A nonparametric efficient evaluation of partial directed coherence.

*(English)*Zbl 1359.92016Summary: Studying the flow of information between different areas of the brain can be performed using the so-called partial directed coherence (PDC). This measure is usually evaluated by first identifying a multivariate autoregressive model and then using Fourier transforms of the impulse responses identified and applying appropriate normalizations. Here, we present another way to evaluate PDCs in multivariate time series. The method proposed is nonparametric and utilizes a strong spectral factorization of the inverse of the spectral density matrix of a multivariate process. To perform the factorization, we have recourse to an algorithm developed by Davis and his collaborators. We present simulations as well as an application on a real data set (local field potentials in a sleeping mouse) to illustrate the methodology. A detailed comparison with the common approach in terms of complexity is made. For long autoregressive models, the proposed approach is of interest.

##### MSC:

92C20 | Neural biology |

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

PDF
BibTeX
XML
Cite

\textit{P.-O. Amblard}, Biol. Cybern. 109, No. 2, 203--214 (2015; Zbl 1359.92016)

Full Text:
DOI

##### References:

[1] | Amblard, PO; Michel, OJJ, On directed information theory and Granger causality graphs, J Comput Neurosci, 30, 7-16, (2011) |

[2] | Amblard, PO; Michel, OJJ, The relation between Granger causality and directed information theory: a review, Entropy, 15, 113-143, (2013) · Zbl 06346067 |

[3] | Anderson BDO, Moore JB (1979) Optimal filtering. Prentice Hall, London |

[4] | Baccalá, LA; Sameshima, K, Partial directed coherence: a new concept in neural structure determination, Biol Cybern, 84, 463-474, (2001) · Zbl 1160.92306 |

[5] | Barnett, L; Seth, AK, The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference, J Neurosci Methods, 223, 50-68, (2014) |

[6] | Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge · Zbl 1058.90049 |

[7] | Brillinger DR (2001) Time series: data analysis and theory. SIAM, Philadelphia · Zbl 0983.62056 |

[8] | Chicharro, D, On the spectral formulation of Granger causality, Biol Cybern, 105, 331-347, (2011) · Zbl 1248.92006 |

[9] | Davis, JH; Dickinson, RG, Spectral factorization by optimal gain iteration, SIAM J Appl Math, 43, 289-301, (1983) · Zbl 0517.65047 |

[10] | Dhamala, M; Rangarajan, G; Ding, M, Analysing information flow in brain networks with nonparametric Granger causality, Neuroimage, 41, 354-362, (2008) |

[11] | Dickinson RG (1978) Iterative methods for matrix spectral factorization. Unpublished master’s thesis, Queen’s University, Kingston, Ontario, Canada |

[12] | Effron B (2010) Large-scale inference: empirical Bayes methods for estimation, testing, and prediction. Cambridge University Press, Cambridge · Zbl 1277.62016 |

[13] | Eichler, M, On the evaluation of information flow in multivariate systems by directed transfer function, Biol Cybern, 94, 469-482, (2006) · Zbl 1138.62048 |

[14] | Eichler M (2011) Graphical modeling of multivariate time series. Probab Theory Relat Fields. doi:10.1007/s00440-011-0345-8 |

[15] | Geweke, J, Measurement of linear dependence and feedback between multiple time series, J Am Stat Assoc, 77, 304-313, (1982) · Zbl 0492.62078 |

[16] | Geweke, J, Measures of conditional linear dependence and feedback between times series, J Am Stat Assoc, 79, 907-915, (1984) · Zbl 0553.62083 |

[17] | Gourévitch, B; Bouquin-Jeannés, RL; Faucon, G, Linear and nonlinear causality between signals: methods, example and neurophysiological applications, Biol Cybern, 95, 349-369, (2006) · Zbl 1161.62429 |

[18] | Granger, CWJ, Testing for causality: a personal viewpoint, J Econ Dyn Control, 2, 329-352, (1980) |

[19] | Hall P (1992) The bootstrap and edgeworth expansion. Srpinger, New York · Zbl 0744.62026 |

[20] | Harris, TJ; Davis, JH, An iterative method for matrix spectral factorization, SIAM J Sci Stat Comput, 13, 531-540, (1992) · Zbl 0748.65041 |

[21] | Lütkepohl H (2005) New introduction to multiple time series analysis. Springer, Berlin · Zbl 1072.62075 |

[22] | Morf, M; Vieira, A; Lee, D; Kailath, T, Recursive multichannel maximum entropy spectral estimation, IEEE Trans Geosci Electron, 16, 85-94, (1978) |

[23] | Percival DB, Walden AT (1993) Spectral analysis for physical application: multitaper and conventional univariate techniques. Cambridge University Press, Cambridge · Zbl 0796.62077 |

[24] | Rozanov YA (1967) Stationary random processes. Holden Day, San Francisco · Zbl 0152.16302 |

[25] | Sameshima, K; Baccalá, LA, Using partial directed coherence to describe neuronal ensemble interactions, J Neurosci Methods, 94, 93-103, (1999) |

[26] | Schelter, B; Timmer, J; Eichler, M, Assessing the strength of directed influences among neural signals using renormalized partial directed coherence, J Neurosci Methods, 179, 121-130, (2009) |

[27] | Schelter, B; Winterhalder, M; Eichler, M; Peifer, M; Hellwig, B; Guschlbauer, B; Lücking, CH; Dahlhaus, R; Timmer, J, Testing for directed influences among neural signals using partial directed coherence, J Neurosci Methods, 152, 210-219, (2005) |

[28] | Sporns O (2010) The networks of the brain. MIT Press, Cambridge · Zbl 1093.92028 |

[29] | Whittaker J (1989) Graphical models in applied multivariate statistics. Wiley, New York · Zbl 1151.62053 |

[30] | Wilson, GT, The factorization of matricial spectral densities, SIAM J Appl Math, 23, 420-426, (1972) · Zbl 0227.65042 |

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.