×

zbMATH — the first resource for mathematics

On the spectral formulation of Granger causality. (English) Zbl 1248.92006
Summary: Spectral measures of causality are used to explore the role of different rhythms in the causal connectivity between brain regions. We study several spectral measures related to Granger causality, comprising the bivariate and conditional Geweke measures, the directed transfer function, and the partial directed coherence. We derive the formulation of dependence and causality in the spectral domain from the more general formulation in the information-theory framework. We argue that the transfer entropy, the most general measure derived from the concept of Granger causality, lacks a spectral representation in terms of only the processes associated with the recorded signals. For all the spectral measures we show how they are related to mutual information rates when explicitly considering the parametric autoregressive representation of the processes. In this way we express the conditional Geweke spectral measure in terms of a multiple coherence involving innovation variables inherent to the autoregressive representation. We also link partial directed coherence with Sims’ criterion of causality. Given our results, we discuss the causal interpretation of the spectral measures related to Granger causality and stress the necessity to explicitly consider their specific formulation based on modeling the signals as linear Gaussian stationary autoregressive processes.

MSC:
92C20 Neural biology
92B25 Biological rhythms and synchronization
62M15 Inference from stochastic processes and spectral analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amblard PO, Michel O (2011) On directed information theory and Granger causality graphs. J Comput Neurosci 30(1): 7–16 · doi:10.1007/s10827-010-0231-x
[2] Baccalá L, Sameshima K (2001) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84(1): 463–474 · Zbl 1160.92306 · doi:10.1007/PL00007990
[3] Baccalá L, Sameshima K, Ballester G, Do Valle A, Timo-Iaria C (1999) Studying the interaction between brain structures via directed coherence and Granger causality. Appl Signal Process 5: 40–48 · doi:10.1007/s005290050005
[4] Barnett L, Barrett AB, Seth AK (2009) Granger causality and transfer entropy are equivalent for Gaussian variables. Phys Rev Lett 103(23): 238701 · doi:10.1103/PhysRevLett.103.238701
[5] Bernasconi C, König P (1999) On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings. Biol Cybern 81(3): 199–210 · Zbl 0957.92005 · doi:10.1007/s004220050556
[6] Bernasconi C, von Stein A, Chiang C, König P (2000) Bi-directional interactions between visual areas in the awake behaving cat. Neuroreport 11(4): 689–692 · doi:10.1097/00001756-200003200-00007
[7] Besserve M, Schoelkopf B, Logothetis NK, Panzeri S (2010) Causal relationships between frequency bands of extracellular signals in visual cortex revealed by an information theoretic analysis. J Comput Neurosci 29(3): 547–566 · doi:10.1007/s10827-010-0236-5
[8] Bressler SL, Seth AK (2011) Wiener Granger causality: a well established methodology. Neuroimage 58(2): 323–329 · doi:10.1016/j.neuroimage.2010.02.059
[9] Bressler SL, Richter CG, Chen Y, Ding M (2007) Cortical functional network organization from autoregressive modeling of local field potential oscillations. Stat Med 26(21): 3875–3885 · doi:10.1002/sim.2935
[10] Bressler SL, Tang W, Sylvester CM, Shulman GL, Corbetta M (2008) Top-down control of human visual cortex by frontal and parietal cortex in anticipatory visual spatial attention. J Neurosci 28(40): 10056–10061 · doi:10.1523/JNEUROSCI.1776-08.2008
[11] Brillinger D (1981) Time series. Data analysis and theory. Holden-Day, San Francisco · Zbl 0486.62095
[12] Brovelli A, Ding M, Ledberg A, Chen Y, Nakamura R, Bressler SL (2004) Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. Proc Natl Acad Sci USA 101: 9849–9854 · doi:10.1073/pnas.0308538101
[13] Buzsáki G (2006) Rhythms of the brain. Oxford University Press, New York · Zbl 1204.92017
[14] Chamberlain G (1982) The general equivalence of Granger and Sims causality. Econometrica 50(3): 569–581 · Zbl 0483.60022 · doi:10.2307/1912601
[15] Chen Y, Bressler S, Ding M (2006) Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. J Neurosci Methods 150(2): 228–237 · doi:10.1016/j.jneumeth.2005.06.011
[16] Cover TM, Thomas JA (2006) Elements of information theory, 2nd ed. Wiley, New York
[17] Dhamala M, Rangarajan G, Ding M (2008) Estimating Granger causality from fourier and wavelet transforms of time series data. Phys Rev Lett 100(1): 018701 · doi:10.1103/PhysRevLett.100.018701
[18] Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and application to neuroscience. In: Schelter B, Winterhalder M, Timmer J (eds) Handbook of time series analysis: recent theoretical developments and applications. Weinheim, Wiley-VCH Verlag, pp 437–460 · Zbl 1268.92079
[19] Eichler M (2006) On the evaluation of information flow in multivariate systems by the directed transfer function. Biol Cybern 94(6): 469–482 · Zbl 1138.62048 · doi:10.1007/s00422-006-0062-z
[20] Florens J (2003) Some technical issues in defining causality. J Econ 112: 127–128
[21] Fries P (2005) A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn Sci 9(10): 474–480 · doi:10.1016/j.tics.2005.08.011
[22] Friston KJ (1994) Functional and effective connectivity in neuroimaging: a synthesis. Hum Brain Mapp 2(5): 56–78 · doi:10.1002/hbm.460020107
[23] Gelfand I, Yaglom A (1959) Calculation of the amount of information about a random function contained in another such function. Am Math Soc Transl Ser 2(12): 199–246 · Zbl 0087.13201
[24] Geweke JF (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77(378): 304–313 · Zbl 0492.62078 · doi:10.1080/01621459.1982.10477803
[25] Geweke JF (1984) Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 79(388): 907–915 · Zbl 0553.62083 · doi:10.1080/01621459.1984.10477110
[26] Gourevitch B, Le Bouquin-Jeannes R, Faucon G (2006) Linear and nonlinear causality between signals: methods, examples and neurophysiological applications. Biol Cybern 95(4): 349–369 · Zbl 1161.62429 · doi:10.1007/s00422-006-0098-0
[27] Gourieroux C, Monfort A, Renault E (1987) Kullback causality measures. Ann Econ Stat 6/7:369–410
[28] Granger CWJ (1963) Economic processes involving feedback. Inf Control 6: 28–48 · Zbl 0123.37502 · doi:10.1016/S0019-9958(63)90092-5
[29] Granger CWJ (1980) Testing for causality: a personal viewpoint. J Econ Dyn Control 2(1): 329–352 · doi:10.1016/0165-1889(80)90069-X
[30] Guo S, Seth AK, Kendrick KM, Zhou C, Feng J (2008a) Partial Granger causality-eliminating exogenous inputs and latent variables. J Neurosci Methods 172(1): 79–93 · doi:10.1016/j.jneumeth.2008.04.011
[31] Guo S, Wu J, Ding M, Feng J (2008) Uncovering interactions in the frequency domain. PLoS Comput Biol 4(5): e1000087 · doi:10.1371/journal.pcbi.1000087
[32] Kaminski M, Blinowska K (1991) A new method of the description of the information flow in the brain structures. Biol Cybern 65(3): 203–210 · Zbl 0734.92003 · doi:10.1007/BF00198091
[33] Kaminski M, Ding M, Truccolo W, Bressler S (2001) Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol Cybern 85(2): 145–157 · Zbl 1160.92314 · doi:10.1007/s004220000235
[34] Kolmogorov A (1939) Sur l’interpolation et extrapolation des suites stationnaires. Comp Rend Acad Sci Paris 208: 2043–2045 · JFM 65.0607.04
[35] Kuersteiner G (2008) Granger-Sims causality. The new palgrave dictionary of economics, 2nd ed. MacMillan, Bedford
[36] Ladroue C, Guo S, Kendrick K, Feng J (2009) Beyond element-wise interactions: identifying complex interactions in biological processes. PLoS ONE 4(9): e6899 · doi:10.1371/journal.pone.0006899
[37] Marko H (1973) Bidirectional communication theory–generalization of information-theory. IEEE Trans Commun 12: 1345–1351 · doi:10.1109/TCOM.1973.1091610
[38] Nedungadi AG, Rangarajan G, Jain N, Ding M (2009) Analyzing multiple spike trains with nonparametric Granger causality. J Comput Neurosci 27(1): 55–64 · Zbl 05784949 · doi:10.1007/s10827-008-0126-2
[39] Pereda E, Quian Quiroga R, Bhattacharya J (2005) Nonlinear multivariate analysis of neurophysiological signals. Prog Neurobiol 77: 1–37 · doi:10.1016/j.pneurobio.2005.10.003
[40] Priestley M (1981) Spectral analysis and time series. Academic Press Inc., San Diego · Zbl 0537.62075
[41] Rissanen J, Wax M (1987) Measures of mutual information and causal dependence between 2 time-series. IEEE Trans Inf Theory 33(4): 598–601 · Zbl 0626.62090 · doi:10.1109/TIT.1987.1057325
[42] Rozanov YA (1967) Stationary random processes. Holden-Day, San Francisco · Zbl 0152.16302
[43] Schelter B, Winterhalder M, Eichler M, Peifer M, Hellwig B, Guschlbauer B, Lucking C, Dahlhaus R, Timmer J (2006) Testing for directed influences among neural signals using partial directed coherence. J Neurosci Methods 152(1-2): 210–219 · doi:10.1016/j.jneumeth.2005.09.001
[44] Schreiber T (2000) Measuring information transfer. Phys Rev Lett 85: 461–464 · doi:10.1103/PhysRevLett.85.461
[45] Sims C (1972) Money, income, and causality. Am Econ Rev 62(4): 540–552
[46] Solo V (2008) On causality and mutual information. In: Proceedings of the 47th IEEE conference on decision and control, pp 4939–4944
[47] Takahashi DY, Baccala LA, Sameshima K (2010) Information theoretic interpretation of frequency domain connectivity measures. Biol Cybern 103(6): 463–469 · Zbl 1403.92046 · doi:10.1007/s00422-010-0410-x
[48] Wiener N (1956) The theory of prediction. In: Beckenbach EF (eds) Modern mathematics for engineers. McGraw-Hill, New York
[49] Winterhalder M, Schelter B, Hesse W, Schwab K, Leistritz L, Klan D, Bauer R, Timmer J, Witte H (2005) Comparison directed of linear signal processing techniques to infer interactions in multivariate neural systems. Signal Process 85(11): 2137–2160 · Zbl 1160.94369 · doi:10.1016/j.sigpro.2005.07.011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.