The EEG is a complex signal, the statistical properties of which depend on both time and space. Regarding the temporal characteristics, it is essential to note that EEG signals are everchanging. However, they can be analytically subdivided into representative epochs (i.e., with more or less constant statistical properties).

Estimates of the length of such epochs vary considerably because of dependence on the subject’s behavioral state. When the latter is kept almost constant, Isaksson and Wennberg (1) found that, over relatively short-time intervals, epochs can be defined that can be considered representative of the subject’s state; in this study, some 90% of the EEG signals investigated had time-invariant properties after 20 seconds, whereas less than 75% remained time invariable after 60 seconds. Empirical observations indicate that EEG records obtained under equivalent behavioral conditions show highly stable characteristics; for example, Dumermuth et al. (2) showed that variations in mean peak (beta activity) of only 0.8 Hz were obtained in a series of 11 EEGs over 29 weeks. In this respect it is interesting to consider the studies of Jansen (3) and Grosveld et al. (4); these authors investigated the possibility of correctly assigning EEG epochs (duration 10.24 seconds) to the corresponding subject by means of multivariate analysis, using half of the EEG epochs recorded from 16 subjects as a training set for the classification algorithm. Using 4 to 10 EEG features, it was found that in 80% to 90% of the cases of EEG epochs were assigned correctly to the corresponding subject. McEwen and Anderson (5) introduced the concept of wide-sense stationarity in EEG analysis; they
proposed a procedure for determining whether a set of signal samples (e.g., an EEG signal) can be considered to belong to a wide-sense stationarity process. Their procedure consisted of calculating the amplitude distributions and power spectra of sample subsets and showing that they do not differ significantly using the Kolmogorov-Smirnov statistic. From this study it was concluded that, for EEG epochs (awake condition or during anesthesia) of less than 32 seconds, the assumption of wide-sense stationarity was valid more than 50% of the time.

On the basis of this type of empirical observation, it can be assumed that relatively short EEG epochs (˜10 seconds) recorded under constant behavioral conditions are quasi-stationary. Elul (6) remarked that the EEG is related to intermittent changes in the synchrony of cortical neurons; thus, he characterized the EEG as a series of short epochs rather than a continuous process.

The fact that EEG signals have different characteristics depending on the place over the head where they are recorded is essential to all EEG recordings. Therefore, in any method of EEG analysis, topographic characteristics have to be taken into account. This means that one should choose EEG montages carefully, in view of the objectives of the analysis. The topographic aspects appear most clearly in the simple case of comparing EEG records from symmetric derivations; indeed, the use of the subject as his or her own control through right-left comparisons is a cornerstone of the neurologic examination. Therefore, right-left comparisons are also paramount in any practical clinical system of EEG analysis.

Some of the underlying assumptions of the most common methods of EEG analysis will be discussed briefly. Gasser (7) has provided a more fundamental discussion of this topic; here, general concepts will suffice.

The exact characteristics of EEG signals are, in general terms, unpredictable. This means that one cannot foresee precisely the amplitude of an EEG graphoelement or the duration of an EEG wave. Therefore, it is said that an EEG signal is a realization of a random or stochastic process. Indeed, it is possible to determine some statistical measures of EEG signals that show considerable regularity, such as an average amplitude or an average frequency. This is a general characteristic of random processes, which are characterized by probability distributions and their moments (e.g., mean, variance, skewness, and kurtosis) or by frequency spectra or correlation functions. Such a description of an EEG signal as a realization of a random process implies a mathematical, but not a biophysical, model.

It should be stressed (8) that the biophysical process underlying EEG generation is not necessarily random in nature, but it may have such a high degree of complexity that only a description in statistical terms is justified. Gasser (7) has also emphasized this point; even in the case of signals that are deterministic (e.g., sinusoids) but very complex (e.g., made of many components), a stochastic approach may be the most adequate.

EEG signals are, of course, time series; they are characterized by a set of values as a function of time. An important problem, however, is whether the general methods for analyzing time series can be applied without restrictions to EEG signals.

In Chapter 4, which discusses EEG dynamics, it was mentioned that modern mathematical tools are being used to analyze EEG signals, assuming that signal generation can be described using sets of nonlinear differential equations. These techniques have been developed within the active field of mathematical research called “deterministic chaos.” In essence, nonlinear dynamical systems such as the neuronal networks generating EEG signals can display chaotic behavior; that is, their behavior can become unpredictable for relatively long periods, and EEG signals may be an expression of chaotic behavior. Since new mathematical tools, based on the analysis of complex nonlinear systems such as the correlation dimension, were introduced in EEG, it became clear that EEG signals may be high-dimensional so that in many cases it is difficult, or even impossible, to distinguish whether these signals are generated by random or by high-dimensional nonlinear deterministic processes (9).

EEG signals are continuous variations of potential as a function of time. However, in most practical cases where quantitative analysis is applied, signals must be digitized so that they can be processed by digital computer. This means that the EEG signal must be processed in such a way that the random variable, potential as a function of time, will have only one set of discrete values at a set of discrete time instances. In technical terms, the process of analog-to-digital (AD) conversion involves sampling combined with the operation of quantizing. According to definitions commonly used (10), sampling is the “process of obtaining a sequence of instantaneous values of a wave at regular or intermittent intervals” and quantization is the “process in which the continuous range of values of an input signal is divided into nonoverlapping subranges and to each subrange a discrete value of the output is uniquely assigned.”

EEG signal sampling must be performed without changing the statistical properties of the continuous signal. Generally, one samples an EEG signal at equidistant time intervals (Δt), thus transforming the continuous signal into a set of impulses with different heights separated by intervals Δt (Fig. 54.1). An important question is the choice of the sampling frequency. This choice is based on the sampling theorem: assuming that a signal x(t) has a frequency spectrum X(f) such that X(f) = 0 for f_N, no information is lost by sampling x(t) at equidistant intervals Δt with f_N = 1/(2 Δt); f_N is called the folding or Nyquist frequency. The sampling frequency, therefore, must be at least equal to 2f_N. A consequence of this theorem is that care has to be taken to ensure that the signal to be sampled has no frequency components above f_N. Therefore, before sampling, all frequency components greater than f_N should be eliminated by low-pass filtering. One should keep in mind that sampling at a frequency below 2f_N is not equivalent to filtering; it would produce aliasing or signal distortion due to folding of frequency components larger than f_N onto lower frequencies (11). The analog voltages of the signal at the sampling moments are converted to a number corresponding to the amplitude subrange or level. Most EEG analysis can be performed using 512 to 2048 amplitude levels (i.e., 9 to 11 bits). Technical details of AD conversion may be found in Susskind (12) and, for the special case of EEG signals, in Lopes da Silva et al. (11) and Steineberg and Paine (13).

The continuous EEG signal is thus replaced by a string of numbers x(t_i) representing the signal amplitude at sequential sample moments; the latter are indicated by the index i along the time axis. The signal is assumed to be a realization of a stationary random process (t_i), which is indicated by underlining the letter (). In general, a collection of EEG signals of a certain length recorded under equivalent conditions is available for analysis. The entire collection of EEG signals is called an ensemble; each member of the ensemble is called a sample function or a realization.

The digitized EEG signal values x(t_i) can be considered realizations of one stochastic variable (t_i) and may be characterized when stationarity is assumed by a histogram; when in an interval 0 > t > T there are n_a sample points in the interval a ± 1/2Δ, n_a/N is called the relative frequency of occurrence of the value a, where N is the total number of samples available. One can define the relative frequencies of all other values similarly. When N becomes infinitely large and Δ infinitely small, n_a/N will tend to a limit value p(x(t_i) = a), called the probability of occurrence of x(t_i) = a. The set of values of p(x(t_i)) is called the signal probability distribution, characterized by a mean and a number of moments. Considering that the discrete random variable × can take any of a set of values from 1 to M, the mean or average of the sample functions is given as follows (E is the symbol for expectation):

Also definable is a class of statistical functions characteristic of the random process: m_n = E[ⁿ(t_i)] with n = 1, 2, 3, …; these functions are called the nth moments of the discrete random variable (t_i). The implicit assumption here and in the following discussion is that the statistical properties of the signal do not change in the interval T. Therefore, the moments are independent of time t_i.

The first moment E[(t_i)] is called the mean of (t_i). It is often preferable to consider the central moments (i.e., the moments around the mean); the second central moment is then:

or σ² or variance of (t_i).

Similarly, the third central moment E[((t_i)-(E((t_i)))³] = m₃ can be defined; from this can be derived the skewness factor β₁ = m₃/(m₂)^3/2. The fourth central moment is E[((t_i)-(E((t_i)))⁴] = m₄, from which can be derived the kurtosis excess: β₂ = m₄/(m₂)². In case of a symmetric amplitude, distribution is β₁ = 0; all odd moments are equal to zero. For a gaussian distribution the even moments have specific values, for example, β₂ = 3; derivatives from this value indicate the peakedness (β₂ > 3) or flatness (β₂ < 3) of the distribution (Fig. 54.2).

In general terms, successive values of a signal, such as an EEG, which result from a stochastic process are not necessarily independent. On the contrary, it is often found that successive discrete values of an EEG signal have a certain degree of interdependence. To describe this interdependence, one may compute the signal joint probability distribution. As an example, consider the definition of the joint probability applied to a pair of values at two discrete moments, (t₁) and (t₂); assume that one disposes of N realizations of the signal; the number of times that at t₁ a value v and at t₂ a value u are encountered is equal to n₁₂. Thus, the joint probability of (t₁) and (t₂) = u may be defined as follows:

A complete description of the properties of the signal generated by a random process can be achieved by specifying the joint probability density function:

for every choice of the discrete time samples t₁, t₂,…, t_n and for every finite value of n. The computation of this function, however, is rather complex. A simpler alternative to this form of description is to compute a number of averages characteristic of the signal, such as convariance, correlations, and spectra. These averages do not necessarily describe a stochastic signal completely, but they may be very useful for a general description of signals such as EEG.

The convariance between two random variables at two time samples (t₁) and (t₂) is given by the following expectation:

Estimating the covariance between any two variables (t₁) and (t₂) requires averaging over a umber of realizations of an ensemble. Another way to estimate the convariance, provided that the signal is stationary and ergodic (for a discussion of these concepts see Ref. 14), is by computing a time average, for one realization of the signal, of the product of the signal and a replica of itself shifted by a certain time τ_k along the time axis. This time average is called the autocorrelation function:

where τ_k = k·Δt.

The following description considers continuous random variables x(t), for the sake of simplifying the formulas. Assuming that every sample function, or realization, is representative of the whole signal being analyzed, it can be shown that for stationary and ergodic processes the time average Φ_xx(τ) for one realization (t) is an estimate of the ensemble average R_xx(τ):

assuming that the signal (t) has mean zero. For the value τ = 0:

which is the signal’s average power or variance σ. An important property of the autocorrelation function is that its Fourier transform (FT) is:

S_xx(f) is called the power density spectrum, or simply the power spectrum, a common method of EEG quantification (Fig. 54.3). The power spectrum S_xx(f) is a function of frequency (Hz); it gives the distribution of the squared amplitude of different frequency components. It should be noted that the word power does not have the meaning of dissipated power in an RC circuit but is used here in another sense. This discussion deals with a question of time series analysis. In general, a stochastic
time function may be expressed in one of the several ways, as a voltage, a length, a velocity, a number of occurrences of a certain event, and so forth. The power spectrum or simply the spectral density of the time function is the FT of the autocorrelation function, the dimension of which is the function’s amplitude dimension squared. In case the signal dimension is in volts, the power spectrum is in (V²·sec) or (V²/Hz). Of course, if the function’s amplitude is in any other dimension, the intensity of the corresponding power spectrum would be yet another dimension. It is useful to keep a clear distinction between electric power dissipated in an electric circuit ( with units [V²·sec/Ω]) and power spectrum.

A function that represents the average correlation between two signals (t) and (t) may be defined in terms equivalent to expression 54.7:

where the signals (t) and (t) are assumed to have means of zero. R_xy(τ) measures the correlation between the two signals and is called the cross-correlation function. Similarly, one can define the FT of R_xy, which is the cross-power spectrum between signals and :

Fundamental discussions of power spectra and related topics are found in many textbooks on signal analysis, for example, Refs. 15 and 14. This discussion cannot go into details about ways of Bendat and Piersol (16) and Otnes and Enochson (17). Application of the frequency analysis principle to EEG signal analysis has a long history, beginning with the pioneering work of Dietsch (18), Grass and Gibbs (19), Knott and Gibbs (20), Drohocki (21), and Walter (22,23). Brazier and Casby (24) and Barlow and Brazier (25) first computed the autocorrelation functions of EEG signals. The general principles on which this work has been based have remained essentially the same since Wiener proposed these signal analysis methods (for a review, see Ref. 26). An important advance in computing power spectra has been achieved with the introduction of a new algorithm for computing the discrete Fourier transform, known as the fast Fourier transform (FFT) (27). In this case, it is assumed that one wants to compute the power spectrum of a discrete EEG signal; the epoch [(t₁)] is considered as a signal sampled at intervals Δt, x(n Δt) with a total of N samples (n = 1…N). By using the discrete FT, the so-called periodogram F(f_i) can be computed:

where f_i = i · Δf with i = 0, 1, 2, …, N. The periodogram can be smoothed by means of a window W(f_k) in order to obtain P_xx(f_k), which is a better estimate of the real power spectrum S_xx(f):

where W(f_k) is the smoothing window with a duration of (2p + 1) samples or data points. Similarly, one can compute a smoothed estimate of the cross-power spectrum (S_xy), which might be called C_xy(f).

The FFT power spectral analysis and its applications are discussed in more detail below.

The close relationship between the concepts of variance σ², autocorrelation (equations 54.6 and 54.7), and power density spectrum (equation 54.9) has already been made apparent; in fact R_xx(0) = σ² and

The autocorrelation function R(τ) and the power density spectrum S(f) correspond thus to the second-order moment of the probability distribution of the random process.

In case the signals are not gaussian, higher order spectra moments must be considered. These can be derived as follows. Assuming that the signal has mean = 0, one can write (as in expression 54.7):

Similar to expression 54.9, the two-dimensional Fourier transform FT₂ of R_xx(τ₁, τ₂) can be defined as the bispectrum or bispectral density:

This discussion cannot go into details about ways of estimating the bispectrum B_xx. For a detailed account of bispectral EEG analysis, refer to Huber et al. (28) and to Dumermuth et al. (29). It is, however, interesting to note that high B_xx values for a couple of frequencies, f₁ and f₂, indicate phase coupling within the frequency triplet f₁, f₂, and (f₁ + f₂). The third moment of the probability distribution, or skewness, is related to the bispectrum. When there exists a sufficiently strong relation between two harmonically related frequency components in a signal, there will exist a significant bispectrum and skewness. The process in such a case is not gaussian; if it were gaussian with mean zero, the bispectrum would be zero. The bispectrum can be used to determine whether the system underlying the EEG generation has nonlinear properties. An example of this form of analysis is given in Figure 54.4.

This section has demonstrated the progression from the basic principles of probability distribution and corresponding moments to the concepts of autocorrelation, power spectra, and high-order spectra. It is also of interest to examine the moments of the spectral density S_xx(f), because this analysis leads to another set of concepts applicable to EEG analysis, the so-called descriptors of Hjorth (30). Thus, one can define the nth spectral moment as follows:

The zero-order moment is then:

which is equal to the variance σ².

It can be shown (for derivation, see Ref. 31) that the second-order moment is defined by the following expression:

and the fourth-order moment is:

In this way the spectral moments relate to the derivatives of the autocorrelation function R_xx(τ) and of the signal (t). The discussion below illustrates how these spectral moments a₀, a₂, and a₄ are related to the descriptors proposed by Hjorth.

An alternative method of EEG signal analysis is based on measuring the distribution of intervals between zero and other level crossings, or between maxima and minima.

A level crossing may be defined in general terms as the time at which a signal (t) passes a certain amplitude level b; b = 0 is a special case referred to as zero crossing (Fig. 54.5). Knowledge of the probability density function of the intervals between successive zero crossings can be useful in characterizing some statistical properties of the signal (t) (mean value 0). p₀(τ) can be called the probability distribution density function of the intervals between any two successive zero crossings and p₁(τ), corresponding to the total time τ between successive zero crossings at which the signal changes in the same direction (i.e., from positive to negative or vice versa). In practice, these functions can be approximated by computing histograms of the interval length between two successive zero crossings or between zero crossings at which the signal has a derivative with the same sign. The moments of the distribution function can also be computed; the simplest case is to compute the average number of zero crossings per time unit (N₀) (e.g., per second) of the signal (t):

This means that the number of zero crossings per time unit N₀ equals the reciprocal of the mean interval length τ.

In some cases it is helpful to determine the probability density function of the intervals between two adjacent zero crossings where the sign of (t) changes from negative to positive or vice versa. In this case, it is necessary to compute additionally the zero crossing interval distribution of the first derivative of (t), (t) (i.e., (t) = d(t)/dt). If the signal (t) to be analyzed is quasi-stationary and has a gaussian distribution, a mathematical relation between N_k, the average rate of zero crossings of the kth derivative of (t), and the power spectrum S_xx(f) can be shown (32, 33 and 34):

In interval analysis, only the values N_k for k = 0, 1, 2 are usually computed. N₀ thus represents the average rate of zero crossings of d(t)/dt (i.e., the rate of intervals between extremes of the signal (t)); N₂ represents the average rate of zero crossings of d²(t)/dt² (i.e., the rate of intervals of the a nonparametric or a parametric approach that is combined inflection points of (t)).

It can be shown (32) that expression 54.22 can separately also be given in terms of the autocorrelation function:

where f_g is the so-called gyrating frequency (35).

These relations between the number of zero crossings per time unit and either spectral moments or the autocorrelation function for EEG signals have been studied in detail by Saltzberg and Burch (36), who concluded that, when the purpose is to monitor long-term changes in the statistical properties of EEG signals, it is legitimate to use average zero-crossing rates to calculate moments of the power spectral density.

Instead of measuring intervals between zero crossings, one can characterize a signal by determining intervals between successive maxima (or minima), which defines a “wave,” or between a maximum and the immediately following minimum or vice versa, which defines a “half-wave.” The section “Mimetic Analysis” considers some of the variants of interval analysis as applied to EEG signals; the straightforward applications of interval analysis are described in the section “Time-Frequency Analysis.”

The previous section considered the statistical properties of EEG signals as realizations of random processes, explaining how such signals can be characterized by the corresponding probability distribution and its moments, by the autocorrelation function or the power spectrum, or by distribution of intervals between level crossings. In all cases, the EEG was treated as a stochastic signal without a specific generation model. Therefore, all the previously described methods and related ones are nonparametric methods. Parametric methods may also be used to analyze EEG signals; in such cases one assumes the EEG signal to be generated by a specific model. For example, assuming that the EEG signal is the output of a linear filter given a white noise input allows characterization of the linear filter by a set of coefficients or parameters (e.g., it may correspond to an autoregressive model as explained below).

Therefore, EEG analysis methods can be divided into two basic categories, parametric and nonparametric. Such a division is conceptually more correct than the more common differentiation between frequency and time domain methods because, as has been explained, such methods as power spectra in the frequency domain and interval analysis in the time domain are closely related; indeed, they represent two different ways of describing the same phenomena. The methods of EEG analysis described here are classified as shown in Table 54.1.

Not all EEG analysis methods can be assigned to one of the two general categories just described. Those having mixed character (i.e., methods that have, as a starting point, a nonparametric or parametric approach that is combined with pattern recognition techniques) must be considered separately. The latter fall into the category of pattern recognition methods. Last, this section shall discuss topographic analysis methods, in which the emphasis is on topographic relations between derivations. Not included here are the evoked potentials, which are discussed elsewhere.

A thorough review of the main techniques currently in use in EEG analysis has been edited by Gevins and Remond (37). For more details on methods of analyzing brain electric signals, the reader is referred to this authoritative handbook.