Limitations of non-linear methods

There are several pitfalls in the application of non-linear methods. One of them is the fact that even for infinite-dimensional, stochastic signals low-dimensional esti- mates may be obtained, which was pointed out by [Theiler et al., 1992]. The temporal coherence of the data may be mistaken for the trace of non-linearity. Therefore be- fore applying non-linear methods one should first check, if indeed the signals have traces of non-linearity, since the non-linear process needs not to produce non-linear time series. The null hypothesis about the stochastic character of the signal may be checked by means of surrogate data technique (Sect. 1.6).

The non-linear methods rely to a large extent on the reconstruction of a phase space. In order to construct the phase space by embedding, rather long stationary

data epochs are needed. In case of biomedical signals long stationary data segments are often hard to find. Another problem is connected with the fact, that the non-linear methods are prone to systematic errors connected with the arbitrary choices (e.g., of the bin length in estimating probabilities, or lag of embedding procedure).

However, the biggest problem in application of non-linear methods is the fact that they are very sensitive to noise. As was pointed out by [Kantz and Schreiber, 2000] for a low-dimensional deterministic signal with a noise component of the order 2- 3%, it will be difficult to reasonably estimate correlation dimension, since in this case a significant scaling region of C(ε) could not be found; the reason being that the artifacts of the noise meet the artifacts of the overall shape of the attractor. Since biomedical signals contain a significant noise component one should approach non- linear methods with caution and apply them only in cases where the non-linearity of a signal is well established and linear methods seem not to work properly.

3

Multiple channels (multivariate) signals

The technological progress in the biomedical field has led to the construction of recording equipment which allows recording of activity from multiple sites. Today a typical dataset contains not only two or four but dozens or hundreds of channels. This is especially the case for EEG, MEG, and sometimes also for ECG and EMG signals. Analysis of multichannel data can give a better insight into the relations be- tween the investigated sites, but it is a challenging task. Besides many experimental and computational difficulties, the problem quite often lies in the proper application of existing mathematical tools. The techniques capitalizing on the covariance struc- ture of the multichannel (multivariate ) data are especially useful in this respect. In this chapter an introduction to basic aspects of multichannel data processing will be presented.

3.1 Cross-estimators: cross-correlation, cross-spectra, coherence

(ordinary, partial, multiple)

Joint moments of the order two: cross-correlation Rxy and cross-covariance Cxy,

were defined in Sect. 1.1 (equations 1.11 and 1.12) by means of ensemble averaging formalism. Under the assumption of ergodicity they can be expressed by the formu- las: Rxy(τ) = Z _∞ −∞x(t)y(t + τ)dt (3.1) Cxy(τ) = Z _∞ −∞(x(t) − μx)(y(t + τ) − μy)dt (3.2)

Cross-correlation and cross-spectrum are bound by means of Fourier transform and inverse Fourier transform (cf. Sect. 2.3.2.1.3):

Sxy( f ) = Z _∞ −∞Rxy(τ)e −i2π f τ_{dτ (3.3)} Rxy(τ) = Z _∞ −∞Sxy( f )e i2π f τ_{d f (3.4)}

Similarly to the power spectrum, cross-spectrum is usually computed by means of the Fourier transforms X and Y of the signals x and y :

Sxy( f ) = lim T→∞

1

TX( f ,T )Y

∗_{( f ,T ) (3.5)}

The fact that we have only a limited data window of length T has the same con- sequences as in cases of power spectra estimation (see Sect. 2.3.2.1). Usually non- rectangular window functions are used and smoothing is applied. Computation of cross-spectrum is implemeted in MATLAB Signal Processing Toolbox as cpsd func- tion. It estimates the cross power spectral density of the discrete-time signals using Welch’s averaged, modified periodogram method of spectral estimation.

Sxy( f ) is a complex value consisting of real and imaginary parts:

Sxy( f ) = Re(Sxy)( f ) + iIm(Sxy)( f ) (3.6)

In polar coordinates it can be expressed by the formula:

Sxy( f ) = |Sxy( f )|eiΦxy( f ), (3.7)

where |Sxy( f )| =

!

Re(Sxy)2( f ) + Im(Sxy)2( f ) is a modulus and Φxy( f ) =

tan−1 Im(Sxy)

Re(Sxy) is a phase.

Coherence is a measure which is often used in biomedical application. It is ex- pressed by the formula:

γxy( f ) =

Sxy( f )

!

Sx( f )Sy( f )

(3.8) where Sxand Syare spectra of signals x and y. Since Sxy( f ) ≤

!

Sx( f )Sy( f ), func-

tion γxy( f ) ≤ 1. Coherence shows the relation between two signals in frequency and

phase. The square of the coherence measures the spectral power in a given frequency common to both signals.

The above formula defines ordinary (bivariate) coherence. If a data set contains more than two channels, the signals can be related with each other in different ways. Namely, two (or more) signals may simultaneously have a common driving input from the third channel. Depending on the character of relations between channels, some of them may be connected directly with each other and some connections can be indirect (through other channels). To distinguish between these situations partial and multiple coherences were introduced.

The construction of partial coherence relies on subtracting influences from all other processes under consideration. For three channels partial coherence is defined as a normalized partial cross spectrum:

κxy|z( f ) = |Sxy|z( f )|

(Sxx|z( f )Syy|z( f )

(3.9)

where partial cross-spectrum is defined as:

For an arbitrary number of channels partial coherence may be defined in terms of the minors of spectral matrix S( f ), which on the diagonal contains spectra and off-diagonal cross-spectra: κi j( f ) = Mi j( f ) ! Mii( f )Mj j( f ) (3.11) where Mi j is a minor of S with the ith row and jth column removed. Its properties

are similar to ordinary coherence, but it is nonzero only for direct relations between channels. If a signal in a given channel can be explained by a linear combination of some other signals in the set, the partial coherence between them will be low.

Multiple coherence is defined by:

Gi( f ) =

'

1− det S( f )

Sii( f )Mii( f ) (3.12)

Its value describes the amount of common components in the given channel and the other channels in the set. If the value of multiple coherence is close to zero then the channel has no common components with any other channel of the set. The high value of multiple coherence for a given channel means that a large part of the variance of that channel is common to all other signals; it points to the strong relation between the signals. Partial and multiple coherences can be conveniently found by means of the autoregressive parametric model MVAR.