Analyzing Attribute Dependencies

1. Analyzing Attribute Dependencies Aleks Jakulin & Ivan Bratko Faculty of Computer and Information Science University of Ljubljana Slovenia

2. Overview • Problem: • Generalize the notion of “correlation” from two variables to three or more variables. • Approach: • Use the Shannon’s entropy as the foundation for quantifying interaction. • Application: • Visualization, with focus on supervised learning domains. • Result: • We can explain several “mysteries” of machine learning through higher-order dependencies.

3. label C importance of attribute A importance of attribute B attribute attribute A B attribute correlation 3-Way Interaction: What is common to A, B and C together; and cannot be inferred from any subset of attributes. 2-Way Interactions Problem: Attribute Dependencies

4. Entropy given C’s empirical probability distribution (p = [0.2, 0.8]). H(C|A) = H(C)-I(A;C) Conditional entropy --- Remaining uncertainty in C after knowing A. H(A) Information which came with knowledge of A H(AB) Joint entropy I(A;C)=H(A)+H(C)-H(AC) Mutual information or information gain --- How much have A and C in common? Approach: Shannon’s Entropy A C

5. Interaction Information I(A;B;C) := I(AB;C) - I(A;C) - I(B;C) = I(A;B|C) - I(A;B) • (Partial) history of independent reinventions: • McGill ‘54 (Psychometrika) - interaction information • Han ‘80 (Information & Control) - multiple mutual information • Yeung ‘91 (IEEE Trans. On Inf. Theory) - mutual information • Grabisch&Roubens ‘99 (I. J. of Game Theory) - Banzhaf interaction index • Matsuda ‘00 (Physical Review E) - higher-order mutual inf. • Brenner et al. ‘00 (Neural Computation) - average synergy • Demšar ’02 (A thesis in machine learning) - relative information gain • Bell ‘03 (NIPS02, ICA2003) - co-information • Jakulin ‘03 - interaction gain

6. Properties • Invariance with respect to attribute/label division: I(A;B;C) = I(A;C;B) = I(C;A;B) = = I(B;A;C) = I(C;B;A) = I(B;C;A). • Decomposition of mutual information: I(AB;C) = I(A;C)+I(B;C)+I(A;B;C) I(A;B;C) is “synergistic information.” • A, B, C are independent  I(A;B;C) = 0.

7. Positive and Negative Interactions • If any pair of the attributes is conditionally independent w/r to a third attribute, the 3-information “neutralizes” the 2-information: I(A;B|C) = 0  I(A;B;C) = -I(A;B) • Interaction information may be positive or negative: • Positive: XOR problem (A = B  C) synergy • Negative: conditional independence, redundant attributes redundancy • Zero: Independence of one of the attributes or a mix of synergy and redundancy.

8. Applications • Visualization • Interaction graphs • Interaction dendrograms • Model construction • Feature construction • Feature selection • Ensemble construction • Evaluation on the CMC domain: predicting contraception method from demographics.

9. Information gain: 100% I(A;C)/H(C) The attribute “explains” 1.98% of label entropy A positive interaction: 100% I(A;B;C)/H(C) The two attributes are in a synergy: treating them holistically may result in 1.85% extra uncertainty explained. A negative interaction: 100% I(A;B;C)/H(C) The two attributes are slightly redundant: 1.15% of label uncertainty is explained by each of the two attributes. Interaction Graphs

10. CMC

11. Application: Feature Construction NBC Model Predictive perf. (Brier score)__ {} 0.2157  0.0013 {Wedu, Hedu} 0.2087  0.0024 {Wedu} 0.2068  0.0019 {WeduHedu} 0.2067  0.0019 {Age, Child} 0.1951  0.0023 {AgeChild} 0.1918  0.0026 {ACWH} 0.1873  0.0027 {A, C, W, H} 0.1870  0.0030 {A, C, W} 0.1850  0.0027 {AC, WH} 0.1831  0.0032 {AC, W} 0.1814  0.0033

12. Alternatives TAN NBC 0.1874  0.0032 0.1849  0.0028 BEST: >100000 models {AC, WH, MediaExp} GBN 0.1811  0.0032 0.1815  0.0029

13. Dissimilarity Measures • The relationships between attributes are to some extent transitive. • Algorithm: • Define a dissimilarity measure between two attributes in the context of the label C: • Apply hierarchical clustering to summarize the dissimilarity matrix.

14. uninformative attribute informative attribute information gain Interaction Dendrogram weakly interacting strongly interacting cluster “tightness” loose tight

15. Application: Feature Selection • Soybean domain: • predict disease from symptoms; • predominantly negative interactions. • Global optimization procedure for feature selection: >5000 NBC models tested (B-Course) • Selected features balance dissimilarity and importance. • We can understand what global optimization did from the dendrogram.

16. Application: Ensembles

17. A A and B are independent. They may both inform us about C, but they have nothing in common. Assumed by: myopic feature importance measures (information gain), discretization algorithms. C B I(AB;C)=I(A;C)+I(B;C) C A and B are conditionally independent given C. If A and B have something in common, it is all due to C. Assumed by: Naïve Bayes, Bayesian networks (A  C  B); B A - I(A;B|C)=0 Implication: Assumptions in Machine Learning

18. Work in Progress • Overfitting: the interaction information computations do not account for the increase in complexity. • Support for numerical and ordered attributes. • Inductive learning algorithms which use these heuristics automatically. • Models that are based on the real relationships in the data, not on our assumptions about them.

19. Summary • There are relationships exclusive to groups of n attributes. • Interaction information is a heuristic for quantification of relationships with entropy. • Two visualization methods: • Interaction graphs • Interaction dendrograms