2.2 Temporal changes in EEG
2.2.5 Parametric models for EEG
EEG time series can be modelled using several different types of models. One of the most common families of models used for EEG is stochastic parametric models such as autoregressive and moving average models and their variants. While AR and MA models are suitable for stationary time series with SRD, their integrated variant autoregressive integrated moving average (ARIMA) model is used for non-stationary time series. EEG has LRTC which can be modelled using a generalisation of ARIMA called autoregressive fractionally integrated moving average model (ARFIMA). The scale-invariant arrhythmic broadband EEG can be considered as a stochastic process and hence can be modelled by a stochastic parametric model such as ARFIMA. ARIMA can model the LRD as well as the SRD in the time series. How ARFIMA can model LRTC is described in the following sections [32]. 2.2.5.1 White noise
The time series without any correlations is called white noise. The samples in white noise are independently identically distributed (iid). The white noise can be represented by equation (2.6).
Xt= εt (2.6)
where εt also known as innovation is randomly drawn from a Gaussian distribution [32]. The
white noise is a stationary process. The power spectrum of white noise is flat with slope of 0 in log-log scale as shown in section 2.2.4.1.
2.2.5.2 Random walk process
The random walk process is an integrated white noise. It is a running sum or cumulative sum of the independent samples. Random walk is given by equation (2.7) and (2.8). The power spectrum of a random walk in log-log scale is a straight line with slope -2 (see 2.2.4.1).
Xt= t
∑
i=0 εi (2.7) Xt= Xt−1+ εt (2.8)The random walk process is non-stationary because its variance increases over time. The current value of the random walk can be defined entirely by the value of its previous sample and current innovation. Taking difference between successive samples gives the random white noise process. This procedure is called differencing. Differencing a random walk changes its slope from -2 to 0 making it a white noise and integrating a white noise changes its slope from 0 to -2.
2.2.5.3 Autoregressive (AR) model
In AR processes, the current value depends only partly on the previous value unlike random walk as given in equation (2.9).
Xt = φ Xt−1+ εt (2.9)
where φ is magnitude of the difference. Such a process has order 1 and is called AR(1) because the currently value is partly dependent only on the previous value. The φ ∈ (−1, 1) keeps the values of the samples within certain bounds preventing drifting to very high or low values. The ACF of AR(1) follows an exponential function indicating that it is an SRD process. The log-log power spectrum of AR(1) is not a straight line, but it is a curve. For AR(1), more power is present in the lower frequency which flattens the power spectrum in these low frequencies. Thus, AR(1) power spectrum has a slope of 0 in lower frequencies and -2 in higher frequencies [32].
The AR process can have higher order such that the current value is dependent on certain number (p) of past values. The AR(p) process is given by equation (2.10).
AR models of several different orders are used to model EEG oscillations. High order AR models usually between 8 to 16 [136, 137, 84, 138] are required for EEG. AR models can model oscillations but cannot give good estimation of broadband EEG because the broadband EEG is non-stationary and AR model is suitable for stationary processes. Moreover, EEG contains LRTC but the AR models can only capture SRD in the time series. Hence, AR models alone cannot model EEG accurately. Further discussion on AR models is included in chapter 6.
The modification of AR models for non-stationary data is adaptive autoregressive mod- els [139, 140] whose parameters are estimated with Kalman filter. Adaptive autoregressive model adaptively changes the order of model to estimate EEG.
2.2.5.4 Moving average (MA) model
The current value of the sample in MA process depends on the current random innovation and the previous random innovation. The equation (2.11) shows the MA(1) process. Higher order MA(q) process is shown in equation (2.12) in which the current value is dependent on the current innovation and q previous innovations. The power spectrum of MA(1) is similar to that of white noise with a very steep slope at highest frequencies.
Xt= εt+ θ εt−1 (2.11)
Xt = εt+ θ1εt−1+ ... + θqεt−q (2.12)
The values of innovations earlier than εt−p do not have any effect on the current value
and are completely unrelated to Xt. Hence MA processes can be used as filters to remove
noise and smooth the time series.
Both AR and MA are linear combinations of the random innovations. The AR model absorbs innovations ε into xtsuch that any previous εt will have an impact on all the future
values of Xt+k. On the other hand, as stated previously, the current value of MA process is
a finite combination of the q previous random innovations. The future values will not be impacted by the innovations previous than the last q innovation values. The invertible MA(1) process (with θ ∈ (−1, 1)) can be represented by an infinite order AR process and vice versa. 2.2.5.5 Autoregressive moving average (ARMA) model
The ARMA(p, q) model is a combinition of pth order AR model and qth order MA model as given by the equation (2.13).
Xt= p
∑
i=1 φiXt−i+ εt+ q∑
i=1 θiεt−i (2.13)ARMA models have also been used to model EEG.
2.2.5.6 Autoregressive integrated moving average (ARIMA) model
The ARIMA(p, d, q) model is a generalisation of ARMA(p, q). The ARFIMA model incorpo- rates non-stationarity in its integration parameter d as given in the equation (2.14). The time series is differenced d times to make it stationary after which, the AR and MA parameters are estimated. 1 − p
∑
i=1 φiBi ! (1 − B)dXt = 1 + q∑
i=1 θiBi ! εt (2.14)where B is the backshift operator such that BXt= Xt−1and BnXt = Xt−n. ARFIMA models
are also used for modelling EEG where EEG is usually differenced once and then the AR and MA parameters are estimated.
2.2.5.7 Autoregressive fractionally integrated moving average (ARFIMA) model for LRTC
While ARIMA(p, d, q) can model non-stationary time series, it is not suitable for LRTC process. The white noise can be modelled with ARIMA(0,0,0) with d = 0 and log-log power spectrum slope of 0. The random walk can be modelled by ARIMA(0,1,0) with d = 1 and log-log power spectrum slope of -2. The process with LRTC has a log-log power spectrum with a slope of approximately -1. The LRTC processes lies between random walk and white noise and hence would need a fractional value for d between 0 and 1 because as mentioned in section 2.2.5.2, differencing once changes the slope of power spectrum from -2 to 0. Hence the fractional differencing with a fractional value of d can model the LRTC part of the time series and the residual stationary SRD could be modelled using AR and MA parameters of ARFIMA(p, d, q) with the equation 2.14. Generally for LRTC, the slope of the power spectrum is between -0.5 and -1.5. An ARIMA model would require an infinite number of parameters to accurately model LRTC process of infinite length [32].
The process of fractional difference is given in chapter 6. For 0 < d < 0.5, the ARFIMA process is stationary and for d > 0.5, the ARFIMA process is non-stationary. The d is estimated from the Hurst exponent using d = H − 0.5. Further details of ARFIMA modelling are comprehensively given in Wagenmakers et al. [32]. Though ARFIMA can model LRTC
and SRD simultaneously, it is a mathematical model that is very descriptive and it does not give any information about the underlying physiological processes. ARFIMA model has been used to generate LRTC process to test different signal analysis approaches [126, 120, 127] but it is not used commonly to model broadband EEG that contains LRTCs as described in this thesis.