Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
From Association Analysis to Causal Discovery Prof Jiuyong Li University of South Australia Association analysis • Diapers -> Beer • Bread & Butter -> Milk Positive correlation of birth rate to stork population • increasing the stork population would increase the birth rate? Further evidence for Causality ≠ Associations Simpson paradox Recovered Not recovered Sum Recover rate Drug 20 20 40 50% No Drug 16 24 40 40% 36 44 80 Female Recovered Not recovered Sum Recover rate Drug 2 8 10 20% No Drug 9 21 30 30% 11 29 40 Male Recovered Not recovered Sum Recover rate Drug 18 12 30 60% No Drug 7 3 10 70% 25 15 40 Association and Causal Relationship • Two variables X and Y. • Prob(Y | X) ≠ P(Y), X is associated with Y (association rules) • Prob(Y | do X) ≠ Prob(Y | X) • How does Y vary when X changes? • The key, How to estimate Prob(Y | do X)? • In association analysis, the relationship of X and Y is analysed in isolation. • However, the relationship between X and Y is affected by other variables. 5 Causal discovery 1 • Randomised controlled trials – Gold standard method – Expensive – Infeasible • Association = causation Causal discovery 2 • Bayesian network based causal inference – Do-calculus (Pearl 2000) – IDA (Maathuis et al. 2009) – To infer causal effects in a Bayesian network. – However – Constructing a Bayesian network is NP hard – Low scalability to large number of variables Leaning causal structures • PC algorithm (Spirtes, Glymour and Scheines) – Not (A ╨ B | Z), there is an edge between A and B. – The search space exponentially increases with the number of variables. CCC B A ABC, ABC, CAB • Constraint based search – CCC (G. F. Cooper, 1997) – CCU (C. Silverstein et. al. 2000) – Efficiently removing noncausal relationships. C CCU B A C ABC Association rules • Many efficient algorithms • Hundreds of thousands to millions of rules. – Many are spurious. • Interpretability – Association rules do not indicate causal effects. Causal rules • Discover causal relationships using partial association and simulated cohort study. • Do not rely on Bayesian network structure learning. The discovery of causal rules also have strong theoretical support. • Discover both single cause and combined causes. • Can be discovered efficiently. • Z. Jin, J. Li, L. Liu, T. D. Le, B. Sun, and R. Wang, Discovery of causal rules using partial association. ICDM, 2012 • J. Li, T. D. Le, L. Liu, J. Liu, Z. Jin, and B. Sun. Mining causal association rules. In Proceedings of ICDM Workshop on Causal Discovery (CD), 2013. Problem Discover causal rules from large databases of binary variables A B C D E F Y #repeats 1 1 1 1 1 1 1 14 1 0 1 1 1 1 1 8 1 1 0 1 0 1 1 15 0 1 1 1 1 1 1 8 0 1 0 0 0 0 0 5 0 0 0 0 1 0 1 6 1 0 0 0 0 1 0 4 1 0 1 1 1 0 0 3 0 1 0 1 1 0 0 3 0 1 0 0 1 0 0 5 AY CY BF Y DE Y Partial association test K K I I J J I K J PA(I, J, K) ³ ca2 Nonzero partial association PA(I, J, K ) = (| å k n11k n00k - n10k n01k 1 2 |- ) n××k 2 n1×k n.1k n0×k n×0k å n2 (n -1) ××k ××k k M. W. Birch, 1964. Partial association test – an example A B C D E F Y G #repeat 1 1 1 1 1 1 1 0 14 1 0 1 1 1 1 1 0 8 1 1 0 1 0 1 1 0 15 0 1 1 1 1 1 1 0 8 0 1 0 0 0 0 0 0 5 0 0 0 0 1 0 1 0 6 1 0 0 0 0 1 0 0 4 1 0 1 1 1 0 0 0 3 1 1 1 1 0 1 1 1 3 0 1 0 0 1 0 0 0 5 n11k n00k n10k n01k 14 3 0 8 1.68 nk 25 4. Partial association test. PA( X , Y , K ) (| k n11k n00k n10k n01k 1 2 | ) nk 2 n1k n.1k n0k n0 k k n2 (n 1) k k PA( BF , Y , ACDE ) n1k n.1k n0k n0 k 14 22 11 3 0.6776 n2k (nk 1) 252 (25 1) Fast partial association test PA(I, J, K ) = (| å k n11k n00 k - n10 k n01k 1 | - )2 n××k 2 n n.1k n0×k n×0 k å n1×k2 (n ××k ××k -1) k • K denotes all possible variable combinations, the number is very large. • Counting the frequencies of the combinations is also time consuming. • Our solution: – Sort data and count frequencies of the equivalence classes. – Only use the combinations existing in the data set. Pruning strategies Definition (Redundant causal rules): Assume that X⊂ W, if X → Y is a causal rule, rule W → Y is redundant as it does not provide new information. Definition (Condition for testing causal rules): We only test a combined causal rule XV → Y if X and Y have a zero association and V and Y have a zero association (cannot pass the quisquare test in step 3). A B D E 1 1 1 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 xG F Algorithm Y #repeats 1 14 1 8 1 15 0 1 8 0 0 0 5 1 0 0 1 6 0 1 0 0 4 0 0 3 1 zero 0 1 1 1 0 association 1 1 1 1 1 1 0 3 0 1 0 0 5 1 C 1 1 1 1 0 positive 1 1 1 1 0 association 0 1 0 1 0 0 0 1 0 Y=1 Y=0 Total 11 12 1. X=0 n21 n22 n2. Total n.1 n.2 n 2 PA ( X , Y , K ) X=1 n n n 2 X ,Y i 2, j 2 i 1, j 1 (nij E (nij )) E (nij ) 2 1. Prune the variable set (support) 2. Create the contingency table for each variable X X2 , Y 3. Calculate the • If X2 , Y 2 go to next step 2 2 • If X , Y move X to a set N 4. Partial association test. • If PA(X, Y, K) is nonzero then XY is a causal rule. 5. Repeat 1-4 for each variable which is the combination of variables in set N Experimental evaluations • We use the Arrhythmia data set in UCI machine learning repository. – We need to classify the presence and absence of cardiac arrhythmia. The data set contains 452 records and each record obtains 279 data attributes and one class attribute • Our results are quite consistent with the results from CCC method. • Some rules in CCC are removed by our method as they cannot pass the partial association test. • Our method can discover the combined rules. CCC and CCU methods are not set to discover these rules. Comparison with CCC and CCU Experimental evaluations Figure 1: Extraction Time Comparison (20K Records) Figure 1: Extraction Time Comparison (100K Records) Summary 1 • Simpson paradox – Associations might be inconsistent in subsets • Partial association test – Test the persistency of associations in all possible partitions. – Statistically sound. – Efficiency in sparse data. • What else? Cohort study 1 Defined population Expose Have a disease Not have a disease Not expose Have a disease • Prospective: follow up. • Retrospective: look back. Historic study. Not have a disease Cohort study 2 • Cohorts: share common characteristics but exposed or not exposed. • Determine how the exposure causes an outcome. • Measure: odds ratio = (a/b) / (c/d) Diseased Healthy Exposed a b Not exposed c d Limitations of cohort study • Need to know a hypothesis beforehand • Domain experts determine the control variables. • Collect data and test the hypothesis. • Not for data exploration. • We need – Given a data set without any hypotheses. – An automatic method to find and validate hypotheses. – For data exploration. Control variables Outcome Cause Other factors • If we do not control covariates (especially those correlated to the outcome), we could not determine the true cause. • Too many control variables result too few matched cases in data. – How many people with the same race, gender, blood type, hair colour, eye colour, education level, …. • Irrelevant variables should not be controlled. – Eye colour may not relevant to the study. Matches • Exact matching – Exact matches on all covariates. Infeasible. • Limited exact matching – Exact matches on a few key covariates. • Nearest neighbour matching – Find the closest neighbours • Propensity score matching – Based on the predicted effect of a treatment of covariates. Method1 Discover causal association rules from large databases of binary variables AY A B C D E F Y 1 1 1 1 1 1 1 A B C D E F Y 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 Fair dataset Methods • A: Exposure variable Fair dataset A B C D E F Y 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 0 • {B,C,D,E,F}: controlled variable set. • Rows with the same color for the controlled variable set are called matched record pairs. A=0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 A=1 Y=1 Y=0 Y=1 n11 n12 Y=0 n21 n22 n12 OddsRatioD f (A ® Y ) = n21 • An association rule A Y is a causal association rule if: OddsRatioD f ( A Y ) 1 Algorithm A B C D E xF 1 1 1 1 1 1 … … 1 1 x G Y 0 1 A B C D E Y 1 1 1 1 1 1 … … 0 1 … … 0 1 0 1 0 1 … 1 1 1 0 … 1. Remove irrelevant variables (support, local support, association) For each association rule (e. g. A Y) 2. Find the exclusive variables of the exposure variable (support, association), i.e. G, F. The controlled variable set = {B, C, D, E}. 3. Find the fair dataset. Search for all matched record pairs 4. Calculate the odds-ratio to identify if the testing rule is causal 5. Repeat 2-4 for each variable which is the combination of 28 variables. Only consider combination of non-causal factors. Experimental evaluations Experimental evaluations CAR Figure 1: Extraction Time Comparison (20K Records) CCC CCU Experimental evaluations Causality – Judea Pearl X1 X2 5.2 … Xn-1 Xn 7.5 6.5 5.2 5.6 7.2 6.6 5.3 … … … … 5.4 7.1 7.1 5.7 5.7 6.9 6.9 5.8 … +1 +0.8 Judea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, 2000. 32 Methods • IDA – Maathuis, H. M., Colombo, D., Kalisch, M., and Buhlmann, P. (2010). Predicting causal effects in largescale systems from observational data. Nature Methods, 7(4), 247–249. 33 Conclusions • Association analysis has been widely used in data mining, but associations do not indicate causal relationships. • Association rule mining can be adapted for causal relationship discovery by combining some statistical methods. – Partial association test – Cohort study • They are efficient alternatives for causal Bayesian network based methods. • They are capable of finding combined causal factors. Discussions • Causality and classification – Estimate prob (Y| do X) instead of prob (Y|X). • Feature section versus controlled variable selection. • Evaluation of causes. – Not classification accuracy – Bayesian networks?? Research Collaborators • • • • • Jixue Liu Lin Liu Thuc Le Jin Zhou Bin-yu Sun Thank you for listening Questions please ??