##
**Linear and nonlinear causality between signals: methods, examples and neurophysiological applications.**
*(English)*
Zbl 1161.62429

Summary: We present and review the most usual methods to detect linear and nonlinear causality between signals: the linear Granger causality test extended to direct causality in the multivariate case (LGC), directed coherence, partial directed coherence (PDC), and the nonlinear Granger causality test extended to direct causality in the multivariate case (partial nonlinear Granger causality, PNGC). All these methods are tested and compared on several ARX, Poisson and nonlinear models, and on neurophysiological data (depth EEG). The results show that LGC, DCOH and PDC are not very robust in relation to nonlinear linkages but they seem to correctly find linear linkages if only the autoregressive parts are nonlinear. The PNGC is extremely dependent on the choice of the parameters. Moreover, LGC and PNGC may give misleading results in the case of causality on a spectral band, which is illustrated by our neurophysiological database.

### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

92C20 | Neural biology |

PDF
BibTeX
XML
Cite

\textit{B. Gourévitch} et al., Biol. Cybern. 95, No. 4, 349--369 (2006; Zbl 1161.62429)

Full Text:
DOI

### References:

[1] | Ashley R, Granger CWJ, Schmalensee R (1980) Advertising and aggregate consumption: an analysis of causality. Econometrica 48:1149–1168 · Zbl 0442.90012 |

[2] | Babloyantz A, Destexhe A (1986) Low-dimensional chaos in an instance of epilepsy. Proc Natl Acad Sci USA 83(10):3513–3517 |

[3] | Baccala LA, Sameshima K (2001) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84(6):463–474 · Zbl 1160.92306 |

[4] | Baek E, Brock W (1992) A general test for nonlinear Granger causality. Working Paper University of Iowa |

[5] | Bernasconi C, Konig P (1999) On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings. Biol Cybern 81(3):199–210 · Zbl 0957.92005 |

[6] | Bernasconi C, von Stein A, Chiang C, Konig P (2000) Bi-directional interactions between visual areas in the awake behaving cat. Neuroreport 11(4):689–692 |

[7] | Boudjellaba H, Dufour JM, Roy R (1992) Testing causality between two vectors in multivariate autoregressive moving average models. J Am Stat Assoc 87:1082–1090 · Zbl 0767.62072 |

[8] | 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(26):9849–9854 |

[9] | Caines P, Chan C (1975) Feedback between stationary stochastic processes. IEEE Trans Automat Control 20(4):498–508 · Zbl 0312.60018 |

[10] | Casdagli MC, Iasemidis LD, Savit RS, Gilmore RL, Roper SN, Sackellares JC (1997) Non-linearity in invasive EEG recordings from patients with temporal lobe epilepsy. Electroencephalogr Clin Neurophysiol 102(2):98–105 |

[11] | Cassidy M, Brown P (2003) Spectral phase estimates in the setting of multidirectional coupling. J Neurosci Methods 127(1):95–103 |

[12] | Chartrand G (1985) Introductory graph theory. Dover, New York |

[13] | Chavez M, Martinerie J, Le Van Quyen M (2003) Statistical assessment of nonlinear causality: application to epileptic EEG signals. J Neurosci Methods 124(2):113–128 |

[14] | Chen Y, Rangarajan G, Feng J, Ding M (2004) Analyzing multiple nonlinear time series with extended Granger causality. Phys Lett A 324(1):26–35 · Zbl 1123.62316 |

[15] | Chen Y, Bressler SL, Ding M (2006) Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. J Neurosci Methods 150(2): 228–237 |

[16] | Dahlhaus R, Eichler M (2003) Causality and graphical models for time series. In: Green P, Hjort N, Richardson S (eds) Highly structured stochastic systems. University Press, Oxford |

[17] | De Clercq W, Lemmerling P, Van Huffel S, Van Paesschen W (2003) Anticipation of epileptic seizures from standard EEG recordings. Lancet 361(9361):971; author reply 971 |

[18] | Diks C, Panchenko V (2005) A note on the Hiemstra–Jones test for Granger non-causality. Stud Nonlinear Dynam Econometrics 9(2), art 4:1–7 · Zbl 1082.62510 |

[19] | Diks CG, Degoede J (2001) A general nonparametric bootstrap test for Granger causality. In: Broer HW, Krauskopf W, Vegter G (eds) Global analysis of dynamical systems. Institute of Physics Publishing, Bristol · Zbl 0996.62041 |

[20] | Ding M, Bressler SL, Yang W, Liang H (2000) Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment. Biol Cybern 83(1):35–45 · Zbl 0966.92004 |

[21] | Elbert T, Ray WJ, Kowalik,ZJ, Skinner JE, Graf KE, Birbaumer N (1994) Chaos and physiology: deterministic chaos in excitable cell assemblies. Physiol Rev 74(1):1–47 |

[22] | Frank GW, Lookman T, Nerenberg MAH, Essex C, Lemieux J, Blume W (1990) Chaotic time series analysis of epileptic seizures. Physica D 46:427–438 · Zbl 0713.92010 |

[23] | Freiwald WA, Valdes P, Bosch J, Biscay R, Jimenez JC, Rodriguez LM, Rodriguez V, Kreiter AK, Singer W (1999) Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex. J Neurosci Methods 94(1):105–119 |

[24] | Geweke J (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77:304–313 · Zbl 0492.62078 |

[25] | Geweke J (1984) Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 79: 907–915 · Zbl 0553.62083 |

[26] | Geweke J, Meese R, Dent W (1983) Comparing alternative tests of causality in temporal systems: analytic results and experimental evidence. J Econometrics 21:161–194 · Zbl 0504.62105 |

[27] | Granger CWJ (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37:424–438 · Zbl 1366.91115 |

[28] | Granger CWJ, Newbold P (1977) Forecasting economic time series. Academic, New York |

[29] | Grassberger P (1988) Finite sample correction to entropy and dimension estimates. Phys Lett A 128:369–373 |

[30] | Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189 · Zbl 0593.58024 |

[31] | Grassberger P, Schreiber T, Schaffrath C (1991) Non-linear time sequence analysis. Int J Bif Chaos 1:521–547 · Zbl 0874.58029 |

[32] | Guéguin M, Le Bouquin-Jeannès R, Faucon F, Chauvel P, Liégeois-Chauvel C (2006) Evidence of functional connectivity between auditory cortical areas revealed by amplitude modulation sound processing. Cerebral Cortex doi:10.1093/cercor/bhj148 |

[33] | Hesse W, Moller E, Arnold M, Witte H, Schack B (2002) Brief causal relations in EEG based on adaptive Granger causality. Biomed Tech (Berl) 47(Suppl 1 Pt 2):510–513 |

[34] | Hesse W, Moller E, Arnold M, Schack B (2003) The use of time-variant EEG Granger causality for inspecting directed interdependencies of neural assemblies. J Neurosci Methods 124(1):27–44 |

[35] | Hiemstra C, Jones J (1994) Testing for linear and nonlinear Granger causality in the stock price-volume relation. J Finance 49:1639–1664 |

[36] | Hoekstra BP, Diks CG, Allessie MA, Degoede J (2001) Non-linear time series analysis: methods and applications to atrial fibrillation. Ann Ist Super Sanita 37(3):325–333 |

[37] | Iasemidis LD, Sackellares JC (1996) Chaos theory and epilepsy. The Neuroscientist. Williams and Wilkins, Baltimore, pp 118–208 |

[38] | Jing H, Takigawa M (2000) Observation of EEG coherence after repetitive transcranial magnetic stimulation. Clin Neurophysiol 111:1620–1631 |

[39] | Jing H, Takigawa M, Okamura H, Doi W, Fukuzako H (2001a) Comparisons of event-related potentials after repetitive transcranial magnetic stimulation. J Neurol 248(3):184–192 |

[40] | Jing H, Takigawa M, Hamada K, Okamura H, Kawaika Y, Yonezawa T, Fukuzako H (2001b) Effects of high frequency repetitive transcranial magnetic stimulation on P300 event-related potentials. Clin Neurophysiol 112:304–313 |

[41] | Kaminski M, Blinowska K, Szelenberger W (1995) Investigation of coherence structure and EEG activity propagation during sleep. Acta Neurobiol Exp (Wars) 55(3):213–219 |

[42] | Kaminski M, Blinowska K, Szclenberger W (1997) Topographic analysis of coherence and propagation of EEG activity during sleep and wakefulness. Electroencephalogr Clin Neurophysiol 102(3):216–227 |

[43] | Kaminski M, Ding M, Truccolo W, Bressler SL (2001) Evaluating relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol Cybern 85:145–157 · Zbl 1160.92314 |

[44] | Kaminski MJ, Blinowska KJ (1991) A new method of the description of the information flow in the brain structures. Biol Cybern 65(3):203–210 · Zbl 0734.92003 |

[45] | Kanzler L (1998) A Study of the efficiency of the foreign exchange market through analysis of ultra-high frequency data. PhD Thesis, University of Oxford |

[46] | Kelly JP (1991) Hearing. Princ Neural Sci |

[47] | Korzeniewska A, Kasicki S, Kaminski M, Blinowska KJ (1997) Information flow between hippocampus and related structures during various types of rat’s behavior. J Neurosci Methods 73(1):49–60 |

[48] | Korzeniewska A, Manczak M, Kaminski M, Blinowska KJ, Kasicki S (2003) Determination of information flow direction among brain structures by a modified directed transfer function (dDTF) method. J Neurosci Methods 125(1-2):195–207 |

[49] | LeBaron B (1997) A fast algorithm for the BDS statistic. Stud Nonlinear Dynam Econometrics 2:53–59 · Zbl 1078.91559 |

[50] | Liang H, Ding M, Nakamura R, Bressler SL (2000) Causal influences in primate cerebral cortex during visual pattern discrimination. Neuroreport 11(13):2875–2880 |

[51] | Liégeois-Chauvel C, Lorenzi C, Trebuchon A, Régis J, Chauvel P (2004) Temporal envelope processing in the human left and right auditory cortices. Cereb Cortex 14:731–740 |

[52] | Ljung L (1999) System identification û theory for the user. Prentice Hall, Englewood Cliffs · Zbl 0949.93509 |

[53] | Lopes da Silva FH, Mars NJI (1987) Parametric methods in EEG analysis. In: Gevins AS, Remond A (eds) Methods of analysis of brain electrical and magnetic signals, EEG handbook, vol 1. Elsevier, Amsterdam, pp 243–260 |

[54] | Martinerie J, Adam C, Le Van Quyen M, Baulac M, Clemenceau S, Renault B, Varela FJ (1998) Epileptic seizures can be anticipated by non-linear analysis. Nat Med 4(10):1173–1176 |

[55] | McSharry PE, Smith LA, Tarassenko L (2003a) Comparison of predictability of epileptic seizures by a linear and a nonlinear method. IEEE Trans Biomed Eng 50(5):628–633 |

[56] | McSharry PE, Smith LA, Tarassenko L (2003b) Prediction of epileptic seizures: are nonlinear methods relevant? Nat Med 9(3):241–242; author reply 242 |

[57] | Moller E, Schack B, Arnold M, Witte H (2001) Instantaneous multivariate EEG coherence analysis by means of adaptive high-dimensional autoregressive models. J Neurosci Methods 105(2):143–158 |

[58] | Natarajan K, Acharya UR, Alias F, Tiboleng T, Puthusserypady SK (2004) Nonlinear analysis of EEG signals at different mental states. Biomed Eng Online 3(1):7 |

[59] | Olofsen E (1991) The identification of strange attractors using experimental time series PhD Thesis, Twente University, The Netherlands |

[60] | Palus M (1996) Nonlinearity in normal human EEG: cycles, temporal asymmetry, nonstationarity and randomness, not chaos. Biol Cybern 75(5):389–396 · Zbl 0861.92007 |

[61] | Palus M (1999) Nonlinear dynamics in the EEG analysis: disappointments and perspectives. In: Pradhan N, Rapp PE, Sreenivasan R (ed) Nonlinear dynamics and brain functioning. Novascience, New York |

[62] | Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33:1065–1076 · Zbl 0116.11302 |

[63] | Picton TW (1987) Human auditory steady state responses. In: Barber C, Blum T (eds) Evoked potentials, vol III. Butterworth, Boston, pp 117–124 |

[64] | Pijn JPM (1990) Quantitative evaluation of EEG signals in epilepsy; nonlinear associations, time delays and nonlinear dynamics. PhD Thesis, University of Amsterdam |

[65] | Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27(3):832–837 · Zbl 0073.14602 |

[66] | Sabesan S, Narayanan K, Prasad A, Spanias A, Sackellares JC, Iasemidis LD (2003) Predictability of epileptic seizures: a comparative study using Lyapunov exponent and entropy based measures. Biomed Sci Instrum 39:129–135 |

[67] | Saito Y, Harashima H (1981) Tracking of information within multichannel EEG record. In: Yamaguchi N, Fujisawa K (eds) Recent advances in EEG and EMG data processing. Elsevier, Amsterdam, pp 133–146 |

[68] | Sameshima K, Baccala LA (1999) Using partial directed coherence to describe neuronal ensemble interactions. J Neurosci Methods 94:93–103 |

[69] | Schack B, Grieszbach G, Arnold M, Bolten J (1995) Dynamic cross-spectral analysis of biological signals by means of bivariate ARMA processes with time-dependent coefficients. Med Biol Eng Comput 33(4):605–610 |

[70] | Schelter B, Winterhalder M, Timmer J (2004a) Detection of coupling directions in multivariate oscillatory systems. In: 8th Experimental chaos conference · Zbl 1141.37400 |

[71] | Schelter B, Winterhalder M, Timmer J (2004b) Time varying causal influences in multivariate time series. In: Workshop ”Recent advances in time series analysis” |

[72] | Schnider SM, Kwong RH, Lenz FA, Kwan HC (1989) Detection of feedback in the central nervous system using system identification techniques. Biol Cybern 60(3):203–212 · Zbl 0667.92004 |

[73] | Schreiber T (2000) Measuring information transfer. Phys Rev Lett 85(2):461–464 |

[74] | Silverman BW (1986) Kernel density estimation techniques for statistics and data analysis. Chapman Hall, London |

[75] | Sims CA (1972) Money, income, and causality. Am Econ Rev 62(4):540–553 |

[76] | Supp GG, Schlgl A, Gunter TC, Bernard M, Pfurtscheller G, Petsche H (2004) Lexical memory search during N400: cortical couplings in auditory comprehension. Neuroreport 15(7):1209–1213 |

[77] | Takigawa M (1988) Rhythmic light therapy for depression and data processing analysis of its effects by directed coherence. Act Nerv Super (Praha) 30(3):177–180 |

[78] | Takigawa M, Wang H, Kawasaki H, Fukuzako H (1996) EEG analysis of epilepsy by directed coherence method. A data processing approach. Int J Psychophysiol 21(2/3):65–73 |

[79] | Takigawa M, Wang H, Hamada K, Shiratani T, Takenouchi K (2000) Directed coherence of EEG on ICSS rats with methamphetamine-induced hyperactivity and stereotyped behavior. Ann N Y Acad Sci 914:311–315 |

[80] | Theiler J (1986) Spurious dimension from correlation algorithms applied to limited time-series data. Phys Rev A 34(3):2427–2432 |

[81] | Theiler J, Rapp PE (1996) Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram. Electroencephalogr Clin Neurophysiol 98(3):213–222 |

[82] | Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD (1992) Testing for nonlinearity in time series: the method of surrogate data. Physica D 58:77–94 · Zbl 1194.37144 |

[83] | Triacca U (2004) Feedback, causality and distance between arma models. Math Comput Simulat 64:679–685 · Zbl 1042.93052 |

[84] | Wang G, Takigawa M (1992) Directed coherence as a measure of interhemispheric correlation of EEG. Int J Psychophysiol 13(2):119–128 |

[85] | Wang G, Yunokuchi K (2002) Causality of frontal and occipital alpha activity revealed by directed coherence. IEICE Trans Inf Syst E85-D:1334–1340 |

[86] | Wang G, Takigawa M, Matsushita T (1992) Correlation of alpha activity between the frontal and occipital cortex. Jpn J Physiol 42(1):1–13 |

[87] | Wiener N, (1956) The theory of prediction. In: Beckenbach EF, editors. Modern Mathematics for Engineers. New York, McGraw-Hill |

[88] | Winterhalder M, Schelter B, Maiwald T, Aschenbrenner- Scheibe R, Brandt A, Schulze-Bonhage A, Timmer J (2004) Nonlinear dynamics in EEG from epileptic patients: Is it possible to predict seizures?. In: Boccaletti S, Gluckman BJ, Kurths J et al (eds) Experimental chaos, vol AIP Conference Proceedings 742. AIP Press, Florence |

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.