Causal Modeling for Anomaly Detection Andrew Arnold Machine Learning Department, Carnegie Mellon University Summer Project with Naoki Abe Predictive Modeling Group, IBM Rick Lawrence, Manager June 23, 2006

2 Contributions Consistent causal structure can be learned from passive observational data Anomalous examples have a quantitatively differentiable causal structure from normal ones Causal structure is a significant contribution to the standard analysis tools of independence and likelihood

3 Outline Motivation & Problem Causation Definition Causal Discovery Causal Comparisson Conclusions & Ongoing Work

4 Motivation Processors: –Detection: Is this wafer good or bad? –Causation: Why is this wafer bad? –Intervention: How can we fix the problem? Business: –Detection: Is this business functioning well or not? –Causation: Why is this business not functioning well? –Intervention: What can IBM do to improve performance?

5 Problem Interventions are expensive and flawed What can passively observed data tell us about the causal structure of a process?

6 Direct Causation X is a direct cause of Y relative to S, iff  z,x 1  x 2 P(Y | X set= x 1, Z set= z)  P(Y | X set= x 2, Z set= z) where Z = S - {X,Y} [Scheines (2005)] Asymmetric Intervene to set Z = z Not just observe Z = z

7 Causal Graphs Causal Directed Acyclic Graph G = {V,E} Each edge X  Y represents a direct causal claim: X is a direct cause of Y relative to V [Scheines (2005)]

8 Probabilistic Independence X and Y are independent iff  x 1  x 2 P(Y | X = x 1 ) = P(Y | X = x 2 ) X and Y are associated iff X and Y are not independent [Scheines (2005)]

9 Causal Structure Probabilistic Independence The Causal Markov Axiom Markov Condition In a Causal Graph: each variable V is independent of its non-effects, conditional on its direct causes. [Scheines (2005)]

10 Causal Structure  Statistical Data [Scheines (2005)]

11 Causal Structure  Statistical Data [Scheines (2005)]

12 Causal Structure  Statistical Data [Scheines (2005)]

13 Causal Discovery Statistical Data  Causal Structure Background Knowledge - Faithfulness - X 2 before X 3 - no unmeasured common causes Statistical Inference [Scheines (2005)]

14 Causal Discovery Algorithm PC algorithm [Spirtes et al., 2000] –Constraint-based search –Only need to know how to test conditional independence –Do not need to measure all causes –Asymptotically correct

15 PC algorithm Begin with the fully connected undirected graph For each pair of nodes, test their independence conditional on all subsets of their neighbors: –i.e., (X _||_ Y | Z)? If independent for any conditioning –remove edge, record subset conditioned upon If dependent for all conditionings –leave edge Orient edges, where possible

16 Independence Tests [Scheines (2005)]

17 Edge Orientation Rule 1: Colliders [Scheines (2005)]

18 More Orientation Rules: Rule 2: Avoid forming new colliders [Scheines (2005)]

19 More Orientation Rules: Rule 3: Avoid forming cycles If there is an undirected edge between X and Y And there is a directed path from X to Y –Then direct X-Y as X  Y Given: OK: BAD (cycle): X Y X Y X Y Z Z Z

20 Our Example Rule 2: Colliders Rule 3: No new V-structures Truth fully recovered [Scheines (2005)]

21 Patterns Often unable to orient all edges Use patterns [Pearl, 2000]: –Represents an equivalence set of DAG’s [Scheines (2005)]

22 How is this causation? Definition of causation Assumptions: –Faithfullness –No common unmeasured causes –Causal Markov condition

23 Results: Key Performance Indicators

24 Results: Key Performance Indicators

25 Causal Structure confirms existing beliefs and suggests new relationships

26 Results: Chip Fabrication

27 Temporal ordering is preserved

28 Using causal structure to explain anomalies Why is one wafer good, and another bad? –Separate data into classes –Form causal graphs on each class –Compare causal structures

29 Classification Support vector machine (SVM): –Max-margin classifier Finds hyperplane maximally separating data Chip data is readily separable: > 95% accuracy on labeled data

30 Form causal graphs Good Train Good Test Bad

31 How to compare? Similarity Score for graphs A and B over common nodes V : –Consider undirected edges as bi-directed –Of all the ordered pairs of variables (x, y) in V, with an arc x  y in either A or B In what percentage is there also x  y in the other graph i.e., (Adj A (x,y) || Adj B (x,y)) && (Adj A (x,y) == Adj B (x,y)) Difference Graph: –If there is an arc x  y in either A or B, but not in both, place the arc x  y in the difference graph –i.e., if (Adj A (x,y) != Adj B (x,y)) then Adj Diff (x,y) = True

32 Comparison Good TestGood Train 59% similar Difference Graph

33 Comparison BadGood Train 37% similar Difference Graph

34 Comparison BadGood Test 35% similar Difference Graph

35 Conclusions Consistent causal structure can be learned from passive observational data Anomalous examples have a quantitatively differentiable causal structure from normal ones Causal structure is a significant contribution to the standard analysis tools of independence and likelihood

36 Ongoing work Comparing to maximum likelihood and minimum description length techniques Looking at time-ordering –How do variables influence each other over time? Using one-class SVM to do clustering –Avoids need for labeled data Relaxing assumptions –Allow latent variables Evaluation is difficult without domain expert Using causal structure to help in clustering

37 References J. Pearl (2000). Causality: Models, Reasoning, and Inference, Cambridge Univ. Press R. Scheines, Causality Slides P. Spirtes, C. Glymour, and R. Scheines (2000). Causation, Prediction, and Search, 2 nd Edition (MIT Press) Thank You ¿ Questions ?