Definition
Neurobiological data are often collected in the form of time series. Spectral analysis is the systematic study of time series in the frequency domain. In parametric spectral analysis, time series are treated as realizations of a stochastic process, and a model, typically an autoregressive model, is fit to the data, from which spectral quantities of interest such as power, coherence, and Granger causality spectra are then derived.
Detailed Description
We start by reviewing the basic concepts in the theory of stochastic processes which is the mathematical foundation of time series analysis. From this review one can clearly see that parametric modeling of time series is not an ad hoc approach but arises naturally from the theory of stochastic processes.
Stochastic Processes
If only one variable is being recorded over time, the time series is said to be univariate. An example of a univariate EEG time series demonstrating the alpha rhythm is shown in Fig. 1. If multiple...
This is a preview of subscription content, log in via an institution to check access.
References
Akaike H (1971) Autoregressive model fitting for control. Ann Inst Stat Math 23(1):163–180
Chatfield C (2004) The analysis of time series: an introduction. Chapman and Hall, Boca Raton
Chen Y, Bressler SL, Ding M (2006) Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. J Neurosci Methods 150:228–237
Dhamala M, Rangarajan G, Ding M (2008a) Estimating Granger causality from Fourier and wavelet transforms of time series data. Phys Rev Lett 100:018701
Dhamala M, Rangarajan G, Ding M (2008b) Analyzing information flow in brain networks with nonparametric Granger causality. Neuroimage 41:354
Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and application to neuroscience. In: Schelter B, Winderhalder M, Timmer J (eds) Handbook of time series analysis. Wiley-VCH, Berlin, pp 437–460
Geweke J (1982) Measurement of linear-dependence and feedback between multiple time-series. J Am Stat Assoc 77:304–313
Geweke J (1984) Measures of conditional linear-dependence and feedback between time-series. J Am Stat Assoc 79:907–915
Granger CWJ (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37(3):424–438
Papoulis A (1985) Levinson algorithm, Wold’s decomposition and spectrum estimation. SIAM Rev 27(3):405–441
Papoulis A (1991) Probability, random variables and stochastic processes. McGraw-Hill, New York
Percival DB, Walden AT (1993) Spectral analysis for physical applications. Cambridge University Press, Cambridge
Priestly MB (1981) Spectral analysis and time series, vols 1, 2. Academic, London
Wen X, Rangarajan G, Ding M (2013) Multivariate Granger causality: an estimation framework based on the factorization of spectral density matrix. Philos Trans R Soc A Math Phys Eng Sci 371:20110610
Wiener N (1956) The theory of prediction. In: Beckenbach EF (ed) Modern mathematics for engineers. McGraw-Hill, New York (Chap 8)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this entry
Cite this entry
Ding, M., Rangarajan, G. (2014). Parametric Spectral Analysis. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_416-1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7320-6_416-1
Received:
Accepted:
Published:
Publisher Name: Springer, New York, NY
Online ISBN: 978-1-4614-7320-6
eBook Packages: Living Reference Biomedicine and Life SciencesReference Module Biomedical and Life Sciences