A simple method for multi-relational outlier detection Sarah Riahi and Oliver Schulte School of Computing Science Simon Fraser University Vancouver, Canada With tools that you probably have around the house lab.

2/13 A simple method for multi-relational outlier detectiona

3/13 System Flow Flach, P. A. (1999), Knowledge representation for inductive learning 'Symbolic and Quantitative Approaches to Reasoning and Uncertainty', Springer, pp Complete Database Population Parameter Values restrict to target individual vector norm outlier score Individual Profile Individual Parameter Values Parameter Learning Algorithm Parameter Learning Algorithm Model.... Input: Model, database, target individual. Output: an outlier score

4/13 Example A simple method for multi-relational outlier detection Model = Markov Logic Network learned for Premier League Season FormulasEstimated Population Parameters Estimated Parameters for P=van Persie SavesMade(P,M)=med AND shotsOnTarget(P,M)=low AND ShotEff(P,M)=low SavesMade(P,M)=med AND shotsOnTarget(P,M)=high AND ShotEff(P,M)=high (331 formulas)....

5/13 Evaluation: Synthetic Data A simple method for multi-relational outlier detection Two Features. Designed so that outliers are easy to distinguish from normals (sanity check). 1. Normals have a strong correlation, outliers none. 2. Outliers have a strong correlation, normals none. 3. Correlations are the same, but marginals are very different.

6/13 Bayesian Network Representation F1=ShotEfficiency F2 =Match_Resullt P(F1=1)= % 50 P(F2=0|F1=0)= % 90 P(F2=1|F1=1)= % 90 Normal=Striker P(F1=1)= % 50 P(F2=1)= % 50 Outlier=MidFielder P(F1=1)= % 50 (a) (b) P(F2=1)= % 50 P(F2=0|F1=0)= % 90 P(F2=1|F1=1)= % 90 F1=ShotEfficiency F2=Match_Resullt Normal=Striker F1=TackleEfficiency F2=Match_Resullt F1=TackleEfficiency F2=Match_Resullt Outlier=MidFielder

7/13 Results AD = Breunig, M.; Kriegel, H.-P.; Ng, R. T. & Sander, J. (2000), LOF: Identifying Density-Based Local Outliers, in ‘ACM SIGMOD'. LOG = Riahi, F.; Schulte, O. & Liang, Q. (2014), 'A Proposal for Statistical Outlier Detection in Relational Structures', AAAI-StarAI Workshop on Statistical-Relational AI. Metric = Area Under Curve ELD = average L1-norm KLD = average difference AD = use single feature marginals only (unit clauses) LOG = outlier score = log-likelihood

8/13 A simple method for multi-relational outlier detection Case Study: Single Features Which formulas/rules influence outlier score the most?  interpretability Which unit clauses influence outlier score the most?

9/13 Novak, P. K.; Webb, G. I. & Wrobel, S. (2009), 'Supervised descriptive rule discovery: A unifying survey of contrast set, emerging pattern and subgroup mining', Journal of Machine Learning Research. Maervoet, J.; Vens, C.; Vanden Berghe, G.; Blockeel, H. & De Causmaecker, P. (2012), 'Outlier Detection in Relational Data: A Case Study', Expert Systems and Applications Case Study: Correlations Which formulas/rules influence outlier score the most?  interpretability Which associations influence outlier score the most? Related to exception mining (Novak et al. 2009) IndividualRuleConfidenc e Individual Confidenc e Class Edin DzekoShotEff = high AND TackleEff = medium  DribbleEff = low 50%38% Van PersieShotEff = high AND TimePlayed = high  ShotsOnTarget = high 70%50% Confidence = conditional probability

10/13 Distribution Divergence Perspective Halpern, “An analysis of first-order logics of probability”, AI Journal de Raedt, L. (2008), Logical and Relational Learning, Springer. Ch.9 Joint Value AssignmentsFrequency for Random Striker Frequency for P=van Persie SavesMade(P,M)=low AND shotsOnTarget(P,M)=low AND ShotEff(P,M)=low 22%10% SavesMade(P,M)=low AND shotsOnTarget(P,M)=high AND ShotEff(P,M)=high 30%62% ….... Outlier Score = Dissimilarity measure between Random Individual and Target Individual. In our work, dissimilarity measure = distribution divergence. Could leverage other distance-type metrics as well.

11/13 Propositionalization for Outlier Detection Lippi, M.; Jaeger, M.; Frasconi, P. & Passerini, A. (2011), 'Relational information gain', Machine Learning 83(2), 219—239. PlayersSavesMade(P,M)=med AND shotsOnTarget(P,M)=low AND ShotEff(P,M)=low SavesMade(P,M)=med AND shotsOnTarget(P,M)=high AND ShotEff(P,M)=high (331 more) Wayne Rooney 13%10%... van Persie 50%62%... ….... Construct 331-dimensional attribute vector for each individual. One frequency/count value for each formula  pseudo-i.i.d data view. Like n-grams. Apply standard single-table analysis methods. Could also use learned weights instead of sufficient statistics.

12/13 Propositionalization Results A simple method for multi-relational outlier detection LowCor = Normals have low correlation. HighCor = Normals have high correlation.

13/13 Summary Outlier detection based on a statistical-relational model. Basic Idea: compare individual profile to entire population. Leverage parameter learning: 1. Learn parameter values for individual. 2. Learn parameter values for entire population. 3. Outlier score = parameter vector difference. E.g. average L1-distance. Leverage relational distance between individuals. In our work, distance ≈ distribution divergence. Outlier score = divergence between individual distribution and population distribution. Another approach: Model-based propositionalization for outlier detection. Attribute-values = frequency counts for patterns in model structure. A simple method for multi-relational outlier detectiona