Association Rule Mining ARM http://www.cs.ndsu.nodak.edu/~rahal/765/ lectures/ Lecture Outline

Association Rule Mining ARM http://www.cs.ndsu.nodak.edu/~rahal/765/ lectures/ Lecture Outline

Association Rule Mining ARM http://www.cs.ndsu.nodak.edu/~rahal/765/ lectures/ Lecture Outline Data Mining and Knowledge Discovery Market Basket Research Models Association Rule Mining Apriori Rule Generation Methods To Improve Aprioris Efficiency Vertical Data Representation What is Data Mining Data mining is the exploration and analysis of large quantities of data in order to discover valid, novel, potentially useful,

and ultimately understandable patterns and knowledge in data. Valid: The patterns hold in general. Novel: We did not know the pattern beforehand. Fargo is in Minnesota ! (live in Fargo) (live in ND) Useful: We can devise actions from the patterns (actionable) Understandable: We can interpret and comprehend the patterns. What Motivated Data Mining? As an evolution in the path of IT

1-Data Collection and Database Creation Primitive File Processing 1960s and earlier 2-Database Management Systems: Hierarchical/Network/Relational database system ERDs SQL Recovery and concurrency control in DBMSs OLTP 1970s-early 1980s 3.1-Advanced Database Systems

Object-oriented/object-relational databases Application-oriented databases Spatial, multimedia, scientific, etc Mid-1980s-present 3.2-Web-based Database Systems XML-based databases systems Web analysis and mining Semantic Web (the whole web as a single XML database) Mid-1990s-present 3.3-Data Warehousing and Data Mining

Multi-dimensional Data warehouse and OLAP technology Data Mining and Knowledge Discovery tools to assist people in their decisionmaking processes Late 1980s-present Why Use Data Mining Today? Market Competition Pressure! The secret of success is to know something that nobody else knows. Aristotle Wal-Mart VS K-Mart Right products, right place, right time, and right

quantities Personalization, CRM Security, homeland defense Analysis of important application data Bioinformatics Stock market data Human analysis skills are inadequate: Volume and dimensionality of the data High data growth rate Storage Computational power Off-the-shelf software Other factors Where Could All Of This Data Be Coming From?

Supermarket scanners Preferred customer cards Sunmarts MoreCards Credit card transactions Call center records ATM machines Demographic data Sensor networks Cameras Web server logs Customer web site trails Biological data (e.g. MicroArray Experiments for expression levels) Image data

Types Of Data/Information Repositories For Data Mining By definition, data mining should be applicable to any kind of information repository Flat files Relational databases data warehouses transactional databases Advanced database systems object-oriented Object-relational Application-oriented databases Multimedia

Text Image Video Audio Heterogeneous databases Appear as centralized Independent components managing different parts of the data How Could We Describe Data Numerical : Domain is ordered and can be represented on the continuous real line (e.g. age, income)

Continuous? Nominal or categorical : Domain is a finite set without any natural ordering (e.g. occupation, marital status, race) Ordinal : Domain is finite and ordered, (e.g.: grade scale, months in a year) The Knowledge Discovery Process Broader than Data Mining Steps: Identify the problem Data mining Action Evaluation and measurement Deployment and integration into reallife processes and/or applications The Data Mining Step in More Detail

Cleaning and integration of various data sources Remove noise and outliers Missing Values (e.g. null values) Noisy data (errors) Inconsistent Data (integration) Selection and transformation of relevant data into appropriate forms Focus on fields of interest Education on salary

Create common units FirstName and F_Name Height in cm and inches Generate new fields Discovery of interesting patterns from the data Pattern evaluation to identify the interesting patterns based on some predefined measures Knowledge presentation to communicate the mined knowledge and information to the user mostly through visualization techniques to provide a better view This process can be repeated as needed Data mining systems are expected to handle large amounts of data Analysis of small datasets is sometimes called

machine learning SDA Statistical data analysis. In other words, data mining must be scalable to large data sets Scalability and efficiency Data Mining Knowledge Patterns Preprocessed Data Target Data Pattern evaluation Knowledge Discovery Original Data Selection and transformation Cleaning and integration presentatio n

Data Mining Tasks Characterization the process of summarizing the general characteristics and features of a specific class of data (usually referred to as the target class) Characterizing the items in a store whose sales have decreased by 50% over a certain period of time. There maybe some common characteristics to all those items which we would like to uncover. Produced by a no-longer trusted producer Discrimination

Discrimination is very similar to characterization in that it reveals the characteristics of a target class in comparison to those characteristics pertaining to one or more other classes. The target and contrasting classes are specified by user and their data is retrieved from the database before the discrimination process starts. As an example, a user might want to discriminate between the characteristics of the items in a store whose sales have increased by 10% over a certain period of time this year sales have increased by 10% over the same period of time last year. Association Rule Mining The process of discovering association rules among attribute values that exist in a given set of data. Market basket research (MBR) where users are usually interested in mining associations

between items in a store by using daily transactions. An example of a rule might be diapersbeer meaning that customers buying diapers are very likely to buy beer. This will give us a good pointer to place diapers next to beer so as to increase sales sometimes people wonder about the strange placement of products in large stores Maternity to infant Classification The process of using a set of training data with known class labels to come up with a model (or function) that predicts the unknown class label of new samples. An example of classification can be found in the banking industry

customer characteristics like age, annual income, marital status, etc are used to predict the possibility of approving loan applications (the loan status is the class label). In an initial step, a dataset containing a certain number of customers with known class labels is used to create a classifier which can then be used to predict the class label of a new application ANN Classification is very similar to regression except that the later is applicable to numerical data while the former is applicable to categorical and numerical data. Clustering The is process of grouping data objects into clusters such that

intra-cluster similarity is maximized inter-cluster similarity is minimized. In other words, objects within the same clusters are very similar and objects in different clusters are not. E.g. studying collective properties of people at different income levels Cluster people based on incomes Study common properties within clusters Lower income related to lower education Outlier detection Through clustering, we can find groups of objects that behave similarly sometimes, we are only interested in those

objects that lie scattered around without behaving similarly to any pattern existing in the data. Those objects are known as outliers as they do not adhere to the patterns defined by the rest of the objects in the dataset. Outlier detection is usually desired in applications where abnormal behavior is of interest such as intrusion detection in networks or terrorist detection in ports of entry not of interest, such as when we clean a dataset from noise Outlier Border Core Eps = 1cm MinPts = 5 Similarity searches given a database of objects, and a query object,

find all similar objects (neighbours) Google search Given a query which a small document Find all similar documents Ranked order them Final Notes on Data Mining Forms the center of a set of research fields and applications dealing with data analysis: databases, statistics, machine learning, artificial intelligence, information sciences/technology and the like

at the same time introduces a lot of new features rendering itself as a separate science. scalability to large datasets Not all types of patterns mined by data mining systems are interesting. Subjective and objective interesting measures. Market Basket Research We will mainly use the Market Basket Research (MBR) application in our ARM description A large set of items, e.g. products sold in a supermarket. A large set of transactions or baskets, each of which contains a small set of the items (called an itemset) bought by a

customer during a single visit to a store. The Set Model Data is organized as a "TRANSACTION TABLE" with 2 attributes: TT(Tid, Itemset) A transaction is a customer transaction at a cash register. Each customer is given an identifier, Tid, for every transaction made Itemset is the set of items in the customer's "basket". Note that tuples in TT are not "flat" (each itemset is a "set") i.e. not relational (why?) a transformation can be made to equivalent but normalized models

TID 1 2 3 4 5 6 7 8 9 10 Atts abc abd abe acd ace ade bcd bce bde cde TID IID The Normalized Set Model

Data is organized as a NORMALIZED TRANSACTION TABLE" with 2 attributes: NTT(Tid,Iid) An itemset is the group of items belonging to the same transaction The TT(Tid, ItemSet) can be "transformed" to NTT(Tid, Iid) and vice versa Could be stored in a database Very deep (10 to 30 tuples) 1 a 6 a 1 b

6 d 1 c 6 e 2 a 7 b 2 b 7 c 2 d

7 d 3 a 8 b 3 b 8 c 3 e 8 e 4 a

9 b 4 c 9 d 4 d 9 e 5 a 10 c 5 c 10 d

5 e 10 e The Boolean Model: "Boolean Transaction Table: 1 1 1 1 0 0 BTT(Tid, Item-1, Item-2,... Itemn) Tid is a transaction identifier Each column is a particular Item (1 column for each item)

2 1 1 0 1 0 3 1 1 0 0 1 4 1 0 1

1 0 5 1 0 1 0 1 a 1 if item is in the basket a 0 if item is not in the basket 6 1 0 0 1 1 7

0 1 1 1 0 8 0 1 1 0 1 9 0 1 0 1

1 10 0 0 1 1 1 TID a b c d e TT, NTT and BTT are equivalent This is the model mostly chosen for ARM Association Rule Mining

Association Rule Mining (ARM) finds interesting associations and/or correlation relationships among large sets of data items. Association rules provide information in the form of "if-then" statements. These rules are computed from the data unlike the if-then rules of logic, association rules are probabilistic in nature strength could be measured An association rule defines a relationship of the form: A C (if A then C) Read as A implies C, where A and C

are sets of items in a data set. A called antecedent and C the consequent Given DB, ARM finds all the ARs D = A data set comprising n records (transactions) and m Boolean valued attributes (BTT model) I = The set of m attributes, {i1,i2, ,im}, represented in D. Itemset = Some subset of I. Each record in D is an itemset For all rules AC: AI, CI, and AC= (A and C are disjoint). An Example DB Items = 5

I = {a,b,c,d,e} Transactions = 10 D = {{a,b,c}, {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {a,d,e}, {b,c,d}, {b,c,e}, {b,d,e}, {c,d,e}} TID 1 2 3 4 5 6 7 8 9 10 Atts abc abd abe acd ace ade bcd

bce bde cde Support of an Itemset Support of an itemset IS is the number of transactions in D containing all items in IS (support of IS={ab} is 3?) Given a support threshold s, sets of items that appear in > s transactions are called frequent itemsets The process is called frequent itemset mining Items={m=milk, c=cheese, p=pepsi, b=bread, j=juice}. Support threshold = 3 transactions. T1 T3 T5 T7

= = = = {m, c, b} {m, b} {m, p, b} {c, b, j} T2 T4 T6 T8 = = = = {m, p, j} {c, j} {m, c, b, j} {b, c} Frequent itemsets: {m}, {c}, {b}, {j}, {m, b}, {c, b}, {j, c}.

Support and Confidence of a Rule AC Support of an itemset IS is the number of transactions containing all items in IS Itemsets are used to derive rules Support of a rule R: AC is the number of transactions in D containing all items in A U C. Frequent rule Significance of a rule Confidence of a rule is Support(R)/ Support(A) Confident rule Strength of a rule Out of those containing A, how many also contain C

Frequent + Confident Strong Example B1 B3 B5 B7 {m, c, b} {m, b} {m, p, b} {c, b, j} B2 = {m, p, j} B4 = {c, j} B6 = {m, c, b, j} B8 = {b, c} An association rule: {m, b} c. What is the confidence? = = = =

support(m, b, c) = 2 Support(m, b) = 4 Confidence = 2/4 = 50%. And so what does that mean? 50% that contain {m, b} also contain c More On The Problem Definition ARM is a two-step process: Find all frequent itemsets: By definition, each of these itemsets will occur at least as frequently as a pre-determined minimum support threshold Generate strong association rules from the frequent itemsets: By definition, these rules must satisfy the minimum support and minimum confidence thresholds A typical question: find all strong association rules with support > s and confidence > c.

Given a database D Find all frequent itemsets (F) using s Produce all strong association rules using c Finding F is the most computationally expensive part, once we have the frequent sets generating ARs is straight forward The Anti-Monotonicity (downwardclosure) of Support Nave: generate all subset itemsets of I and test each The number of potential subset itemsets 2m If m=5, #potential itemsets = 32 If m=20, #potential itemsets 1,048,576 Imagine what would supermarkets have? m = 10,000? Conclusion?

Breakthrough: If an itemset A has support greater than s then all its subsets must also be have support greater than s Nave approach is infeasible example Alternatively if an itemset A is not frequent then none of its supersets will be supported. Proposed by Agrawal 1993 from IBM Almaden Research Centerits started ARM and the field of data mining Apriori Proposed by Agrawal Apriori Uses the downward-closure of support

to reduce the number of itemsets that need to be counted (called candidate frequent itemsets C) Works on a level-by-level basis (i.e. uses frequent itemsets L from the previous to generate frequent itemsets at this level) Ck and Lk At every level k generates Ck from Lk-1and counts their frequency in the database Two steps are performed to generate C k Join Step: C is generated by joining L with itself Prune Step: all itemsets in Ck whose k-1 subsets k k-1 are not ALL frequent (i.e. present in Lk-1) are removed How many subsets does an itemset of size k have? 2k

E.g. k=3 How many subsets of size k-1 does an itemset of size k have? k The Apriori Algorithm Pseudo-code: Ck: Candidate frequent itemset of size k Lk : frequent itemset of size k L1 = {frequent items}; for (k = 1; Lk !=; k++) do begin Ck+1 = candidates generated from Lk; Remove any itemset from Ck+1 that has at least one infrequent k subset for each transaction t in database do increment the counts of all candidates in Ck+1 that are contained in t (count the frequency of each itemset in Ck+1) Lk+1 = candidates in Ck+1 with min_support end return k Lk; Example of Generating Candidates

Suppose the items in all itemsets are listed in some order L3={abc, abd, acd, ace, bcd} Self-joining: L3*L3 Combine any two itemsets in Lk if they only differ by the last item abcd from abc and abd acde from acd and ace C4 = {abcd , acde} Pruning: abcd: abc, abd, acd, bcd

acde: acd, ace, ade, cde C4={abcd} How To Generate Candidates? Lk Ck+1 Step 1: self-joining Lk insert into Ck+1 select p.item1, p.item2, , p.itemk, q.itemk from Lk p, Lk q where p.item1=q.item1, , p.itemk-1=q.itemk-1, p.itemk < q.itemk Step 2: pruning forall itemsets c in Ck+1 do forall k-subsets s of c do An Example Support 2 Database D TID 100 200 300

400 itemset sup. C1 {1} 2 {2} 3 Scan D {3} 3 {4} 1 {5} 3 Items 134 235 1235 25 C2 L2 itemset sup {1 3} {2 3} {2 5} {3 5} 2 2 3

2 C3 itemset {2 3 5} L1 itemset sup. {1} {2} {3} {5} C2 itemset sup {1 2} 1 {1 3} 2 {1 5} 1 Scan D {2 3} 2 {2 5} 3 {3 5} 2 Scan D L3 itemset sup {2 3 5} 2

2 3 3 3 itemset {1 2} {1 3} {1 5} {2 3} {2 5} {3 5} Generation of Association Rules Given all frequent itemsets Every frequent itemset I of size > 2 is divided into a candidate head Y and a body X such that X intersection Y = {}. This process starts with Y = {}, resulting in the rule I {}

always holds with 100% confidence (why?) After that, the algorithm iteratively generates candidate heads of size k + 1, starting with k = 0 Is Apriori Fast Enough? Performance Bottlenecks The core of the Apriori algorithm: Uses frequent (k 1)-itemsets to generate candidate frequent kitemsets Uses databases scan to collect counts for the candidate itemset 1 scan per level 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 scans, n is the length of the longest pattern One scan per level Improving Apriori Transaction reduction Reducing the number of transactions scanned in future iterations A transaction that does not contain any frequent k-itemsets cannot contain any frequent (k+1)-itemsets. E.g. Frequent 1 itemsets {1, 3, 5} Trans = {2,4}

As a result, we need not consider it further for subsequent scans of D for l-itemsets where l>k. Saves on scanning times Partitioning Using this approach we only need two database scans to generate all frequent itemsets Good when original DB cant fit in memory First, we divided D, into n non-overlapping partitions such that each can easily fits into memory. The minimum support threshold (referred to local support threshold) for itemsets in each partition is minsuppxN/|D| (where N is the number of transactions in that partition). For each partition, all frequent itemsets within that partition are found. These are called local frequent itemsets.

For each itemset, we record tids of the transactions containing the items in the itemset. As a result, we could find the local frequent itemsets in just one database scan. Local frequent itemsets may not be frequent with respect to the entire database, D; however, any frequent itemset in D must occur as a local frequent itemset in at least one partition Therefore we could use the local frequent itemsets as candidates with respect to D. Second, we scan D to get the support of all

candidate itemsets (which have already been generated using the partitions). Partition size and number of partitions are set so that each partition can fit into main memory and therefore be read only once in each phase. Good when original DB cant fit in memory Sampling This is statistical-based approach the principle that since we can not deal with the whole population, we can get a representative sample (usually random) whose size is much smaller than the population and work with that. The accuracy of approaches used this idea depends on how representative the chosen sample is. In short, we select a sample S form D and generate all frequent itemsets in S usually using a lower support threshold than minsupp. Some approaches that follow this idea claim

that they can mine all rules using samples. Tries Another data structure that is commonly used is a trie (or prefix-tree). The first approach to ever use tries in ARM is Frequent Pattern Growth (FPGrowth) by Han et al. The idea here is to view each transaction as an ordered string of items. The idea is compress by maximizing overlap between transactions Every k-itemset is attached to its k - 1-prefix. Every node stores the last item in the itemset it represents, its support, and its branches Vertical Data Representation Each item, I, is represented by a bit

vector, VI The support of an item is the count of 1s in its vector The support of an itemset {a,b} is the count of 1s in Va & Vb An Example TT Layout TID 100 200 300 400 BTT Layout Items 134 235 1235 25 TID 100 200 300

400 12345 10110 01101 11101 01001 Binary Vertical (BV) Layout D TID 100 200 300 400 12345 10110 01101 11101 01001 Database D TID

12345 100 10110 Support(3) = 3 200 01101 Support (3,5) = 2 300 11101 Support (1,3,5) = 1 400 01001 Just ANDing operations Could be optimized by compression through P-trees Saves time References - 2000

R. Agarwal, C. Aggarwal, and V. V. V. Prasad. A tree projection algorithm for generation of frequent itemsets. In Journal of Parallel and Distributed Computing (Special Issue on High Performance Data Mining), 2000. R. Agrawal, T. Imielinski, and A. Swami. Mining association rules between sets of items in large databases. SIGMOD'93, 207-216, Washington, D.C. R. Agrawal and R. Srikant. Fast algorithms for mining association rules. VLDB'94 487-499, Santiago, Chile. R. Agrawal and R. Srikant. Mining sequential patterns. ICDE'95, 3-14, Taipei, Taiwan. R. J. Bayardo. Efficiently mining long patterns from databases. SIGMOD'98, 85-93, Seattle, Washington. S. Brin, R. Motwani, and C. Silverstein. Beyond market basket: Generalizing association rules to correlations. SIGMOD'97, 265-276, Tucson, Arizona. S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket analysis. SIGMOD'97, 255-264, Tucson, Arizona, May 1997. K. Beyer and R. Ramakrishnan. Bottom-up computation of sparse and iceberg cubes. SIGMOD'99, 359-370, Philadelphia, PA, June 1999. D.W. Cheung, J. Han, V. Ng, and C.Y. Wong. Maintenance of discovered association rules in large databases: An incremental updating technique. ICDE'96, 106-114, New Orleans, LA. M. Fang, N. Shivakumar, H. Garcia-Molina, R. Motwani, and J. D. Ullman. Computing iceberg queries efficiently. VLDB'98, 299-310, New York, NY, Aug. 1998. References (2)

G. Grahne, L. Lakshmanan, and X. Wang. Efficient mining of constrained correlated sets. ICDE'00, 512-521, San Diego, CA, Feb. 2000. Y. Fu and J. Han. Meta-rule-guided mining of association rules in relational databases. KDOOD'95, 39-46, Singapore, Dec. 1995. T. Fukuda, Y. Morimoto, S. Morishita, and T. Tokuyama. Data mining using two-dimensional optimized association rules: Scheme, algorithms, and visualization. SIGMOD'96, 13-23, Montreal, Canada. E.-H. Han, G. Karypis, and V. Kumar. Scalable parallel data mining for association rules. SIGMOD'97, 277-288, Tucson, Arizona. J. Han, G. Dong, and Y. Yin. Efficient mining of partial periodic patterns in time series database. ICDE'99, Sydney, Australia. J. Han and Y. Fu. Discovery of multiple-level association rules from large databases. VLDB'95, 420-431, Zurich, Switzerland.

J. Han, J. Pei, and Y. Yin. Mining frequent patterns without candidate generation. SIGMOD'00, 112, Dallas, TX, May 2000. T. Imielinski and H. Mannila. A database perspective on knowledge discovery. Communications of ACM, 39:58-64, 1996. M. Kamber, J. Han, and J. Y. Chiang. Metarule-guided mining of multi-dimensional association rules using data cubes. KDD'97, 207-210, Newport Beach, California. M. Klemettinen, H. Mannila, P. Ronkainen, H. Toivonen, and A.I. Verkamo. Finding interesting rules from large sets of discovered association rules. CIKM'94, 401-408, Gaithersburg, Maryland. References (3) F. Korn, A. Labrinidis, Y. Kotidis, and C. Faloutsos. Ratio rules: A new paradigm for fast, quantifiable data mining. VLDB'98, 582-593, New York, NY. B. Lent, A. Swami, and J. Widom. Clustering association rules. ICDE'97, 220-231, Birmingham, England. H. Lu, J. Han, and L. Feng. Stock movement and n-dimensional inter-transaction association rules. SIGMOD Workshop on Research Issues on Data Mining and Knowledge Discovery (DMKD'98), 12:1-12:7, Seattle, Washington. H. Mannila, H. Toivonen, and A. I. Verkamo. Efficient algorithms for discovering association rules. KDD'94, 181-192, Seattle, WA, July 1994.

H. Mannila, H Toivonen, and A. I. Verkamo. Discovery of frequent episodes in event sequences. Data Mining and Knowledge Discovery, 1:259-289, 1997. R. Meo, G. Psaila, and S. Ceri. A new SQL-like operator for mining association rules. VLDB'96, 122-133, Bombay, India. R.J. Miller and Y. Yang. Association rules over interval data. SIGMOD'97, 452-461, Tucson, Arizona. R. Ng, L. V. S. Lakshmanan, J. Han, and A. Pang. Exploratory mining and pruning optimizations of constrained associations rules. SIGMOD'98, 13-24, Seattle, Washington. N. Pasquier, Y. Bastide, R. Taouil, and L. Lakhal. Discovering frequent closed itemsets for association rules. ICDT'99, 398-416, Jerusalem, Israel, Jan. 1999. References (4)

J.S. Park, M.S. Chen, and P.S. Yu. An effective hash-based algorithm for mining association rules. SIGMOD'95, 175-186, San Jose, CA, May 1995. J. Pei, J. Han, and R. Mao. CLOSET: An Efficient Algorithm for Mining Frequent Closed Itemsets. DMKD'00, Dallas, TX, 11-20, May 2000. J. Pei and J. Han. Can We Push More Constraints into Frequent Pattern Mining? KDD'00. Boston, MA. Aug. 2000. G. Piatetsky-Shapiro. Discovery, analysis, and presentation of strong rules. In G. Piatetsky-Shapiro and W. J. Frawley, editors, Knowledge Discovery in Databases, 229-238. AAAI/MIT Press, 1991. B. Ozden, S. Ramaswamy, and A. Silberschatz. Cyclic association rules. ICDE'98, 412-421, Orlando, FL. J.S. Park, M.S. Chen, and P.S. Yu. An effective hash-based algorithm for mining association rules. SIGMOD'95, 175-186, San Jose, CA. S. Ramaswamy, S. Mahajan, and A. Silberschatz. On the discovery of interesting patterns in association rules. VLDB'98, 368-379, New York, NY.. S. Sarawagi, S. Thomas, and R. Agrawal. Integrating association rule mining with relational database systems: Alternatives and implications. SIGMOD'98, 343-354, Seattle, WA. A. Savasere, E. Omiecinski, and S. Navathe. An efficient algorithm for mining association rules in large databases. VLDB'95, 432-443, Zurich, Switzerland. A. Savasere, E. Omiecinski, and S. Navathe. Mining for strong negative associations in a large database of customer transactions. ICDE'98, 494-502, Orlando, FL, Feb. 1998.

References (5) C. Silverstein, S. Brin, R. Motwani, and J. Ullman. Scalable techniques for mining causal structures. VLDB'98, 594-605, New York, NY. R. Srikant and R. Agrawal. Mining generalized association rules. VLDB'95, 407-419, Zurich, Switzerland, Sept. 1995. R. Srikant and R. Agrawal. Mining quantitative association rules in large relational tables. SIGMOD'96, 1-12, Montreal, Canada. R. Srikant, Q. Vu, and R. Agrawal. Mining association rules with item constraints. KDD'97, 67-73, Newport Beach, California. H. Toivonen. Sampling large databases for association rules. VLDB'96, 134-145, Bombay, India, Sept. 1996. D. Tsur, J. D. Ullman, S. Abitboul, C. Clifton, R. Motwani, and S. Nestorov. Query flocks: A generalization of association-rule mining. SIGMOD'98, 1-12, Seattle, Washington.

K. Yoda, T. Fukuda, Y. Morimoto, S. Morishita, and T. Tokuyama. Computing optimized rectilinear regions for association rules. KDD'97, 96-103, Newport Beach, CA, Aug. 1997. M. J. Zaki, S. Parthasarathy, M. Ogihara, and W. Li. Parallel algorithm for discovery of association rules. Data Mining and Knowledge Discovery, 1:343-374, 1997. M. Zaki. Generating Non-Redundant Association Rules. KDD'00. Boston, MA. Aug. 2000. O. R. Zaiane, J. Han, and H. Zhu. Mining Recurrent Items in Multimedia with Progressive Resolution Refinement. ICDE'00, 461-470, San Diego, CA, Feb. 2000. Questions ? Thank you !!!

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