# New method for analysis of nonstationary signals

- Robert A Stepien
^{1}Email author

**5**:3

https://doi.org/10.1186/1753-4631-5-3

© Stepien; licensee BioMed Central Ltd. 2011

**Received: **3 January 2011

**Accepted: **22 June 2011

**Published: **22 June 2011

## Abstract

### Background

Analysis of signals by means of symbolic methods consists in calculating a measure of signal complexity, for example informational entropy or Lempel-Ziv algorithmic complexity. For construction of these entropic measures one uses distributions of symbols representing the analyzed signal.

### Results

We introduce a new signal characteristic named *sequential spectrum* that is suitable for analysis of the wide group of signals, including biosignals.

The paper contains a brief review of analyses of artificial signals showing features similar to those of biosignals. An example of using sequential spectrum for analyzing EEG signals registered during different stages of sleep is also presented.

### Conclusions

Sequential spectrum is an effective tool for general description of nonstationary signals and it its advantage over Fourier spectrum. Sequential spectrum enables assessment of pathological changes in EEG-signals recorded in persons with epilepsy.

## Keywords

## Introduction

There are many ways of using symbolic dynamics for time series analysis and all of them need coding i.e. conversing of the analyzed time series into symbols series. The differences between different symbolic methods are in coding procedure and/or in calculated complexity measure, such as entropy [1–3] or Lempel-Ziv complexity [4, 5]. These characteristics describe dynamics of the process generating the analyzed signal. Complexity measures are usually scalars containing only general information about complexity of a process generating the analyzed signal. The comparison of processes about the similar dynamic complexity is ineffective, it is a major inconvenience of these measures.

We introduce a new symbolic measure that is similar to frequency characteristics and we call it a sequential spectrum (*seq-spectrum*) in analogy to frequency spectrum. Sequential spectrum like Lempel-Ziv complexity, belongs to methods applying short ordered sequences of symbols (*tuples*). The important difference is that in the case of seq-spectrum it is the values of first derivative of the signal that is encoded. Moreover, in seq-spectrum only *mono-sequences* i.e. tuples containing only one kind of symbol are considered and lengths of mono-sequences correspond to frequencies.

However, in spite of analogies to frequency spectrum, sequential spectrum is not a transformation of the signal to the frequency space. It does not exist a reverse procedure enabling reconstruction of the signal from its seq-spectrum.

## Methods

*binary occupancy*is calculated i.e. relative contribution of the mono-sequences of length

*N*into the analyzed symbols series.

*x*(

*i*) we calculate the first range differences and represent them by the symbols from the two-elements set {0,1}:

As a result of signal's encoding we obtain binary symbol series, *P*, for example [1,1,1,0,0,0,1,1,1,1,0,0,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0].

*N*x

*s*], i.e. mono-sequences of length

*N*consisting only one type of symbol (

*s*≡0 or

*s*≡1), are counted in series

*P*. As the result we obtain the cardinality,

*L*[

*N*x

*s*], and we repeat counting procedure for all possible values of

*N*(limited by the length of symbols series

*P*), so obtaining the distribution of cardinalities. Knowing cardinalities

*L*[

*N*x

*s*] we calculate

*binary occupancy*,

*O*[

*N*x

*s*], for mono-sequences, [

*N*x

*s*], in the binary symbol series

*P*,

where *I* is the length of symbol series *P*, i.e. the total number of symbols in this series. In other words, binary occupancy characterizes distribution of monotonic intervals of length *N* in the analyzed time series - decreasing (s≡ 0) or increasing (s≡ 1) intervals.

*f*and the length,

*N*, of the mono-sequence [

*N*x

*s*] (Figure 2). If the sampling frequency of signal is

*f*

_{ s }and

*N*is the length of a mono-sequence then the related frequency is:

Similarly to Fourier spectrum, a width of seq-spectrum depends on the sampling frequency and it is usually wider for higher sampling frequencies. However, with increasing sampling frequency seq-spectrum extends towards longer sequences i.e. towards lower frequencies unlike Fourier spectrum that extends toward higher frequencies i.e. shorter wavelengths (shorter sequences). But it is not always the case, because when sampling frequency increases additional local maxima and minima appear in the time series representing the analyzed signal so mono-sequences might become shorter.

The small resolution of seq-spectrum in the range of short mono-sequences is a consequence of the fact that the length *N* is a positive integer number.

The shape of seq-spectrum and its width contains information about general properties of the analyzed signal.

## Results

Before using seq-spectrum for analysis of biosignals, the features of seq-spectrum were tested on well defined artificial signals. Knowing the results of analysis of artificial signals with different characteristics we could start to analyze complex natural signal like EEG.

### Analysis of artificial signals

*N*x0] and [

*N*x1] (Figure 3).

The plots on the Figure 3 show that the shape and width of seq-spectrum of stationary noise signal depend on the fluctuation exponent, α, that is on level of correlation in the signal - stronger correlation leads to wider seq-spectrum. Seq-spectrums for both kinds of mono-sequences [*N*x0] and [*N*x1] are identical for the same fluctuation exponent α.

where *a* = 1 and *b* = 0.

*N*x0] and [

*N*x1] are different from each other (Figure 4). It is a consequence of shortening of the period of the signal. So, for the chirp based on the sinus function, each negative arm is always shorter than the preceding positive arm i.e. the mono-sequences consisting of the symbols 0 are shorter than the preceding mono-sequences consisting of the symbol 1; it is opposite for the chirp based on the cosine function

The plot of seq-spectrum for chirp (Figure 4) shows linear relation between lengths of mono-sequences and binary occupancy. This result illustrates an advantage of seq-spectrum over classical Fourier spectrum because Fourier spectrum calculated for chirp does not show linear relation arising from (Eq.5).

We obtained identical seq-spectra for the mono-sequences [*N*x0] and [*N*x1] when comparing the results for signal with the same level of added noise. The added noise makes seq-spectrum wider then seq-spectrum of the signal without noise. Pattern of changes of binary occupancy in seq-spectrum is regular - the values of occupancy either increase or decrease with increasing noise amplitude for the same length of mono-sequences.

Another feature of biosignals is their nonstationarity. Nonstationary signals are usually analyzed using moving windows technique. Seq-spectrum is more useful than Fourier spectrum for nonstationary signals.

*N*= 1 and

*N*= 2, contribute to the seq-spectrum (Figure 8a). In the case of mono-sequences composed of symbol 1 the seq-spectrum is much wider, however dominating contributions are of mono-sequences of lengths

*N*= 1 and

*N*= 2 (Figure 8b). So, sequential spectrum of logistic map depends on the symbol of which the mono-sequences are composed.

### Seq-spectrums of real biosignals

Seq-spectrum can be useful for sleep staging - the values of binary occupancy as represented by seq-spectrum changes depending on the stage of sleep. To demonstrate this we used hypnograms analyzed by medical doctors and we traced the seq-spectra in different sleep stages as marked on these medical hypnograms.

_{ RW }([

*N*x0]), i.e. we calculate the differences between pairs of seq-spectrum values in all sleep stages and the respective value for the chosen stage. Figure 10. shows relative seq-spectrum calculated relatively to the state of vigilance; the straight horizontal line is the seq-spectrum of the reference state. In the case of the slow-wave sleep (stages 3 and 4) this relative spectrum is negative for mono-sequencecs of the length from 1 to 9, while for longer mono-sequences the relative spectrum is positive. For stage S2 the relative spectrum is similar to those for stages 3 and 4.

In most cases in the spectrum it is also possible to distinguish in the relative seq-spectrum two ranges but for REM stage the values are either only positive or only negative. However, it is not the rule - the curve of the relative spectrum for REM is sometimes similar to that for stage S1.

For stage S1 situation is more complicated. It is possible to distinguish three or four ranges in which the difference of binary occupancies between stage S1 and the state of vigilance is positive or negative. Border of these ranges are not usually sharp - for short mono-sequences (for lengths 1, 2) the value of the relative cover is positive. The range in which the values are negative usually comes after this range (within the limits from 3-4 to 9). The next range contains again positive values.

The question arises whether universal ranges of the values of the binary occupancy do exist. This would make it a tool for automatic sleep staging. But it is doubtful since EEG signals show strong individual diversification.

Comparing the seq-spectra O([*N*x0]) and O([*N*x1], we see that for the normal activity choice of the symbol is unimportant - seq-spectra are identical. It is interesting that for strong seizures differences between seq-spectra are not significant, while for the case of weak seizures they really are.

## Discussion and Conclusions

Sequential spectrum is an effective tool for general description of nonstationary signals. The results of analysis of EEG-signals during sleep and epilepsy show that sequential spectrum is useful characteristic for analysis of biosignals. The spectral analysis we can widen on the analysis time-sequential (seq-spectrogram) after the fashion time-frequency methods.

Sequential spectrum can also be used to categorize processes generating signals. Between seq-spectra O([*N*x0]) and O([*N*x1]) of stationary random signals high congruity exists. It decreases when autocorrelation of the signal grows or the signal is non-stationary, or chaotic. This requires further research because asymmetry between spectra O([*N*x0]) and O([*N*x1]) appears also for signals in which epochs appear about the constant amplitude. It results from the way of coding the signal (Eq.1); when the signal is constant there is an assigned symbol "1".

## Declarations

### Acknowledgements

I thank Prof. Wlodzimierz Klonowski for fruitful discussions. This work was supported by Nalecz Institute of Biocybernetics and Biomedical Engineering Polish Academy of Sciences statutory activity 4.4/st/11.

## Authors’ Affiliations

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## Copyright

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