Time-Frequency Causal Inference Uncovers Anomalous Events in Environmental Systems
Abstract
Causal inference in dynamical systems is a challenge for different research areas. So far it is mostly about understanding to what extent the underlying causal mechanisms can be derived from observed time series. Here we investigate whether anomalous events can also be identified based on the observed changes in causal relationships. We use a parametric time-frequency representation of vector autoregressive Granger causality for causal inference. The use of time-frequency approach allows for dealing with the nonstationarity of the time series as well as for defining the time scale on which changes occur. We present two representative examples in environmental systems: land-atmosphere ecosystem and marine climate. We show that an anomalous event can be identified as the event where the causal intensities differ according to a distance measure from the average causal intensities. The driver of the anomalous event can then be identified based on the analysis of changes in the causal effect relationships.
Keywords
Time-frequency causality analysis Vector Autoregressive Granger Causality Attribution of anomalous eventsNotes
Acknowledgments
The authors thank the Carl Zeiss Foundation for the financial support within the scope of the program line “Breakthroughs: Exploring Intelligent Systems” for “Digitization—explore the basics, use applications”. This work used eddy covariance data acquired and shared by the FLUXNET community.
References
- 1.Barnett, L., Seth, A.K.: The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference. J. Neurosci. Methods 223, 50–68 (2014)CrossRefGoogle Scholar
- 2.Wan, E.A., Nelson, A.T.: Dual Extended Kalman Filter Methods, pp. 123–173. Wiley-Blackwell (2002). https://doi.org/10.1002/0471221546.ch5. Chapter 5
- 3.Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19(6), 716–723 (1974). https://doi.org/10.1109/TAC.1974.1100705MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Anderson, T.: The Statistical Analysis of Time Series. Wiley Classics Library. Wiley, New York (1994)CrossRefGoogle Scholar
- 5.Attanasio, A., Pasini, A., Triacca, U.: Granger causality analyses for climatic attribution. Atmos. Clim. Sci. 3(4), 515–522 (2013). https://doi.org/10.4236/acs.2013.34054CrossRefGoogle Scholar
- 6.Baccalá, L.A., Sameshima, K., Takahashi, D.: Generalized partial directed coherence. In: 15th International Conference on Digital Signal Processing, pp. 163–166. IEEE (2007)Google Scholar
- 7.Barnett, L., Seth, A.K.: Behaviour of Granger causality under filtering: theoretical invariance and practical application. J. Neurosci. Methods 201(2), 404–419 (2011). https://doi.org/10.1016/j.jneumeth.2011.08.010CrossRefGoogle Scholar
- 8.Barz, B., Guanche, Y., Rodner, E., Denzler, J.: Maximally divergent intervals for extreme weather event detection. In: MTS/IEEE OCEANS Conference Aberdeen, pp. 1–9 (2017). https://doi.org/10.1109/OCEANSE.2017.8084569
- 9.Eichler, M.: Graphical modelling of multivariate time series. Probab. Theory Relat. Fields 153(1), 233–268 (2012). https://doi.org/10.1007/s00440-011-0345-8MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Faes, L., Porta, A., Nollo, G.: Testing frequency-domain causality in multivariate time series. IEEE Trans. Biomed. Eng. 57(8), 1897–1906 (2010)CrossRefGoogle Scholar
- 11.Faghmous, J.H., Kumar, V.: A big data guide to understanding climate change: the case for theory-guided data science. Big Data 2(3), 155–163 (2014)CrossRefGoogle Scholar
- 12.Feldhoff, J., Donner, R.V., Donges, J.F., Marwan, N., Kurths, J.: Detection of coupling directions by means of inter-system recurrence networks. Phys. Lett. A 376, 3504–3513 (2012)CrossRefGoogle Scholar
- 13.Frank, P.: Analytical and qualitative model-based fault diagnosis - a survey and some new results. Eur. J. Control 2(1), 6–28 (1996). https://doi.org/10.1016/S0947-3580(96)70024-9CrossRefzbMATHGoogle Scholar
- 14.Geweke, J.: Measurement of linear dependence and feedback between multiple time series. J. Am. Stat. Assoc. 77(378), 304–313 (1982)MathSciNetCrossRefGoogle Scholar
- 15.Granger, C.W.J.: Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37(3), 424–438 (1969). http://www.jstor.org/stable/1912791CrossRefGoogle Scholar
- 16.Granger, C.W.: Investigating causal relations by econometric models and cross-spectral methods. Econometrica J. Econometric Soc. 37, 424–438 (1969)CrossRefGoogle Scholar
- 17.Haykin, S.: Adaptive Filter Theory, 3rd edn. Prentice-Hall Inc., Upper Saddle River (1996)zbMATHGoogle Scholar
- 18.Mahecha, M.D., et al.: Detecting impacts of extreme events with ecological in situ monitoring networks. Biogeosciences 14(18), 4255–4277 (2017). https://doi.org/10.5194/bg-14-4255-2017CrossRefGoogle Scholar
- 19.Marinazzo, D., Liao, W., Chen, H., Stramaglia, S.: Nonlinear connectivity by Granger causality. NeuroImage 58(2), 330–338 (2011). https://doi.org/10.1016/j.neuroimage.2010.01.099CrossRefGoogle Scholar
- 20.Papagiannopoulou, C., et al.: A non-linear Granger-causality framework to investigate climate-vegetation dynamics. Geoscientific Model Dev. 10(5), 1945–1960 (2017). https://doi.org/10.5194/gmd-10-1945-2017CrossRefGoogle Scholar
- 21.Peters, J., Janzing, D., Schölkopf, B.: Elements of Causal Inference - Foundations and Learning Algorithms. Adaptive Computation and Machine Learning Series. The MIT Press, Cambridge (2017)zbMATHGoogle Scholar
- 22.Rambal, S., Joffre, R., Ourcival, J.M., Cavender-Bares, J., Rocheteau, A.: The growth respiration component in Eddy CO\(_2\) flux from a quercus ilex mediterranean forest. Glob. Change Biol. 10(9), 1460–1469 (2004). https://doi.org/10.1111/j.1365-2486.2004.00819.xGoogle Scholar
- 23.Reichstein, M., Camps-Valls, G., Stevens, B., Jung, M., Denzler, J., Carvalhais, N., Prabhat: Deep learning and process understanding for data-driven earth system science. Nature 195–204 (2019). https://doi.org/10.1038/s41586-019-0912-1
- 24.Schwarz, G.: Estimating the dimension of a model. Ann. Statist. 6(2), 461–464 (1978). https://doi.org/10.1214/aos/1176344136MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Seth, A.K., Barrett, A.B., Barnett, L.: Granger causality analysis in neuroscience and neuroimaging. J. Neurosci. 35(8), 3293–3297 (2015). https://doi.org/10.1523/JNEUROSCI.4399-14.2015CrossRefGoogle Scholar
- 26.Shadaydeh, M., Garcia, Y.G., Mahecha, M., Reichstein, M., Denzler, J.: Causality analysis of ecological time series: a time-frequency approach. In: Chen, C., Cooley, D., Runge, J., Szekely, E. (eds.) Climate Informatics Workshop 2018, pp. 111–114 (2018)Google Scholar
- 27.Solo, V.: State-space analysis of Granger-Geweke causality measures with application to fMRI. Neural Comput. 28(5), 914–949 (2016). https://doi.org/10.1162/NECO_a_00828. pMID: 26942749MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Takahashi, D.Y., Baccal, L.A., Sameshima, K.: Connectivity inference between neural structures via partial directed coherence. J. Appl. Stat. 34(10), 1259–1273 (2007). https://doi.org/10.1080/02664760701593065MathSciNetCrossRefGoogle Scholar
- 29.Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., Farmer, J.D.: Testing for nonlinearity in time series: the method of surrogate data. Physica D 58(1), 77–94 (1992). https://doi.org/10.1016/0167-2789(92)90102-SCrossRefzbMATHGoogle Scholar
- 30.Trifunov, V.T., Shadaydeh, M., Runge, J., Eyring, V., Reichstein, M., Denzler, J.: Nonlinear causal link estimation under hidden confounding with an application to time series anomaly detection. In: German Conference on Pattern Recognition (2019)Google Scholar
- 31.Zhong, M., Xue, T., Ding, S.X.: A survey on model-based fault diagnosis for linear discrete time-varying systems. Neurocomputing 306, 51–60 (2018). https://doi.org/10.1016/j.neucom.2018.04.037CrossRefGoogle Scholar