Temporal Causal Modeling with Graphical Granger Methods

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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)

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Talk Outline

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

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A Motivating Example Key Performance Indicator
Data (KPI)in Corporate Index Management SP
Variables
Company HAL HAL HAL HAL HAL HAL HAL
Year 1999 2000 2000 2000 2000 2001 2001
Quarter 4 1 2 3 4 1 2
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
Time
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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 HAL HAL HAL HAL HAL HAL
Year 1999 2000 2000 2000 2000 2001 2001
Quarter 4 1 2 3 4 1 2
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
Time
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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

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Variable Space Expansion Feature Space Mapping
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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

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Exhaustive Granger vs. Lasso Granger
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Baseline methods SIN and VAR

• SIN
• VAR

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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

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Experiment 1A Performance vs. Factors- Random
sampling all factors -
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Experiment 1s Efficiency
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Experiment 1B Performance vs. Factors- Fixing
other factors -
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Experiment 1C Performance vs. Factors- Detail
Parametric Conditioning -
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Experiment 2 Learned Graphs
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Experiment 3 Real World DataOutput Graphs on
the Corporate KPI Data