single level vs. multiple-level analysis (Based on the levels of abstractions involved in the rule set)

Terminologies Item Itemset 1-Itemset 2-Itemset I1, I2, I3, …
A, B, C, … Itemset {I1}, {I1, I7}, {I2, I3, I5}, … {A}, {A, G}, {B, C, E}, … 1-Itemset {I1}, {I2}, {A}, … 2-Itemset {I1, I7}, {I3, I5}, {A, G}, …

Terminologies K-Itemset Frequent K-Itemset Association Rule
If the length of the itemset is K Frequent K-Itemset If the length of the itemset is K and the itemset satisfies a minimum support threshold. Association Rule If a rule satisfies both a minimum support threshold and a minimum confidence threshold

Analysis The number of itemsets of a given cardinality tends to grow exponentially

Mining Association Rules: Apriori Principle
Min. support 50% Min. confidence 50% For rule A  C: support = support({A  C}) = 50% confidence = support({A  C})/support({A}) = 66.6% The Apriori principle: Any subset of a frequent itemset must be frequent

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

Example

Apriori Algorithm

Example of Generating Candidates
L3={abc, abd, acd, ace, bcd} Self-joining: L3*L3 abcd from abc and abd acde from acd and ace Pruning: acde is removed because ade is not in L3 C4={abcd}

Another Example 1 Database D 1 3 4 2 3 5 1 2 3 5 2 5 scan D count C1
2 5 scan D count C1 C1 count generate L1 L1 1 2 3 5 scan D count C2 C2 count generate L2 L2 13 23 25 35 C2 12 15 generate C2 scan D count C3 C3 count generate L3 L3 235 C3 generate C3

Another Example 2

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 to collect counts for the candidate itemsets The bottleneck of Apriori: 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

Demo-IBM Intelligent Minner

Demo Database

Methods to Improve Apriori’s Efficiency
Hash-based itemset counting: A k-itemset whose corresponding hashing bucket count is below the threshold cannot be frequent Transaction reduction: A transaction that does not contain any frequent k-itemset is useless in subsequent scans Partitioning: Any itemset that is potentially frequent in DB must be frequent in at least one of the partitions of DB Sampling: mining on a subset of given data, lower support threshold + a method to determine the completeness

Partitioning

Hash-Based Itemset Counting
= * +

Compare Apriori & DHP (Direct Hash & Pruning)

Compare Apriori & DHP DHP

DHP: Database Trimming

DHP (Direct Hash & Pruning)
A database has four transactions Let min_sup = 50%

Example: Apriori

Example: DHP

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!

Construct FP-tree from a Transaction DB

Construction Steps Scan DB once, find frequent 1-itemset (single item pattern) Order frequent items in frequency descending order Sorting DB according to the frequency descending order Scan DB again, construct FP-tree

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) Compression ratio could be over 100

Frequent Pattern Growth
Order frequent items in frequency descending order For I5 {I1, I5}, {I2, I5}, {I2, I1, I5} For I4 {I2, I4} For I3 {I1, I3}, {I2, I3}, {I2, I1, I3} For I1 {I2, I1}

Frequent Pattern Growth
For I5 {I1, I5}, {I2, I5}, {I2, I1, I5} For I4 {I2, I4} For I3 {I1, I3}, {I2, I3}, {I2, I1, I3} For I1 {I2, I1} Sub DB Trimming Databases Sub DB Sub DB FP-tree Sub DB Conditional FP-tree from Conditional Pattern-Base

Conditional FP-tree Conditional FP-tree from Conditional Pattern-Base
for I3

Mining Results Using FP-tree
For I5 (不產生NULL) Conditional Pattern Base {(I2I1:1), (I2I1I3:1)} Conditional FP-tree Generate Frequent Itemsets I2:2 Rule: I2I5:2 I1:2 Rule: I1I5:2 I2I1:2 Rule: I2I1I5:2 ◎ NULL: 2 Item ID Support count Node Link I2 2 I1 ◎ “I2”: 2 ◎ “I1”: 2

For I4 Conditional Pattern Base {(I2I1:1), (I2:1)} Conditional FP-tree Generate Frequent Itemsets I2:2 Rule: I2I4:2 ◎ NULL: 2 Item ID Support count Node Link I2 2 ◎ “I2”: 2

For I3 Conditional Pattern Base {(I2I1:2), (I2:2), (I1:2)} Conditional FP-tree ◎ NULL: 4 Item ID Support count Node Link I2 4 I1 ◎ “I2”: 4 ◎ “I1”: 2 ◎ “I1”: 2

For I1/I3 Conditional Pattern Base {(NULL:2), (I2:2)} Conditional FP-tree Generate Frequent Itemsets Null:4 Rule: I1I3:4 I2:2 Rule: I2I1I3:2 ◎ NULL: 4 Item ID Support count Node Link I2 2 ◎ “I2”: 2

For I2/I3 Conditional Pattern Base {(NULL:4)} Conditional FP-tree Generate Frequent Itemsets Null Rule: I2I3:4 ◎ NULL: 4

For I1 Conditional Pattern Base {(NULL:2), (I2:4)} Conditional FP-tree Generate Frequent Itemsets I2:4 Rule: I2I1:4 ◎ NULL: 6 Item ID Support count Node Link I2 4 ◎ “I2”: 4

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)

Major Steps to Mine FP-tree
Construct conditional pattern base for each item 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

Virtual Items in Association Mining
Different region exhibit different selling patterns. Thus, including as virtual item the information on the location or the type of stores (existing or new) where the purchase was made will enable the comparisons between locations or types within a single chain Virtual item may include information on whether the purchase was made with cash, a credit card or check. The inclusion of such virtual item allows to analyze the association between the payment method and items purchased. Virtual item may include information on the day of the week or the time of the day the transaction occurred. The inclusion of such virtual item allows to analyze the association between the transaction time and items purchased

Virtual Items: An Example

Dissociation Rules A dissociation rule is similar to an association rule except that it can have “not item-name” in the condition or the result of the rule A and not B ® C A and D ® not E Dissociation rules can be generated by a simple adaptation of the association rule analysis

Discussions The size of a typical transaction grows because it now includes inverted items The total number of items used in the analysis doubles Since the amount of computation grows exponentially with the number of items, doubling the number of items seriously degrades performance The frequency of the inverted items tends to be much larger than the frequency of the original items. So, it tends to produce rules in which all items are inverted. These rules are less likely to be actionable. not A and not B ® not C It is useful to invert only the most frequent items in the set used for analysis. It is also useful to invert some items whose inverses are of interest.

Interestingness Measurements
Subjective Measures A rule (pattern) is interesting if it is unexpected (surprising to the user) actionable (the user can do something with it) Objective Measures Two popular measurements: Support confidence

Criticism to Support and Confidence
Example 1 Among 5000 students 3000 play basketball 3750 eat cereal 2000 both play basket ball and eat cereal play basketball  eat cereal [40%, 66.7%] is misleading because the overall percentage of students eating cereal is 75% which is higher than 66.7%.

Example 2 X and Y: positively correlated, X and Z, negatively related support and confidence of X=>Z dominates We need a measure of dependent or correlated events

Improvement (Correlation) Taking both P(A) and P(B) in consideration P(A^B)=P(B)*P(A), if A and B are independent events A and B negatively correlated, if the value is less than 1 otherwise A and B positively correlated When improvement is less than 1, negating the result produces a better rule X => NOT Z

Multiple-Level Association Rules
Items often form hierarchy Items at the lower level are expected to have lower support Rules regarding itemsets at appropriate levels could be quite useful Transaction database can be encoded based on dimensions and levels We can explore multi-level mining

Transaction Database

Concept Hierarchy

Mining Multi-Level Associations
A top_down, progressive deepening approach First find high-level strong rules milk ® bread [20%, 60%] Then find their lower-level “weaker” rules 2% milk ® wheat bread [6%, 50%] Variations at mining multiple-level association rules. Cross-level association rules (Generalized Asso. Rules) 2% milk ® Wonder wheat bread Association rules with multiple, alternative hierarchies 2% milk ® Wonder bread

Multi-level Association: Uniform Support vs. Reduced Support
Uniform Support: the same minimum support for all levels One minimum support threshold is needed. Lower level items do not occur as frequently. 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 There are 4 search strategies: Level-by-level independent Level-cross filtering by k-itemset Level-cross filtering by single item Controlled level-cross filtering by single item

Uniform Support Optimization Technique
 Optimization Technique The search avoids examining itemsets containing any item whose ancestors do not have minimum support.

Uniform Support : An Example
min_sup=50% L1: 2,3,4 L2: 23,24,34 L3: 234 min_sup=60% L1: 2,3,6,8,9 L2: 23,68,69,89 L3: 689 min_sup=50% L1: 9 min_sup=60%

All Cheese Crab Milk 3 1 2 1 2 6 3 5 King’s Crab Sunset Milk Dairyland Milk Dairyland Cheese Best Cheese Bread Apple Pie 4 5 6 4 9 7 10 8 Best Bread Wonder Bread Goldenfarm Apple Westcoast Bread Tasty Pie

min_sup=50% Apriori/DHP FP Growth (1) (2) min_sup=50% L1: 2,3,6,8,9 Scan DB C2: 23,36,29,69,28,68,39,89 C3: 239,369,289,689 (3) min_sup=50% Scan DB L2: 23,68,69,89 L3: 689

Reduced Support Each level of abstraction has its own minimum support threshold.

Search Strategies for Reduced Support
There are 4 search strategies: Level-by-level independent Full-Breadth Search No pruning No background knowledge of frequent itemsets is used for pruning Level-cross filtering by single item An item at the ith level is examined if and only if its parent node at the (i-1)th level is frequent. Level-cross filtering by k-itemset An k-itemset at the ith level is examined if and only if its corresponding parent k-itemset at the (i-1)th level is frequent. Controlled level-cross filtering by single item

Level-Cross Filtering by Single Item

Reduced Support : An Example
min_sup=60% Apriori/DHP FP Growth (1) (1) min_sup=50% L1: 2,3,6,9 Scan DB (2) min_sup=50% L2: 23,69 Apriori/DHP FP Growth (3)

Level-Cross Filtering by K-Itemset
 

min_sup=60% Apriori/DHP FP Growth (1) (2) min_sup=50% L1: 2,3,6,9 Scan DB C2: 23,36,29,69,39 C3: 239,369 (3) min_sup=50% Scan DB L2: 23,69

Reduced Support Level-by-level independent
It is very relaxed in that it may lead to examining numerous infrequent items at low levels, finding associations between items of little importance. Level-cross filtering by k-itemset It allows the mining system to examine only the children of frequent k-itemsets. This restriction is very strong in that there usually are not many k-itemsets. Many valuable patterns may be filtered out. Level-cross filtering by single item A compromise between the above two approaches This method may miss associations between low level items that are frequent based on a reduced minimum support, but whose ancestors do not satisfy minimum support.

Controlled Level-Cross Filtering by Single Item

min_sup=60% Apriori/DHP FP Growth level_passage_sup=50% L1: 2,3,4,5,6 (1) (1) min_sup=50% L1: 2,3,6,8,9 Scan DB (2) L2: 23,68,69,89 L3: 689 min_sup=50% Apriori/DHP FP Growth (3)

Multi-Dimensional Association
Single-Dimensional (Intra-Dimension) Rules: Single Dimension (Predicate) with Multiple Occurrences. buys(X, “milk”)  buys(X, “bread”) Multi-Dimensional Rules:  2 Dimensions Inter-dimension association rules (no repeated predicates) age(X,”19-25”)  occupation(X,“student”)  buys(X,“coke”) hybrid-dimension association rules (repeated predicates) age(X,”19-25”)  buys(X, “popcorn”)  buys(X, “coke”)

Summary Association rule mining Some Important Issues
Probably the most significant contribution from the database community in KDD A large number of papers have been published Some Important Issues Generalized Association Rules Multiple-Level Association Rules Association Analysis in Other Types of Data Spatial Data, Multimedia Data, Time Series Data, etc. Weighted Association Rules Quantitative Association Rules

Weighted Association Rules
Why Weighted Association Analysis? In previous work, all items in a transactional database are treated uniformly Items are given weights to reflect their importance to the user The weights may correspond to special promotions on some products, or the profitability of different items Some products may be under promotion and hence are more interesting, or some products are more profitable and hence rules concerning them are of greater values

Weighted Association Rules
A simple attempt to solve this problem is to eliminate the items with small weights However, a rule for a heavy weighted item may also consist of low weighted items Is Apriori algorithm feasible? Apriori algorithm depends on the downward closure property which governs that subsets of a frequent itemset are also frequent However, it is not true for the weighted case

Weighted Association Rules: An Example
Total Benefits: 500 Benefits for the First Transaction: ( )=160 Benefits for the Second Transaction: ( )=130 Benefits for the Third Transaction: ( )= 110 Benefits for the Fourth Transaction: ( )=100 Suppose Weighted_Min_Sup = 40% Minimum Benefits = 500 * 40% = 200

An Example Minimum Benefits = 500 * 40% = 200 Itemset {3,5,6,7}
Support Count (Frequency): 3 70 * 3 = 210 >= 200 {3,5,6,7}is a Frequent Itemset Itemset {3,5,6} Benefits: 60 Support Count ( Frequency): 3 60 * 3 = 180 < 200 {3,5,6} is not a Frequent Itemset Apriori Principle can not be applied!

K-Support Bound If Y is a frequent q-itemset Example
Support_Count(Y)  (Weighted_Min_Sup * Total_Benefits) / Benefits(Y) Example {3,5,6,7} is a Frequent 4-Itemset Support_Count({3,5,6,7}) = 3  (40% * 500) / Benefits({3,5,6,7}) = (40% * 500) / 70 = 2.857 If X is a frequent k-itemset containing q-itemset Y Minimum_Support_Count(X)  (Weighted_Min_Sup * Total_Benefits) / (Benefits(Y) + (k-q) Maximum Remaining Weights) X is a Frequent 5-Itemset containing {3,5,6,7} Minimum_Support_Count(X)  (40% * 500) / ( ) = 1.81 K-Support Bound

K-Support Bound Itemset {1,2}
Benefits: 70 Support_Count({1,2}) = 1 < (40% * 500) / Benefits({1,2}) = (40% * 500) / 70 = 2.857 {1,2} is not a Frequent Itemset If X is a frequent 3-itemset containing {1,2} Minimum_Support_Count(X)  (40% * 500) / ( ) = 2 But, Maximum_Support_Count(X) = 1 No frequent 3-itemsets containing {1,2} If X is a frequent 4-itemset containing {1,2} Minimum_Support_Count(X)  (40% * 500) / ( ) = 1.667 No frequent 4-itemsets containing {1,2} Similarly, no frequent 5, 6, 7-itemsets containing {1,2} The algorithm is designed based on this k-support bound

MINWAL Algorithm

Step by Step Input: Product & Transactional Databases
Weighted_Min_Sup = 50% Total Profits: 1380

Step 2, 7 Search(D) Counting(D, w)
This subroutine finds out the maximum transaction size in that transactional database D Size = 4 in this case Counting(D, w) This subroutine cumulates the support counts of the 1-itemsets The k-support bounds of each 1-itemset will be calculated, and the 1-itemsets with support counts greater than any of the k-support bounds will be kept in C1

K-Support Bound  (50% * 1380) / (10 + 90) = 6.9
Step 7  K-Support Bound  (50% * 1380) / ( ) = 6.9

Step 11 Join(Ck-1) The Join step generates Ck from Ck-1 as Apriori Algorithm If we have {1, 2, 3}, {1, 2, 4} in Ck-1 {1, 2, 3, 4} will be generated in Ck In this case, C1 = {1 (4), 2 (5), 4 (6), 5 (7)} C2 = Join(C1) = {12, 14, 15, 24, 25, 45} Support_Count

Estimated_Support_Count
Step 12 Prune(Ck) The itemset will be pruned in either of the following cases A subset of the candidate itemset in Ck does not exist in Ck-1 Estimate an upper bound on the support count (SC) of the joined itemset X, which is the minimum support count among the k different (k-1)-subsets of X in Ck-1. If the estimated upper bound on the support count shows that the itemset X cannot be a subset of any large itemset in the coming passes (from the calculation of k-support bounds for all itemsets), that itemset will be pruned In this case, C2 = Prune(C2) = {12 (4), 14 (4), 15 (4), 24 (5), 25 (5), 45 (6)} Using K-Support-Bound Estimated_Support_Count

Step 12 Prune(Ck) Using K-Support-Bound (No one is pruned)

Step 13 Checking(Ck, D) Scan DB & Generate Lk   C2 L2

Step 11, 12 Join(C2) Prune(C3) C2 = {15 (4), 24 (5), 25 (5), 45 (6)}
C3 = Join(C2) = {245} Prune(C3) C3 = Prune(C3) = {245 (5)} Using K-Support-Bound (No one is pruned)

Step 13 Checking(C3, D): Scan DB C3 = L3 = {245}
Finally, L = {45, 245}

Step 15 Generate Rules for L = {45, 245} 4  5 (confidence = 100%)
Min_Conf=90%

Generalized Association Rules

Quantitative Association Rules
Let min_sup = 50%, we have {A, B} 60% {B, D} 70% {A, B, D} 50% {A(1..2), B(3)} % {A(1..2), B(3..5)} % {A(1..2), B(1..5)} % {B(1..5), D(1..2)} % {B(3..5), D(1..3)} % {B(1..5), D(1..3)} 70% {B(1..3), D(1..2)} 50% {B(3..5), D(1..2)} 50% {A(1..2), B(3..5), D(1..2)} 50% 

Advanced Topics in Data Mining: Association Rules

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