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Relating the Network Graphs of State-Space Representations to Granger Causality Conditions

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Hybrid and Networked Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 493))

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Abstract

In this chapter, we will discuss the problem of estimating the network graphs of state-space representations based on observed data: we observe the output generated by each node of a network of state-space representations, and we would like to reconstruct the communication graph of the network, i.e., we would like to find out which nodes exchange information. The potential exchange of information is not assumed to be observable, i.e., it may take place via hidden internal states. We present an approach based on the notion of Granger causality. The essence of this approach is that there exists a communication link between two nodes, if the outputs generated by the corresponding nodes are related by Granger causality. More precisely, we show an equivalence between the existence of state-space representation in which subsystems corresponding to certain nodes exchange information, and the presence of Granger causality relation between the outputs generated by those subsystems. Since Granger causality can be checked based on observed data, these results open up the possibility of data-driven reverse engineering of the communication graph. We will discuss the case of stochastic linear time-invariant systems, and then the case of stochastic bilinear/LPV/switched systems.

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Notes

  1. 1.

    Stationarity implies that the (co)variance matrices are time-independent.

  2. 2.

    For any other symmetric solution \(\tilde{\Sigma }\), the matrix \(\tilde{\Sigma }\!-\!\Sigma \) is positive definite.

  3. 3.

    See [34, Definition 9.4.1] for the definition of coercivity.

References

  1. Amblard, P.-O., Michel, O.J.: On directed information theory and granger causality graphs. J. Comput. Neurosci. 30(1), 7–16 (2011)

    Article  MathSciNet  Google Scholar 

  2. Barnett, L., Barrett, A.B., Seth, A.K.: Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett. 103, 238701 (2009)

    Article  Google Scholar 

  3. Bolognani, S., Bof, N., Michelotti, D., Muraro, R., Schenato, L.: Identification of power distribution network topology via voltage correlation analysis. In: 52nd IEEE Conference on Decision and Control, pp. 1659–1664 (2013)

    Google Scholar 

  4. Bombois, X., Colin, K., Van Den Hof, P.M.J., Hjalmarsson, H.: On the informativity of direct identification experiments in dynamical networks. Working paper or preprint, August 2021

    Google Scholar 

  5. Caines, P.E.: Weak and strong feedback free processes. IEEE Trans. Autom. Control 21(5), 737–739 (1976)

    Article  MathSciNet  Google Scholar 

  6. Caines, P.E., Chan, C.: Feedback between stationary stochastic processes. IEEE Trans. Autom. Control 20(4), 498–508 (1975)

    Article  MathSciNet  Google Scholar 

  7. Caines, P.E., Deardon, R., Wynn, H.P.: Conditional orthogonality and conditional stochastic realization. In: Rantzer, A., Byrnes, C.I. (eds.) Directions in Mathematical Systems Theory and Optimization, vol. 286, pp. 71–84. Springer, Berlin, Heidelberg (2003)

    Google Scholar 

  8. Caines, P.E., Deardon, R., Wynn, H.P.: Bayes nets of time series: stochastic realizations and projections. In: Pronzato, L., Zhigljavsky, A. (eds.) Optimal Design and Related Areas in Optimization and Statistics. Springer Optimization and its Applications, vol. 28, pp. 155–166. Springer, New York (2009)

    Chapter  Google Scholar 

  9. Caines, P.E.: Linear Stochastic Systems. John Wiley and Sons, New-York (1998)

    Google Scholar 

  10. Dankers, A.G.: System identification in dynamic networks. Ph.D. thesis, Delft University of Technology (2014)

    Google Scholar 

  11. David, O.: fMRI connectivity, meaning and empiricism: comments on: Roebroeck et al. the identification of interacting networks in the brain using fMRI: model selection, causality and deconvolution. NeuroImage 58(2), 306–309 (2011)

    Google Scholar 

  12. Eichler, M.: Granger causality and path diagrams for multivariate time series. J. Econ. 137(2), 334–353 (2007)

    Article  MathSciNet  Google Scholar 

  13. Eichler, M.: Graphical modelling of multivariate time series. Probab. Theory Relat. Fields 153(1), 233–268 (2012)

    Article  MathSciNet  Google Scholar 

  14. Friston, K.J., Harrison, L., Penny, W.: Dynamic causal modeling. NeuroImage 19(4), 1273–1302 (2003)

    Article  Google Scholar 

  15. Gevers, M.R., Anderson, B.: On jointly stationary feedback-free stochastic processes. IEEE Trans. Autom. Control 27(2), 431–436 (1982)

    Article  Google Scholar 

  16. Goebel, R., Roebroeck, A., Kim, D.-S., Formisano, E.: Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping. Magn. Reson. Imaging 21, 1251–1261 (2003)

    Article  Google Scholar 

  17. Gonçalves, J., Howes, R., Warnick, S.: Dynamical structure functions for the reverse engineering of LTI networks. In: 46th IEEE Conference on Decision and Control, pp. 1516–1522 (2007)

    Google Scholar 

  18. Granger, C.W.J.: Economic processes involving feedback. Inf. Control 6(1), 28–48 (1963)

    Article  MathSciNet  Google Scholar 

  19. Havlicek, M., Roebroeck, A., Friston, K., Gardumi, A., Ivanov, D., Uludag, K.: Physiologically informed dynamic causal modeling of fMRI data. NeuroImage 122, 355–372 (2015)

    Article  Google Scholar 

  20. Howes, R., Eccleston, L., Gonçalves, J., Stan, G.-B., Warnick, S.: Dynamical structure analysis of sparsity and minimality heuristics for reconstruction of biochemical networks. In: 47th IEEE Conference on Decision and Control (2008)

    Google Scholar 

  21. Hsiao, C.: Autoregressive modelling and causal ordering of econometric variables. J. Econ. Dyn. Control 4, 243–259 (1982)

    Article  Google Scholar 

  22. Isidori, A.: Direct construction of minimal bilinear realizations from nonlinear input-output maps. IEEE Trans. Autom. Control 626–631 (1973)

    Google Scholar 

  23. Jozsa, M.: Relationship between Granger non-causality and network graphs of state-space representations. Ph.D. thesis, University of Groningen (2019)

    Google Scholar 

  24. Jozsa, M., Petreczky, M., Camlibel, M.K.: Causality based graph structure of stochastic linear state-space representations. In: 56th IEEE Conference on Decision and Control, pp. 2442–2447 (2017)

    Google Scholar 

  25. Jozsa, M., Petreczky, M., Camlibel, M.K.: Relationship between causality of stochastic processes and zero blocks of their joint innovation transfer matrices. In: 20th World Congress of the International Federation of Automatic Control, pp. 4954–4959 (2017)

    Google Scholar 

  26. Jozsa, M., Petreczky, M., Camlibel, M.K.: Relationship between Granger non-causality and network graph of state-space representations. IEEE Trans. Autom. Control 64(8), 912–927 (2019)

    Article  Google Scholar 

  27. Jozsa, M., Petreczky, M., Camlibel, M.K.: Causality and network graph in general bilinear state-space representations. IEEE Trans. Autom. Control 65(8), 3623–3630 (2020)

    Article  MathSciNet  Google Scholar 

  28. Julius, A.A., Zavlanos, M., Boyd, S., Pappas, G.J.: Genetic network identification using convex programming. IET Syst. Biol. 3, 155–166 (2009)

    Google Scholar 

  29. Kang, T., Moore, R., Li, Y., Sontag, E.D., Bleris, L.: Discriminating direct and indirect connectivities in biological networks. Natl. Acad. Sci. USA 112, 12893–12898 (2015)

    Article  Google Scholar 

  30. Katayama, T.: Subspace Methods for System Identification. Springer-Verlag (2005)

    Google Scholar 

  31. Kempker, P.L.: Coordination control of linear systems. Ph.D. thesis, Amsterdam: Vrije Universiteit (2012)

    Google Scholar 

  32. Kempker, P.L., Ran, A.C.M., van Schuppen, J.H.: Construction and minimality of coordinated linear systems. Linear Algebr. Appl. 452, 202–236 (2014)

    Article  MathSciNet  Google Scholar 

  33. Kramer, G.: Directed information for channels with feedback. Ph.D. thesis, Swiss Federal Institute of Technology Zürich (1998)

    Google Scholar 

  34. Lindquist, A., Picci, G.: Linear Stochastic Systems: a Geometric Approach to Modeling. Estimation and Identification, Springe, Berlin (2015)

    Book  Google Scholar 

  35. Massey, J.L.: Causality, feedback and directed information. In: International Symposium on Information Theory and its Applications, pp. 27–30 (1990)

    Google Scholar 

  36. Nordling, T.E.M., Jacobsen, E.W.: On sparsity as a criterion in reconstructing biochemical networks. In: 18th IFAC World Congress, pp. 11672–11678 (2011)

    Google Scholar 

  37. Pambakian, N.: LQG coordination control. Master’s thesis, Delft University of Technology (2011)

    Google Scholar 

  38. Pearl, J.: Causality: Models, Reasoning and Inference, 1st edn. Cambridge University Press (2000)

    Google Scholar 

  39. Penny, W.D., Stephan, K.E., Mechelli, A., Friston, K.J.: Comparing dynamic causal models. NeuroImage 22(3), 1157–1172 (2004)

    Article  Google Scholar 

  40. Petreczky, M., Vidal, R.: Realization theory for a class of stochastic bilinear systems. IEEE Trans. Autom. Control 63(1), 69–84 (2017)

    Article  MathSciNet  Google Scholar 

  41. Ran, A.C.M., van Schuppen, J.H.: Coordinated linear systems. In: Coordination Control of Distributed Systems. Lecture Notes in Control and Information Sciences, vol. 456, pp. 113–121 (2014)

    Google Scholar 

  42. Roebroeck, A., Formisano, E., Goebel, R.: Reply to Friston and David: after comments on: the identification of interacting networks in the brain using fMRI: model selection, causality and deconvolution. NeuroImage 58(2), 310–311 (2011)

    Article  Google Scholar 

  43. Roebroeck, A., Seth, A.K., Valdes-Sosa, P.A.: Causal time series analysis of functional magnetic resonance imaging data. J. Mach. Learn. Res. Proc. Track 12, 65–94 (2011)

    Google Scholar 

  44. Valdes-Sosa, P.A., Roebroeck, A., Daunizeau, J., Friston, K.J.: Effective connectivity: influence, causality and biophysical modeling. NeuroImage 58(2), 339–361 (2011)

    Article  Google Scholar 

  45. Van den Hof, P.M.J., Dankers, A., Heuberger, P.S.C., Bombois, X.: Identification of dynamic models in complex networks with prediction error methods-basic methods for consistent module estimates. Automatica 49(10), 2994–3006 (2013)

    Article  MathSciNet  Google Scholar 

  46. Weerts, H.H.M.: Identifiability and identification methods for dynamic networks. Ph.D. thesis, Eindhoven University of Technology (2018)

    Google Scholar 

  47. Weerts, H.H.M., Van den Hof, P.M.J., Dankers, A.G.: Identifiability of linear dynamic networks. Automatica 89, 247–258 (2018)

    Article  MathSciNet  Google Scholar 

  48. Yuan, Y., Glover, K., Gonçalves, J.: On minimal realisations of dynamical structure functions. Automatica 55, 159–164 (2015)

    Article  MathSciNet  Google Scholar 

  49. Zhang, W., Liu, W., Zang, C., Liu, L.: Multi-agent system based integrated solution for topology identification and state estimation. IEEE Trans. Ind. Inform. 13(2), 714–724 (2017)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Engineering, Control, University of Cambridge, Cambridge, CB2 1PZ, UK

    Mónika Józsa

  2. University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000, Lille, France

    Mihály Petreczky

  3. Johann Bernoulli Institute for Mathematics and Computer Science, and Artificial Intelligence, University of Groningen, 9700 AK, Groningen, The Netherlands

    M. Kanat Camlibel

Authors
  1. Mónika Józsa
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  2. Mihály Petreczky
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  3. M. Kanat Camlibel
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Corresponding author

Correspondence to Mónika Józsa .

Editor information

Editors and Affiliations

  1. Department Contrôle Identification Diagnostic, CNRS, Université de Lorraine, CRAN, Vandoeuvre-lès-Nancy, France

    Romain Postoyan

  2. DANCE INRIA, Gipsa-lab, CNRS, Saint Martin d’Hères, Isère, France

    Paolo Frasca

  3. MODESTY, L2S CNRS, CentraleSupélec, Gif-sur-Yvette, France

    Elena Panteley

  4. MAC, LAAS-CNRS, Toulouse, France

    Luca Zaccarian

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Józsa, M., Petreczky, M., Camlibel, M.K. (2024). Relating the Network Graphs of State-Space Representations to Granger Causality Conditions. In: Postoyan, R., Frasca, P., Panteley, E., Zaccarian, L. (eds) Hybrid and Networked Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 493. Springer, Cham. https://doi.org/10.1007/978-3-031-49555-7_4

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  • DOI: https://doi.org/10.1007/978-3-031-49555-7_4

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