Abstract
The goal of this paper is to develop a measure for characterizing complex dependence between time series that cannot be captured by traditional measures such as correlation and coherence. Our approach is to use copula models of functionals of the Fourier coefficients which is a generalization of coherence. Here, we use standard parametric copula models with a single parameter from both elliptical and Archimedean families. Our approach is to analyze changes in activity in local field potentials in the rat cortex prior to and immediately following the onset of stroke. We present the necessary theoretical background, the multivariate models and an illustration of our methodology on these local field potential data. Simulations with nonlinear dependent data reveal that there is information that is missed by not taking into account dependence on specific frequencies. Moreover, these simulations demonstrate how our proposed method captures more complex nonlinear dependence between time series. Finally, we illustrate our copula-based approach in the analysis of local field potentials of rats.
Similar content being viewed by others
References
Aas K, Berg D (2009) Models for construction of multivariate dependence—a comparison study. Eur J Finance 15(7–8):639–659
Akaike H (1987) Factor analysis and AIC. In: Selected papers of Hirotugu Akaike, Springer, pp 371–386
De la Pena V, Ibragimov R, Sharakhmetov S et al (2006) Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series. In: Optimality, Institute of Mathematical Statistics, pp 183–209
Deheuvels P (1979) La fonction de dépendance empirique et ses propriétés. académie royale de belgique. Bull Cl Sci 65(5):274–292
Fiecas M, Ombao H (2011) The generalized shrinkage estimator for the analysis of functional connectivity of brain signals. Ann Appl Stat 5:1102–1125
Fiecas M, Ombao H (2016) Modeling the evolution of dynamic brain processes during an associative learning experiment. J Am Stat Assoc 111(516):1440–1453
Freyermuth J, Ombao H, von Sachs R (2010) Spectral estimation from replicated time series: an approach using the tree-structured wavelets mixed effects model. J Am Stat Assoc 105:634–646
Gao X, Shen W, Shahbaba B, Fortin N, Ombao H (2016) Evolutionary state-space model and its application to time-frequency analysis of local field potentials. Preprint arXiv:1610.07271
Genest C, Favre A-C (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368
Genest C, Rivest L-P (1993) Statistical inference procedures for bivariate archimedean copulas. J Am Stat Assoc 88(423):1034–1043
Gijbels I, Mielniczuk J (1990) Estimating the density of a copula function. Commun Stat Theory Methods 19(2):445–464
Gotman J (1982) Automatic recognition of epileptic seizures in the eeg. Electroencephalogr Clin Neurophysiol 54(5):530–540
Guevara MA, Corsi-Cabrera M (1996) Eeg coherence or eeg correlation? Int J Psychophysiol 23(3):145–153
Ibragimov R (2005) Copula-based dependence characteriztions and modeling for time series. Harvard Institute of Economic Research Discussion Paper No. 2094
Joe H (1997) Multivariate models and multivariate dependence concepts. CRC Press, Boca Raton
Jordanger LA, Tjøstheim D (2014) Model selection of copulas: aic versus a cross validation copula information criterion. Stat Probab Lett 92:249–255
Kendall MG (1948) Rank correlation methods. Griffin, Spokane Valley
Li X, Mikusiński P, Taylor MD (1998) Strong approximation of copulas. J Math Anal Appl 225(2):608–623
Long C, Brown EN, Manoach D, Solo V (2004) Spatiotemporal wavelet analysis for functional MRI. NeuroImage 23(2):500–516
Lowin JL (2010) The Fourier copula: theory & applications. SSRN. https://doi.org/10.2139/ssrn.1804664
Matousek M (1973) Review of various methods of eeg analysis. In: International EEG handbook. Elsevier, Amsterdam, vol 5(part A), pp 137–138
Motta G, Ombao H (2012) Evolutionary factor analysis of replicated time series. Biometrics 68(3):825–836
Nelsen RB (2007) An introduction to copulas. Springer, Berlin
Ngo D, Ombao H, Sun Y, Genton MG, Wu J, Srinivasan R, Cramer S (2015) An exploratory data analysis of electroencephalograms using the functional boxplots approach. Front Neurosci 9:282
Nunez PL, Srinivasan R et al (2006) Electric fields of the brain: the neurophysics of EEG. Oxford University Press, USA
Ombao H, Van Bellegem S (2006) Coherence analysis of nonstationary time series: a linear filtering point of view. IEEE Trans Signal Process 56:2259–2266
Ombao H, Von Sachs R, Guo W (2005) Slex analysis of multivariate nonstationary time series. J Am Stat Assoc 100(470):519–531
Ombao H, Lindquist M, Thompson W, Aston J (2016) Handbook of statistical methods for neuroimaging. CRC Press, Boca Raton
Ombao H, Fiecas M, Ting C-M, Low YF (2018) Statistical models for brain signals with properties that evolve across trials. NeuroImage 180:609–618
Purdon PL, Solo V, Weisskoff RM, Brown EN (2001) Locally regularized spatiotemporal modeling and model comparison for functional MRI. NeuroImage 14(4):912–923
Sancetta A, Satchell S (2004) The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econ Theory 20(3):535–562
Shaw J (1981) An introduction to the coherence function and its use in EEG signal analysis. J Med Eng Technol 5(6):279–288
Shaw JC (1984) Correlation and coherence analysis of the eeg: a selective tutorial review. Int J Psychophysiol 1(3):255–266
Shumway R, Stoffer D (2017) Time series analysis and its applications: with R examples, 4th edn. Springer, Berlin
Sklar M (1959) Fonctions de repartition an dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231
Van der Vaart AW (1998) Asymptotic statistics, vol 3. Cambridge University Press, Cambridge
Wann EG (2017) Large-scale spatiotemporal neuronal activity dynamics predict cortical viability in a rodent model of ischemic stroke. Ph.D. Dissertation, UC Irvine
Acknowledgements
Hernando Ombao was supported by KAUST Baseline Funds, and Ron D. Frostig was supported by the Leducq Foundation (15CVD02).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fontaine, C., Frostig, R.D. & Ombao, H. Modeling dependence via copula of functionals of Fourier coefficients. TEST 29, 1125–1144 (2020). https://doi.org/10.1007/s11749-020-00703-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-020-00703-5