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Frontiers in Computational and Systems Biology pp 83–111Cite as

Granger Causality: Theory and Applications

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Part of the book series: Computational Biology ((COBO,volume 15))

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Notes

  1. 1.

    A lag operator ℒ is such that ℒX t =X t−1. Thus, applying the operator k times yields: ℒk X t =X t−k . The auto-regressive model X t +A 1 X t−1+A 2 X t−2+⋅⋅⋅=ε t can be represented as (Id+A 1ℒ+A 2ℒ2+⋅⋅⋅)X t =ε t .

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Acknowledgements

This work is supported by grants from EPRSC (UK, CARMEN EP/E002331/1) and EU grant (BION) and NSFC (China).

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Authors and Affiliations

  1. Mathematics and Computer Science College, Hunan Normal University, Changsha, 410081, P.R. China

    Shuixia Guo & Jianfeng Feng

  2. Department of Computer Science and Mathematics, Warwick University, Coventry, CV4 7AL, UK

    Christophe Ladroue & Jianfeng Feng

Authors
  1. Shuixia Guo
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  2. Christophe Ladroue
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  3. Jianfeng Feng
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Corresponding author

Correspondence to Jianfeng Feng .

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Editors and Affiliations

  1. Centre for Scientific Computing, Dept. Computer Science & Mathematics, Warwick University, Coventry, CV4 7AL, United Kingdom

    Jianfeng Feng

  2. Dept. Epidemiology, Michigan State University, East Lansing, 48824, Michigan, USA

    Wenjiang Fu

  3. Dept. Biological Sciences, University of Southern California, Los Angeles, 90089, California, USA

    Fengzhu Sun

Appendix: Estimating the Error Covariance Matrix

Appendix: Estimating the Error Covariance Matrix

The main quantity of interest for calculating Granger causality is the covariance matrix of the error term ε in the d-dimensional auto-regressive (AR) model

$$X_t=\sum_{i=1}^pA_iX_{t-i}+\varepsilon (t)$$

The coefficients of the matrices A i need to be estimated from the data. Typically, they are found by minimising the variance of the error between the prediction X t and the observation at the same time t. Morf’s procedure [33] provides a fast and robust way of estimating A i and the covariance matrix of ε via a recursive algorithm. Unless prior knowledge informs us about the likely order p of the AR model (for example by knowing at which time-scale one should expect interactions to take place), it also has to be estimated from the data. The goodness of fit alone is not sufficient for selecting the optimal order: adding a new order implies adding d 2 more unknowns to the system, which considerably improves the fitting simply by increasing the degrees of freedom. The Akaike Information Criterion (AIC, [2]) provides a trade-off between the model complexity (a function of p) and the fit. For an AR model of dimension d, of order p and with T observations, this quantity can be written as

$$\texttt{AIC}(p)=2d^2p+d(T-p)\log \left(2\pi\sum_{\underset{i=1\ldots d}{t=p+1\ldots T}}\frac{ \varepsilon ^2_i(t)}{d(T-p)}\right)$$

The optimal order p is then defined as the one that minimises the AIC. A point estimate of the Granger causality is often not sufficient for making a strong conclusion about the data and a confidence interval is required. In some cases ([25] for partial Granger causality), it is possible to derive the confidence interval in closed form. In general, it is estimated via a bootstrap procedure [32]—given the optimal AR model, an ensemble of signals are generated which each produce a value for the Granger causality. The final estimation of the Granger causality and its confidence interval are defined as the average and the standard error of this set of estimates.

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Guo, S., Ladroue, C., Feng, J. (2010). Granger Causality: Theory and Applications. In: Feng, J., Fu, W., Sun, F. (eds) Frontiers in Computational and Systems Biology. Computational Biology, vol 15. Springer, London. https://doi.org/10.1007/978-1-84996-196-7_5

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