×

zbMATH — the first resource for mathematics

Quantitative analysis of directional strengths in jointly stationary linear multivariate processes. (English) Zbl 1266.92002
Summary: Identification and analysis of directed influences in multivariate systems is an important problem in many scientific areas. Recent studies in neuroscience have provided measures to determine the network structure of the process and to quantify the total effect in terms of energy transfer. These measures are based on joint stationary representations of a multivariate process using vector auto-regressive (VAR) models. A few important issues remain unaddressed though. The primary outcomes of this study are (i) a theoretical proof that the total coupling strength consists of three components, namely, the direct, indirect, and the interference produced by the direct and indirect effects, (ii) expressions to estimate/calculate these effects, and (iii) a result which shows that the well-known directed measure for linear systems, partial directed coherence (PDC) only aids in structure determination but does not provide a normalized measure of the direct energy transfer. Simulation case studies are shown to illustrate the theoretical results.

MSC:
92B05 General biology and biomathematics
92C20 Neural biology
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
92-08 Computational methods for problems pertaining to biology
Software:
astsa
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akaike H (1968) On the use of a linear model for the identification of feedback systems. Ann Inst Stat Math 20: 425–439 · Zbl 0198.51202 · doi:10.1007/BF02911655
[2] Baccala L, Sameshima K (2001a) Overcoming the limitations of correlation analysis for many simultaneously processed neural structures. Prog Brain Res 130: 34–47
[3] Baccala L, Sameshima K (2001b) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84: 463–474 · Zbl 1160.92306 · doi:10.1007/PL00007990
[4] Baccala L, Sameshima K, Takahashi D (2007) Generalized partial directed coherence. In: Proceedings of the 15th international conference on digital signal processing (DSP 2007), pp 163–166
[5] Battaglia F (1984) Inverse covariances of a multivariate time series. Metron 42(3–4): 117–129 · Zbl 0579.62079
[6] Born M, Wolf E (1980) Principles of optics. Pergamon Press, London
[7] Dahlhaus R (2000) Graphical interaction models for multivariate time series. Metrika 51: 157–172 · Zbl 1093.62571 · doi:10.1007/s001840000055
[8] Eichler M (2006) On the evaluation of information flow in multivariate systems by the directed transfer function. Biol Cybern 94: 469–482 · Zbl 1138.62048 · doi:10.1007/s00422-006-0062-z
[9] Gevers M, Anderson B (1981) Representations of jointly stationary stochastic feedback processes. Int J Control 33(5): 777–809 · Zbl 0511.93059 · doi:10.1080/00207178108922956
[10] Gigi S, Tangirala AK (2009) Quantification of directed influences in multivariate systems by time-series modelling. In: Proceedings of the international conference INCACEC-2009, vol 2, pp 1017–1023
[11] Granger CWJ (1969) Investigating causal relationships by econometric models and cross-spectral methods. Econometrica 37(3): 424–438 · Zbl 1366.91115 · doi:10.2307/1912791
[12] Jackson JE (1991) A user’s guide to principal components. John Wiley, New York · Zbl 0743.62047
[13] Kaminski M, Blinowska K (1991) A new method of the description of the information flow in the brain structures. Biol Cybern 65: 203–210 · Zbl 0734.92003 · doi:10.1007/BF00198091
[14] Lutkepohl H (2005) New introduction to multiple time series analysis. Springer, New York
[15] Narasimhan S, Shah S (2008) Model identification and error covariance matrix estimation from noisy data using PCA. Control Eng Prac 16: 146–155 · doi:10.1016/j.conengprac.2007.04.006
[16] Pereda E, Quiroga R, Bhattacharya J (2005) Nonlinear multivariate analysis of neurophysiological signals. Prog Neurobiol 77: 1–37 · doi:10.1016/j.pneurobio.2005.10.003
[17] Priestley M (1981) Spectral analysis and time series. Academic Press, London · Zbl 0537.62075
[18] Rantala S, Jokinen H, Jokipii M, Saarela O, Suoranta R (1988) Process dynamics study based on multivariate ar-modelling. IECON 1988, pp 320–325
[19] Saito Y, Harashima H (1981) Tracking of information within multichannel EEG record-causal analysis in EEG. In: Yamaguchi N, Fujisawa K (eds) Recent advances in eeg and meg data processing. Elsevier, Amsterdam
[20] Schelter B, Timmer J, Eichler M (2009) Assessing the strength of directed influences among neural signals using renormalized partial directed coherence. J Neurosci Methods 179(1): 121–130 · doi:10.1016/j.jneumeth.2009.01.006
[21] Shumway RH, Stoffer DS (2000) Time series analysis and its applications. Springer, New York
[22] Smith SW (1997) Scientist and engineer’s guide to digital signal processing. California Technical Publishing, San Diego, CA
[23] Stefani RT, Shahian B, Savant CJ, Hostetter GH (2002) Design of feedback control systems. Oxford University Press, New York
[24] Strang G (2006) Linear algebra and its applications. Thomson, Belmont, CA · Zbl 0338.15001
[25] Winterhalder M, Schelter B, Hesse W, Schwab K, Leistriz L, Klan D, Bauer R, Timmer J, Witte H (2005) Comparison of linear signal processing techniques to infer directed interactions in multivariate neural systems. Signal Processing 85: 2137–2160 · Zbl 1160.94369 · doi:10.1016/j.sigpro.2005.07.011
[26] Yamashita O, Sadato N, Okada T, Ozaki T (2005) Evaluating frequency-wise directed connectivity of bold signals applying relative power contribution with the linear multivariate time-series models. NueroImage 25: 478–490 · doi:10.1016/j.neuroimage.2004.11.042
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.