1. Granger Causality for Time-Series Anomaly Detection By Zhangzhou

2. Introduction&Background Time-Series Data Conception & Examples & Features

3. Time-Series Model Static model Y t = β 0 + β z t + μ t Finite Distributed Lag model,FDL gfr t = α 0 + ξ 0 pe t + ξ 1 pe t-1 + ξ 2 pe t-2 + μ t

4. Multivariate time series Vector Auto-Regression(VAR)

5. Granger Causality For a VAR(p)

6. Problem Definition There usually exist two types of anomalies in multivariate time-series data : “univariate anomaly” and “dependency anomaly” Solution : investigate Granger graphical models,which uncover the temporal dependencies between variables

7. The Lasso Granger Method λ is the penalty parameter, the Xi Granger causers Xj if at least one value in βis nonzero by statistical significant tests.

8. Granger Graphical Models for Anomaly Detection

9. Detection of dependency anomaly(GGM) Learning temporal causal graph of D(b) by regularization Computing the anomaly scores of D(b) using KL-divergence Determining anomaly by threshold cutoff

10. Learning temporal causal graphs Null hypothesis : the temporal causal graphs of reference set and test set are the same, we can use the temporal graphs as additional constraint in Lasso-Granger algorithm

11. Procedure Lasso-Granger(X,T)

12. Computing anomaly scores Kullback-Leibler(KL) divergence, for a particular time-series Xi, we can define its anomaly score as follows:

13. Determine anomaly by threshold cutoff and Slide a window through the reference data and calculate the anomaly scores for each window. Then use the scores to approximate the distribution of the anomaly scores and use the α-quantile of this distribution as threshold cutoff.

14. Experiments