# Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models
## 1 Introduction  
The visual examination of electroencephalograms (EEGs) is limited by potential subjectivity due to limited protocols and the inability to identify hidden patterns, characteristics or relationships in large amounts of data [1, 2]. The introduction of quantitative methods in the analysis of EEGs attempt to overcome these limitations by introducing objective measures of EEG characteristics [1]. These methods give both the clinician and/or researcher access to the valuable information within the EEG concerning the dynamics of brain activity. Much of the quantitative research in epilepsy is based on the detection of changes (spikes, spike-waves, sharp waves, etc.,) in the properties of an EEG; the changes in the waveforms are associated with changes in behavioral and mental states. During an epileptic seizure, the EEG exhibits robust changes [3, 4]. The amplitude of the EEG increases during an epileptic seizure due to episodic brief neuronal synchronous discharges; the seizure at onset may be seen in only a few EEG channels (local or partial seizure) or in all EEG channels (generalized seizure) [2]. The information obtained from EEGs using quantitative methods has been used in such areas as attempting to predict/anticipate the occurrence of seizures, attempting to detect the onset/occurrence of seizures, and in modeling seizure propagation/dynamics. The EEG, like many physiological time series, is widely believed to be non-stationary due to its time-varying properties; the non-stationarity may arise from the fact that the observation time is shorter than the characteristic time scale of the EEG [3]. The non-stationarity of the EEG limits the use of classical time series analysis methods. However, early attempts at analyzing EEG signals in the time domain were based on a linear approach using autoregressive moving average (ARMA) modeling [5]. Even though the use of ARMA models in the analysis of EEG signals is limited because of the EEG's time-varying properties, their usage contributed to the understanding of EEG signals [6]. [PERSON] et al. [7], citing studies by [PERSON] et al., [PERSON] et al., and [PERSON] et al., found changes in the autoregressive parameters seconds prior to the onset of a seizure as well as differences in characteristics between one-minute epochs prior to a seizure and control subjects. There are also studies on characterization of EEG time series using spectral and wavelet analysis [8; 9; 10; 11; 12; 13; 14]. Some EEG analyses have moved away from linear time series analysis methods and toward the use of non-linear time series analysis. Measures such as Lyapunov exponents, correlation dimension, correlation density, entropy, and dynamic similarity have been employed [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. Additional techniques that have been used in quantitative EEG analysis either alone or combined with other methods include, but are not limited to, spatial-temporal modeling, wavelets, genetic algorithms, data mining, morphological filters, discriminant analysis, neural networks and support vector machines [29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44]. Convolutional Neural Networks have also been used for feature classification and seizure detection [45; 46; 47; 48]. Recent research has also focused on high-frequency oscillations (HFOs) in epileptic EEGs [45; 48; 49; 50; 51; 52; 53]  
While the above listed methods have been used in EEG analysis, the use of GARCH models in characterizing volatility has not been explored. Conditional volatility models originated in finance with the ARCH (autoregressive conditional heteroscedastistic) model of [PERSON] and the GARCH (generalized autoregressive conditional heteroscedastistic) model of [PERSON] [54; 55]. These models and their extensions have been used to characterize the temporal behavior and clustering of volatility in financial markets.