Data Mining Association Rules: Advanced Concepts and Algorithms Lecture Notes for Chapter 7 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Continuous and Categorical Attributes Example of Association Rule: {Number of Pages  [5,10)  (Browser=Mozilla)}  {Buy = No} How to apply association analysis formulation to non- asymmetric binary variables?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Handling Categorical Attributes l Transform categorical attribute into asymmetric binary variables l Introduce a new “item” for each distinct attribute- value pair –Example: replace Browser Type attribute with  Browser Type = Internet Explorer  Browser Type = Mozilla  Browser Type = Netscape

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Handling Categorical Attributes l Potential Issues: l What if attribute has many possible values  Example: attribute country has more than 200 possible values  Many of the attribute values may have very low support –Potential solution: Aggregate the low-support attribute values l What if distribution of attribute values is highly skewed  Example: 95% of the visitors have Buy = No  Most of the items will be associated with (Buy=No) item –Potential solution: drop the highly frequent items

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Handling Continuous Attributes l Different kinds of rules: –Age  [21,35)  Salary  [70k,120k)  Buy –Salary  [70k,120k)  Buy  Age:  =28,  =4 l Different methods: –Discretization-based –Statistics-based –Non-discretization based  minApriori

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Handling Continuous Attributes l Use discretization l Unsupervised: –Equal-width binning –Equal-depth binning –Clustering l Supervised: Classv1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 Anomalous Normal bin 1 bin 3 bin 2 Attribute values, v

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Discretization Issues l Size of the discretized intervals affect support & confidence –If intervals too small  may not have enough support –If intervals too large  may not have enough confidence l Potential solution: use all possible intervals {Refund = No, (Income = $51,250)}  {Cheat = No} {Refund = No, (60K  Income  80K)}  {Cheat = No} {Refund = No, (0K  Income  1B)}  {Cheat = No}

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Discretization Issues l Execution time –If intervals contain n values, there are on average O(n 2 ) possible ranges l Too many rules {Refund = No, (Income = $51,250)}  {Cheat = No} {Refund = No, (51K  Income  52K)}  {Cheat = No} {Refund = No, (50K  Income  60K)}  {Cheat = No}

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Approach by Srikant & Agrawal l Preprocess the data –Discretize attribute using equi-depth partitioning  Use partial completeness measure to determine number of partitions  Merge adjacent intervals as long as support is less than max-support l Apply existing association rule mining algorithms l Determine interesting rules in the output

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Approach by Srikant & Agrawal l Discretization will lose information –Use partial completeness measure to determine how much information is lost C: frequent itemsets obtained by considering all ranges of attribute values P: frequent itemsets obtained by considering all ranges over the partitions P is K-complete w.r.t C if P  C,and  X  C,  X’  P such that: 1. X’ is a generalization of X and support (X’)  K  support(X) (K  1) 2.  Y  X,  Y’  X’ such that support (Y’)  K  support(Y) Given K (partial completeness level), can determine number of intervals (N) X Approximated X

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Interestingness Measure l Given an itemset: Z = {z 1, z 2, …, z k } and its generalization Z’ = {z 1 ’, z 2 ’, …, z k ’} P(Z): support of Z E Z’ (Z): expected support of Z based on Z’ –Z is R-interesting w.r.t. Z’ if P(Z)  R  E Z’ (Z) {Refund = No, (Income = $51,250)}  {Cheat = No} {Refund = No, (51K  Income  52K)}  {Cheat = No} {Refund = No, (50K  Income  60K)}  {Cheat = No}

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Interestingness Measure l For S: X  Y, and its generalization S’: X’  Y’ P(Y|X): confidence of X  Y P(Y’|X’): confidence of X’  Y’ E S’ (Y|X): expected support of Z based on Z’ l Rule S is R-interesting w.r.t its ancestor rule S’ if –Support, P(S)  R  E S’ (S) or –Confidence, P(Y|X)  R  E S’ (Y|X)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistics-based Methods l Example: Browser=Mozilla  Buy=Yes  Age:  =23 l Rule consequent consists of a continuous variable, characterized by their statistics –mean, median, standard deviation, etc. l Approach: –Withhold the target variable from the rest of the data –Apply existing frequent itemset generation on the rest of the data –For each frequent itemset, compute the descriptive statistics for the corresponding target variable  Frequent itemset becomes a rule by introducing the target variable as rule consequent –Apply statistical test to determine interestingness of the rule

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistics-based Methods l How to determine whether an association rule interesting? l Compare the statistics for segment of population covered by the rule vs segment of population not covered by the rule: A  B:  versus A  B:  ’ l Statistical hypothesis testing:  Null hypothesis: H0:  ’ =  +   Alternative hypothesis: H1:  ’ >  +   Z has zero mean and variance 1 under null hypothesis  n1 are supporting records, n2 are non-supporting records  s1 is std for supporting records, s2 is std for non-supporting records

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Statistics-based Methods l Example: r: Browser=Mozilla  Buy=Yes  Age:  =23 l Rule is interesting if difference between  and  ’ is greater than 5 years (i.e.,  = 5) l For r, suppose n1 = 50, s1 = 3.5 l For r’ (complement): n2 = 250, s2 = 6.5 l For 1-sided test at 95% confidence level, critical Z-value for rejecting null hypothesis is l Since Z is greater than 1.64, r is an interesting rule ’ ’

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Min-Apriori (Han et al) Example: W1 and W2 tends to appear together in the same document Document-term matrix:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Min-Apriori l Data contains only continuous attributes of the same “type” –e.g., frequency of words in a document l Notice that the table values are not actually continuous –Typically the values are normalized; range from 0 to 1 –E.g. the normalized value for W1 in D1 is 2/5 = 0.4

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Min-Apriori l Discretization does not apply as users want association among words not ranges of words l Potential solution: –Convert into 0/1 matrix (using a threshold) and then apply existing algorithms  Have to find a good threshold  lose word frequency information

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Min-Apriori l How to determine the support of a word? –If we simply sum up its frequency, support count will be greater than total number of documents!  Normalize the word vectors – e.g., using L 1 norm  Each word has a support equals to 1.0 Normalize

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Min-Apriori l New definition of support: Example: Sup(W1,W2,W3) = = 0.17

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Anti-monotone property of Support Example: Sup(W1) = = 1 Sup(W1, W2) = = 0.9 Sup(W1, W2, W3) = = 0.17

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Concept Hierarchies How do concept hierarchies relate to the preceding material????

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multi-level Association Rules l Why should we incorporate concept hierarchy? l Rules at lower levels may not have enough support to appear in any frequent itemsets l Rules at lower levels of the hierarchy are overly specific  e.g., skim milk  white bread, 2% milk  wheat bread, skim milk  wheat bread, etc. are indicative of association between milk and bread l In other words, the same reason that we discretized continuous variables earlier

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multi-level Association Rules l How do support and confidence vary as we traverse the concept hierarchy? If X is the parent item for both X1 and X2, then  (X) ≥  (X1) +  (X2) l If  (X1  Y1) ≥ minsup, and X is parent of X1, Y is parent of Y1 then  (X  Y1) ≥ minsup,  (X1  Y) ≥ minsup  (X  Y) ≥ minsup l If conf(X1  Y1) ≥ minconf, thenconf(X1  Y) ≥ minconf

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multi-level Association Rules l Approach 1: –Extend current association rule formulation by augmenting each transaction with higher level items Original Transaction: {skim milk, wheat bread} Augmented Transaction: {skim milk, wheat bread, milk, bread, food} l Issues: –Items that reside at higher levels have much higher support counts  if support threshold is low, too many frequent patterns involving items from the higher levels –Increased dimensionality of the data Low for what items?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multi-level Association Rules l Approach 2: –Generate frequent patterns at highest level first –Then, generate frequent patterns at the next highest level, and so on l Issues: –I/O requirements will increase dramatically because we need to perform more passes over the data –May miss some potentially interesting cross-level association patterns How many more passes?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Sequence Data ObjectTimestampEvents A102, 3, 5 A206, 1 A231 B114, 5, 6 B172 B217, 8, 1, 2 B281, 6 C141, 8, 7 Sequence Database:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Examples of Sequence Data Sequence Database SequenceElement (Transaction) Event (Item) CustomerPurchase history of a given customer A set of items bought by a customer at time t Books, diary products, CDs, etc Web DataBrowsing activity of a particular Web visitor A collection of files viewed by a Web visitor after a single mouse click Home page, index page, contact info, etc Event dataHistory of events generated by a given sensor Events triggered by a sensor at time t Types of alarms generated by sensors Genome sequences DNA sequence of a particular species An element of the DNA sequence Bases A,T,G,C Sequence E1 E2 E1 E3 E2 E3 E4 E2 Element (Transaction) Event (Item)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Formal Definition of a Sequence l A sequence is an ordered list of elements (transactions) s = –Each element contains a collection of events (items) e i = {i 1, i 2, …, i k } –Each element is attributed to a specific time or location l Length of a sequence, |s|, is given by the number of elements of the sequence l A k-sequence is a sequence that contains k events (items)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Examples of Sequence l Web sequence: l Sequence of initiating events causing the nuclear accident at 3-mile Island: ( l Sequence of books checked out at a library:

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Formal Definition of a Subsequence l A sequence is contained in another sequence (m ≥ n) if there exist integers i 1 < i 2 < … < i n such that a 1  b i1, a 2  b i1, …, a n  b in l The support of a subsequence w is defined as the fraction of data sequences that contain w l A sequential pattern is a frequent subsequence (i.e., a subsequence whose support is ≥ minsup) Data sequenceSubsequenceContain? Yes No Yes

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Sequential Pattern Mining: Definition l Given: –a database of sequences –a user-specified minimum support threshold, minsup l Task: –Find all subsequences with support ≥ minsup

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Sequential Pattern Mining: Challenge l Given a sequence: –Examples of subsequences:,,, etc. l How many k-subsequences can be extracted from a given n-sequence? n = 9 k=4: Y _ _ Y Y _ _ _ Y

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Sequential Pattern Mining: Example Minsup = 50% Examples of Frequent Subsequences: s=60% s=80% s=60% How was the support of determined to be 60%? What constitutes a sequence in this example?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Extracting Sequential Patterns Given n events: i 1, i 2, i 3, …, i n l Candidate 1-subsequences:,,, …, l Candidate 2-subsequences:,, …,,, …, l Candidate 3-subsequences:,, …,,, …,,, …,,, … Huh?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Generalized Sequential Pattern (GSP) l Step 1: –Make the first pass over the sequence database D to yield all the 1- element frequent sequences l Step 2: Repeat until no new frequent sequences are found –Candidate Generation:  Merge pairs of frequent subsequences found in the (k-1)th pass to generate candidate sequences that contain k items –Candidate Pruning:  Prune candidate k-sequences that contain infrequent (k-1)-subsequences –Support Counting:  Make a new pass over the sequence database D to find the support for these candidate sequences –Candidate Elimination:  Eliminate candidate k-sequences whose actual support is less than minsup

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Candidate Generation l Base case (k=2): –Merging two frequent 1-sequences and will produce two candidate 2-sequences: and –Ok?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Candidate Generation General case (k>2): l A frequent (k-1)-sequence w 1 is merged with another frequent (k-1)-sequence w 2 to produce a candidate k-sequence if the subsequence obtained by removing the first event in w 1 is the same as the subsequence obtained by removing the last event in w 2 l The resulting candidate after merging is given by the sequence w 1 extended with the last event of w 2. –If the last two events in w 2 belong to the same element, then the last event in w 2 is part of the last element in the merged sequence. –Otherwise, the last event in w 2 becomes a separate element appended to the end of w 1 in the merged sequence.

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Candidate Generation Examples l Merging the sequences w 1 = and w 2 = will produce the candidate sequence because the last two events in w 2 (4 and 5) belong to the same element l Merging the sequences w 1 = and w 2 = will produce the candidate sequence because the last two events in w 2 (4 and 5) do not belong to the same element l We do not have to merge the sequences w 1 = and w 2 = to produce the candidate because if the latter is a viable candidate, then it can be obtained by merging w 1 with

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ GSP Example How did we prune down to 1 candidate? A k-sequence is infrequent if any of its k-1 sequences are infrequent. How can we tell if a k-1 sequence is infrequent? For each pruned candidate, propose a k-1 infrequent sequence.

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Timing Constraints (I) {A B} {C} {D E} <= m s <= x g >n g x g : max-gap n g : min-gap m s : maximum span Data sequenceSubsequenceContain? Yes No Yes No x g = 2, n g = 0, m s = 4

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Mining Sequential Patterns with Timing Constraints l Approach 1: –Mine sequential patterns without timing constraints –Postprocess the discovered patterns l Approach 2: –Modify GSP to directly prune candidates that violate timing constraints –Question:  Does Apriori principle still hold? l Possibly

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Apriori Principle for Sequence Data Suppose: x g = 1 (max-gap) n g = 0 (min-gap) m s = 5 (maximum span) minsup = 60% support = 40% but support = 60% Problem exists because of max-gap constraint No such problem if max-gap is infinite Example where a priori principle is violated How???

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ How to avoid violating a priori principle? New Concept: Contiguous Subsequence

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Contiguous Subsequences l s is a contiguous subsequence of w = … if any of the following conditions hold: 1.s is obtained from w by deleting an item from either e 1 or e k 2.s is obtained from w by deleting an item from any element e i that contains more than 2 items 3.s is a contiguous subsequence of s’ and s’ is a contiguous subsequence of w (recursive definition) l Examples: s = (Details next slide) –is a contiguous subsequence of,, and –is not a contiguous subsequence of and

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Contiguous Subsequences l Examples: s = –is a contiguous subsequence of,, and –is not a contiguous subsequence of and,, and Can’t delete {3} in either sequence

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Modified Candidate Pruning Step l Without maxgap constraint: –A candidate k-sequence is pruned if at least one of its (k-1)-subsequences is infrequent (violates a priori) l With maxgap constraint: –A candidate k-sequence is pruned if at least one of its contiguous (k-1)-subsequences is infrequent (doesn’t violate a priori)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Timing Constraints (II) {A B} {C} {D E} <= m s <= x g >n g <= ws x g : max-gap n g : min-gap ws: window size m s : maximum span Data sequenceSubsequenceContain? No (Why?) Yes Yes x g = 2, n g = 0, ws = 1, m s = 5 Min-gap requirement is violated

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Modified Support Counting Step l Given a candidate pattern: –Any data sequences that contain, ( where time({c}) – time({a}) ≤ ws) (where time({a}) – time({c}) ≤ ws) will contribute to the support count of candidate pattern

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Other Formulation l In some domains, we may have only one very long time series –Example:  monitoring network traffic events for attacks  monitoring telecommunication alarm signals l Goal is to find frequent sequences of events in the time series –This problem is also known as frequent episode mining E1 E2 E1 E2 E1 E2 E3 E4 E3 E4 E1 E2 E2 E4 E3 E5 E2 E3 E5 E1 E2 E3 E1 Pattern:

General Support Counting Schemes Assume: x g = 2 (max-gap) n g = 0 (min-gap) ws = 0 (window size) m s = 2 (maximum span) With overlap allowed With no overlap allowed

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Frequent Subgraph Mining l Extend association rule mining to finding frequent subgraphs l Useful for –Web Mining How? –Computational chemistry. How? –Bioinformatics. How? –Spatial data sets. How? l What is the complexity of subgraph finding?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Frequent Subgraph Mining l Examples of graphs where subgraph mining could be useful

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Graph Definitions

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Representing Transactions as Graphs l Each transaction is a clique of items

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Representing Graphs as Transactions

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Challenges l Node may contain duplicate labels –How is this possible? l Support and confidence –How to define them? l Additional constraints imposed by pattern structure –Support and confidence are not the only constraints –Assumption: frequent subgraphs must be connected l Apriori-like approach: –Use frequent k-subgraphs to generate frequent (k+1) subgraphs  What is k?

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Challenges… l Support: –number of graphs that contain a particular subgraph l Apriori principle still holds l Level-wise (Apriori-like) approach: –Vertex growing:  k is the number of vertices –Edge growing:  k is the number of edges

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Vertex Growing

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Edge Growing

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Apriori-like Algorithm l Find frequent 1-subgraphs l Repeat In practice, it is not as easy. There are many other issues Candidate generation Use frequent (k-1)-subgraphs to generate candidate k-subgraph Candidate pruning Prune candidate subgraphs that contain infrequent (k-1)-subgraphs Support counting Count the support of each remaining candidate Eliminate candidate k-subgraphs that are infrequent

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Example: Dataset

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Example

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Candidate Generation l In Apriori: –Merging two frequent k-itemsets will produce a candidate (k+1)-itemset l In frequent subgraph mining (vertex/edge growing) –Merging two frequent k-subgraphs may produce more than one candidate (k+1)-subgraph

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multiplicity of Candidates (Vertex Growing)

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multiplicity of Candidates (Edge growing) l Case 1: identical vertex labels

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multiplicity of Candidates (Edge growing) l Case 2: Core contains identical labels Core: The (k-1) subgraph that is common between the joint graphs

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Multiplicity of Candidates (Edge growing) l Case 3: Core multiplicity

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Adjacency Matrix Representation The same graph can be represented in many ways

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Graph Isomorphism l A graph is isomorphic if it is topologically equivalent to another graph

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Graph Isomorphism l Test for graph isomorphism is needed: –During candidate generation step, to determine whether a candidate has been generated –During candidate pruning step, to check whether its (k-1)-subgraphs are frequent –During candidate counting, to check whether a candidate is contained within another graph

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Graph Isomorphism l Use canonical labeling to handle isomorphism –Map each graph into an ordered string representation (known as its code) such that two isomorphic graphs will be mapped to the same canonical encoding –Example:  Lexicographically largest adjacency matrix String: Canonical: