Efficient and Effective Itemset Pattern Summarization: Regression-based Approaches Ruoming Jin Kent State University Joint work with Muad Abu-Ata, Yang Xiang, and Ning Ruan (KSU)

Problem Definition Given a large collection of frequent itemsets and their supports, how we can concisely represent them? –Coverage criterion The Spanning Set Approach [F. Arati, A. Gionis, Mannila, Approximating a collection of frequent sets, KDD’04]. –Frequency criterion The Profile-based Approach [X. Yan, H. Cheng, J. Han, and D. Xin, Summarizing itemset patterns, a profile-based approach, KDD’05.] The Markov Random Field Approach [C. Wang and S. Parthasarathy, Summarizing itemset patterns using probabilistic models, KDD’06.]

Frequency Criterion The restoration function of a set of itemsets S is a function The restoration error: We use 2-norm in this study.

Probabilistic Restoration Function Applying the independence probabilistic model for a set of itemsets S: An example,

Problem 1: Optimal Parameters What are the optimal parameters, p(S),p(a),p(c),p(d), minimizing the restoration error:

Non-Linear Regression We introduce the independent variable We have |S| data points.

Linear Regression Approximation Using Taylor expansion, we show the restoration error from linear regression is very close to the error by using the non-linear regression!

Problem 2: Optimal Partition To reduce the restoration error, we adopt the partition strategy –Partition the entire collection of frequent itemsets into K disjoint subsets, and build the restoration function for each subset How to optimally partition a set of itemsets into K disjoint subsets so that the total restoration error can be minimized?

Our Approaches NP-hard problem Two heuristic algorithms –K-Regression –Tree Regression

K-Regression A k-means type clustering procedure: 1.Random partition the set of itemsets S into K partition 2.[Regression Step] Apply regression to find the optimal parameters on each partition 3.[Re-assignment Step] For each itemset, assign it to the partition which minimizes its restoration error based on the optimal parameters discovered by Step 2 4.Repeat 2 and 3 until the total restoration error does not increase or the improvement is small Just as k-means, k-regression is guaranteed to converge!

Tree Regression Using Regression to find optimal parameters for each subset of itemsets S={{a},{b},{c},{d},{a,b},{a,c},{b,c},{a,d},{c,d}, {a,b,c},{a,b,d},{a,c,d}}

Tree Regression Construction A Decision-type of construction algorithm –Question 1: How to find K subsets of itemsets? –Question 2: How to find the optimal splitting? Answer for Q1 –Maintain a queue for the “current” leaf node, and always pick up the leaf nodes with the maximal average restoration error to split Answer for Q2 –Maximally reduce the total restoration error Min E(S)-E(S_1)-E(S_2)

An Interesting Connection Jerome H. Friedman’s 1977 paper, “A tree-structured Approach to nonparametric multiple regression”. Unfortunately, this work seems never got enough attention. However, it seems part of the inspiration for the CART (regression tree) and MARS (Multivariate Adaptive Regression Spline).

Experimental Results

Chess Restoration Error

BMS-POS Restoration Error

BMS-POS Running Time

Conclusion Using linear regression to identify optimal parameters of the probabilistic restoration function (based on the independence assumption) for a set of itemsets Two algorithms to optimally partition the set of itemsets into K parts –K-regression –Tree regression

Thanks!!