# Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models
## 3 Conditional Volatility Models
### Generalized Autoregressive Conditional Heteroscedasticity Models  
One limitation of an ARMA model is the assumption of constant conditional variance (\(Var(Y_{t}|F_{t-1})=\sigma_{Y}^{2}\) where \(F_{t-1}\) is the set of information available at time \(t-1\), mathematically, a \(\sigma\)-field generated from \(\{Y_{t-1}\), \(Y_{t-2}\),... \(\}\)), and, thus, is not satisfactory for use in modeling time series with time varying variance [56]. Variance that changes of over time may affect the inferential validity and efficiency of the parameters of an ARMA model [58]. Models that account conditionally for non-constant volatility (non-constant conditional variance) allow for better predictions of (local) variability and better prediction intervals [59]. Time series with non-constant volatility are common in finance, and the most common family of volatility models were developed to model financial time series; their use in biomedical research has been rarely utilized [60; 61; 62]. The Autoregressive Conditional Heteroscedasticity (ARCH) model was developed by [PERSON] and generalized (GARCH) by [PERSON] [54; 55]. The GARCH(\(p\),\(q\)) model for a time series \(Y_{t}=[y_{1},y_{2},\dots,y_{T}]\) is  
\[ \begin{split} Y_{t}&=E_{t-1}[Y_{t}]+\varepsilon_{t} \\ \varepsilon_{t}&=z_{t}\sigma_{t}\\ \sigma_{t}^{2}&=a_{0}+\sum\limits_{i=1}^{p}a_{i} \varepsilon_{t-i}^{2}+\sum\limits_{j=1}^{q}b_{i}\sigma_{t-j}^{2}\end{split}  \tag{2}\]  
where \(E_{t-1}[Y_{t}]\) is the expectation conditional on information available at time \(t-1\), \(z_{t}\) are i.i.d. zero mean random variables with unit variance, \(a_{0}>0\), \(a_{i},b_{i}\geq 0\) and  
\[\sum\limits_{i=1}^{\max(p_{d})}(a_{i}+b_{i})<1 \tag{3}\]  
[63]. These constraints ensure positive and finite unconditional variance for \(\varepsilon_{t}\), with \(\sum\limits_{i=1}^{\max(p,q)}(a_{i}+b_{i})<1\) a necessary and sufficient condition for weak stationarity. A less restrictive necessary and sufficient condition for strict stationarity is that \(E\big{(}\log\!\left(a_{1}\sigma_{t}^{2}+b_{1}\right)\big{)}<0\), with certain conditions on the \(z_{t}\). This allows the sum \(a_{1}+b_{1}\) to equal or exceed one [55; 64; 65; 66]. The \(\varepsilon_{t}\), often referred to as'shocks,' are serially uncorrelated, but dependent. The \(a_{i}\) and \(b_{i}\) in the conditional variance equation are the ARCH and GARCH parameters respectively, with \(E(\varepsilon_{t})=0\) and the unconditional variance of \(\varepsilon_{t}\) is  
\[E\big{(}\varepsilon_{t}^{2}\big{)}=\frac{a_{0}}{1-\left(\sum\limits_{i=1}^{p} a_{i}+\sum\limits_{j=1}^{q}b_{i}\right)}, \tag{4}\]  
which will be positive due to the constraints. A GARCH model with \(q=0\) is an ARCH(\(p\)) model and in order to identify the parameters, \(A(\varepsilon)=\sum\limits_{i=1}^{p}a_{i}\varepsilon^{i}\) and \(B(\varepsilon)=1-\sum\limits_{i=1}^{p}b_{i}\varepsilon^{i}\) are co-primes [67]. If the conditional variance equation follows an ARCH(1) process, \(\sigma_{t}^{2}=a_{0}+a_{1}\varepsilon_{t-1}^{2}\), large values of \(\varepsilon_{t-1}\) result in a larger deviation from \(E_{t-1}(\varepsilon_{t})=0\), and the conditional variance of \(\varepsilon_{t}\) is larger as a result. This volatility will propagate since a large deviation of \(\varepsilon_{t}\) makes \(\sigma_{t+1}^{2}\) and \(\varepsilon_{t+1}^{2}\) large (similarly for small values of \(\varepsilon_{t}\)). Thus, the volatility will persist, but will eventually revert to the unconditional variance since \(a_{1}<1\). For ARCH(\(p\)) processes, large past shocks \(\big{\{}\varepsilon_{t-i}^{2}\big{\}}_{i=1}^{p}\) lead to a large conditional variance for and a large value of \(\varepsilon_{t}\). Thus, the probability of a large shock following a large shock is greater than the probability of small shock. This leads to the volatility clustering often seen in financial time series [56; 68]. The development of GARCH(\(p\),\(q\)) models were motivated by the shortcomings in the ARCH(\(p\)) models and provide more flexibility than ARCH(\(p\)) models.