Selection of the optimal order for multivariate autoregressive model of electroencephalograms for patients with epilepsy
Keywords:autoregressive model, electroencephalography, epilepsy, the order of statistical model, epileptic seizures
AbstractIntroduction. Brain electrical activity signals (or EEG) by their very nature are non-stationary time series. This basically allows applying a set of mathematical-statistical analysis methods to them. One of the most common methods for signal analyzing is the construction of autoregressive mathematical models and analysis of their parameters in order to obtain additional information about the signal itself or causality between signals. In multivariate autoregressive (MVAR) modeling of EEG, the main issue is the optimal choice of model order. In this work, the approach for selecting the optimal order of MVAR models of brain electrical activity signals of subjects diagnosed with epilepsy is proposed.
MVAR modeling. The autoregressive model assumes that the current sample of the discrete signal can be linearly predicted as a weighted sum of its previous samples. MVAR model extends this assumption to multiple time series so that the vector of current samples of all signals is modeled as a linear sum of their previous samples. MVAR models of EEG signals essentially are formed by solving systems of linear equations. The Yule-Walker method of linear equations systems solving is used in this paper. The accuracy of EEG modeling depends on the order of model. Each model order influenced by the amount of delay between current samples and last previous samples used to generate the model. To assess the order and quality of models the Schwarz-Bayes information criterion (SBC) is used in this work taking into account the covariance matrix of the residuals. Additionally, the quality is assessed by Pearson's correlation coefficient between the real and simulated data. In this paper, the MVAR modeling and statistical analysis of the models' optimal orders of the input signal periods before, during and after an epileptic seizure is carried out.
Experimental results. Two sets of EEG data with generalized and focal epileptic seizures are used. The first group of patients with focal seizures consists of 26 people and more than 100 epileptic seizures. The second group with generalized seizures consists of 11 people and about 50 epileptic seizures. For EEG signals modeling, values of orders in a range from 1 to 22 are used. Consequently, for each investigated period of signal (before, during and after a seizure), 22 different MVAR models are constructed and compared. After modeling, the obtained models for each order value are evaluated using the SBC criterion.
Conclusions. According to the results, it is recommended to choose the order of MVAR models of EEG signals in the predefined range of orders from 11 to 13. Since the sampling rate of the signals used in these experiments is 250 Hz, the specified range of order values indicates that MVAR-modelling of one signal includes information that contains all other signals with a delay of 44-52 ms. Therefore, theoretically, it is possible to allocate functional characteristics of brain electrical activity for patients with epilepsy that occur synchronously in different parts of the brain and spread at an average of 50 ms. Moreover, the ways of further research of electrical brain activity and functional connections of brain regions during epileptic activity are indicated.
Faes L. Testing Frequency Domain Analysis of CausalInteractions in Physiological Time Series / L. Faes, G. Nollo. - IEEE. - 2010. - No 57(8). - pp. 1897-1906.
Van Mierlo P. Changes in connectivity patterns in the kainate model of epilepsy / P. van Mierlo, S. Assecondi, P. Boon, I. Lemahieu, eds. - Berlin: Springer. - 2009. - pp. 35-54.
Seth A. Causal connectivity of evolved neural networks during behavior / A. Seth. - Network: Computation in Neural Systems. - 2005. - № 16(1). - pp. 360-363
Lütkepohl H. New introduction to multiple time series analysis / A. Massaro. - Berlin: Springer. - 2005. - 764 p.
Жаринов И.О. К вопросу о выборе порядка авторегрессионных моделей сигналов электроэнцефалограмм человека (в медицинском приборостроении) / И.О. Жаринов // Научно-технический вестник информационных технологий, механики и оптики. - 2006. - № 33. - 12 р.
Hurvich C. Regression and Time Series Model Selection in Small Samples / C. Hurvich, CL. Tsai // Biometrika. - 1989. - Vol 76, No 2. - pp. 297.
Porcaro C. Choice of multivariate autoregressive model order affecting real network functional connectivity estimate / C. Porcaro, F. Zappasodi, PM. Rossini, F. Tecchio // Biometrika. - 2009. - Vol. 120, No 2. - pp. 436-448.
Chatfield C. The Analysis of Time Series: An Introduction / C. Chatfield. - CRC Press, 1980. - 11 p.
Vrieze S. Model selection and psychological theory: A discussion of the differences between the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) / S. Vrieze // Psychological Methods. - 2012. - Vol. 17, No 2. - pp. 228-243.
Панічев О. Ю. Порівняння результатів прогнозування епілептичних нападів при використанні різних схем відведення ЕЕГ / О.Ю. Панічев, А.О. Попов, В.І. Харитонов // Вісник НТУУ «КПІ». Серія Радіотехніка. Радіоапаратобудування. - 2017. - № 68. - с. 54-58
Popov A. Heart beat-to-beat intervals classification for epileptic seizure prediction / A. Popov, O. Panichev, Y. Karplyuk, Y. Smirnov, S. Zaunseder and V. Kharytonov // SPSympo. - 2017
Smirnov Y. Epileptic seizure prediction based on singular value decomposition of heart rate variability features / A.Y. Smirnov, A. Popov, O. Panichev, Y. Karplyuk and V. Kharytonov // SPSympo. - 2017.
Faes L., Porta A. and Nollo G. (2010) Testing Frequency-Domain Causality in Multivariate Time Series. IEEE Transactions on Biomedical Engineering, Vol. 57, Iss. 8, pp. 1897-1906. DOI: 10.1109/tbme.2010.2042715
Van Mierlo P., Assecondi S., Staelens S., Boon P. and Lemahieu I. (2009) Changes in connectivity patterns in the kainate model of epilepsy. IFMBE Proceedings, pp. 360-363. DOI: 10.1007/978-3-540-89208-3_85
Seth A.K. (2005) Causal connectivity of evolved neural networks during behavior. Network: Computation in Neural Systems, Vol. 16, Iss. 1, pp. 35-54. DOI: 10.1080/09548980500238756
Lütkepohl H. (2005) Introduction. New Introduction to Multiple Time Series Analysis, Springer, pp. 1-7. DOI: 10.1007/978-3-540-27752-1_1
Zharinov I.O. (2006) K voprosu o vybore poryadka avtoregressionnykh modelei signalov elektroentsefalogramm cheloveka (v meditsinskom priborostroenii) [On the choice of the order of autoregressive models of signals of human electroencephalograms]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, No 33, p. 121-132.
Hurvich C.M. and Tsai C. (1989) Regression and Time Series Model Selection in Small Samples. Biometrika, Vol. 76, Iss. 2, pp. 297. DOI: 10.2307/2336663
Porcaro C., Zappasodi F., Rossini P.M. and Tecchio F. (2009) Choice of multivariate autoregressive model order affecting real network functional connectivity estimate. Clinical Neurophysiology, Vol. 120, Iss. 2, pp. 436-448. DOI: 10.1016/j.clinph.2008.11.011
Chatfield C. (1980) Introduction. The Analysis of Time Series: An Introduction, CRC Press, pp. 1-11. DOI: 10.1007/978-1-4899-2923-5_1
Vrieze S.I. (2012) Model selection and psychological theory: A discussion of the differences between the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Psychological Methods, Vol. 17, Iss. 2, pp. 228-243. DOI: 10.1037/a0027127
Panichev O., Popov A. and Kharytonov V. (2017) Comparison of epileptic seizure prediction performance for different EEG derivation schemes. Visn. NTUU KPI, Ser. Radioteh. radioaparatobuduv., Vol. 68, pp. 54-58. DOI: 10.20535/radap.2017.68.54-58
Popov A., Panichev O., Karplyuk Y., Smirnov Y., Zaunseder S. and Kharytonov V. (2017) Heart beat-to-beat intervals classification for epileptic seizure prediction. 2017 Signal Processing Symposium (SPSympo). DOI: 10.1109/sps.2017.8053647
Smirnov Y., Popov A., Panichev O., Karplyuk Y. and Kharytonov V. (2017) Epileptic seizure prediction based on singular value decomposition of heart rate variability features. 2017 Signal Processing Symposium (SPSympo). DOI: 10.1109/sps.2017.8053648
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).