Temporal Causal Modeling with Graphical Granger Methods SIGKDD 07 August 13, 2007 Andrew Arnold (Carnegie Mellon University) Yan Liu (IBM T.J. Watson Research) Naoki Abe (IBM T.J. Watson Research)

Talk Outline Introduction and motivation Overview of Granger causality Graphical Granger methods Exhaustive Granger Lasso Granger SIN Granger Vector auto-regression (VAR) Experimental results

A Motivating Example: Key Performance Indicator Data (KPI) in Corporate Index Management [S&P] Variables Company HAL Year 1999 2000 2001 Quarter 4 1 2 3 Revenue ($M) 6.24 6.54 5.82 3.89 4.1 4.41 3.6 Revenue-to-RD 2.185704 1.734358 1.381822 0.416212 0.843057 0.906083 0.930714 Revenue-to-RD CAGR -0.61429 -0.47757 -0.32646 Innovation Index 0.517621 0.578062 0.567874 0.98624 0.696722 0.679335 .734627 Innovation Index CAGR 0.346008 0.175194 .229845 CapEx to Revenue 0.152292 0.258789 0.111111 0.63592 1.33114 1.389658 0.009722 compound annual growth rate Time

KPI Case Study: Temporal Causal Modeling for Identifying Levers of Corporate Performance How can we leverage information in temporal data to assist causal modeling and inference ? Key idea: A cause necessarily precedes its effects… Variables Company HAL Year 1999 2000 2001 Quarter 4 1 2 3 Revenue ($M) 6.24 6.54 5.82 3.89 4.1 4.41 3.6 Revenue-to-RD 2.185704 1.734358 1.381822 0.416212 0.843057 0.906083 0.930714 Revenue-to-RD CAGR -0.61429 -0.47757 -0.32646 Innovation Index 0.517621 0.578062 0.567874 0.98624 0.696722 0.679335 .734627 Innovation Index CAGR 0.346008 0.175194 .229845 CapEx to Revenue 0.152292 0.258789 0.111111 0.63592 1.33114 1.389658 0.009722 compound annual growth rate Time

Granger Causality Granger causality Introduced by the Nobel prize winning economist, Clive Granger [Granger ‘69] Definition: a time series x is said to “Granger cause” another time series y, if and only if: regressing for y in terms of past values of both y and x is statistically significantly better than regressing y on past values of y only Assumption: no common latent causes

Variable Space Expansion & Feature Space Mapping Fewer edges Fewer variables

Graphical Granger Methods Exhaustive Granger Test all possible univariate Granger models independently Lasso Granger Use L1-normed regression to choose sparse multivariate regression models [Meinshausen & Buhlmann, ‘06] SIN Granger Do matrix inversion to find correlations between features across time [Drton & Perlman, ‘04] Vector auto-regression (VAR) Fit data to linear-normal time series model [Gilbert, ‘95] Too many variables. How can we condense.

Exhaustive Granger vs. Lasso Granger Too many variables. How can we condense.

Baseline methods: SIN and VAR

Empirical Evaluation of Competing Methods Evaluation by simulation Sample data from synthetic (linear normal) causal model Learn using a number of competing methods Compare learned graphs to original model Measure similarity of output graph to original graph in terms of Precision of predicted edges Recall of predicted edges F1 of predicted edges Parameterize performance analysis Randomly sample graphs from parameter space Lag; Features; Affinity; Noise; Samples per feature; Samples per feature per lag Conditioning to see interaction effects E.g. Effect of # features when samples_per_feature_per_lag is small vs large

Experiment 1A: Performance vs. Factors - Random sampling all factors -

Experiment 1’s Efficiency

Experiment 1B: Performance vs. Factors - Fixing other factors - 13 13

Experiment 1C: Performance vs Experiment 1C: Performance vs. Factors - Detail: Parametric Conditioning -

Experiment 2: Learned Graphs

Experiment 3: Real World Data Output Graphs on the Corporate KPI Data