Multivariate approach for estimating the local spectral F-test and its application to the EEG during photic stimulation

https://doi.org/10.1016/j.cmpb.2018.05.010Get rights and content

Highlights

  • The sum of N local spectral F-tests is proposed as new evoked response detector.

  • The detector performance always improves using simulated signals.

  • Results with EEG data also indicate better performance in detection for N > 2.

Abstract

Background and objective

The local spectral F-test (SFT) corresponds to a statistical way of assessing whether the spectrum of a signal is flat in the vicinity of a specific frequency. The power of this univariate test (comparing one frequency component  against its neighbours using only one signal) depends on the signal-to-noise ratio, which is fixed in the case of electroencephalogram (EEG) analysis. However, this limitation could be overcome by considering more signals in the analysis. Thus, this work presents an alternative multivariate approach for estimating the local SFT.

Methods

Probabilities of detection and false alarm studies were performed for this new detector using Monte Carlo simulations and theoretically whenever possible. The application was illustrated in recorded EEG data collected during photic stimulation.

Results

The results showed that it is worth using more channels if available, since the probability of detecting a response tends to increase with increasing number of signals. In the application to the EEG during photic stimulation, the best results were obtained by using N > 2 signals (around 30% more accurate when compared with the univariate case. The false positive levels were maintained below 5%).

Conclusion

Consequently, it is conjectured that it is always better to apply the proposed method if more than one EEG signal with the same signal-to-noise ratio (SNR) is available. For the case where the SNRs are different, a guideline has been given to improve the detection.

Introduction

The local spectral F-test (SFT) corresponds to a statistical way of assessing whether the spectrum of a signal is flat in the vicinity of a specific frequency [1]. It differs from another spectral F-test that aims at assessing whether two spectral estimates pertain to a population with same theoretical spectrum [2]. The local SFT has been used for detecting auditory evoked responses embedded in the electroencephalogram (EEG) [1] and visual steady-state evoked responses [3].

The statistics is obtained by calculating the ratio between the spectral estimate at a given frequency and the average of the spectral estimates at the neighbouring frequencies. The sampling distribution of the test under the null hypothesis of flat spectrum in Gaussian signals is obtained as a F (Fisher–Snedecor) distribution, since both numerator and denominator are distributed as a chi-squared distribution and are independent. The alternative hypothesis of a spectral peak at a given frequency has been investigated by [3], leading to a type I, non-central F distribution with noncentrality parameter proportional to the signal-to-noise ratio (SNR). The distribution under alternative hypothesis is useful in order to theoretically predict the power of the statistical test.

However, the power of this univariate test (comparing one frequency component against its neighbours using only one signal) depends on the SNR, which is fixed in the case of EEG analysis. Therefore, averaging techniques must be used in order to allow the detection of evoked responses to periodic stimulation in signals with low SNR, which may lead to long stretches of EEG signals to be used. This may constitute a limitation, for example, in intra-surgical monitoring – where the delays before alarms could be raised – or in paediatric audiometric tests – where the children become restless.

In order to overcome this limitation, [4] proposed a bivariate SFT, obtained as the sum of two local SFT. The performance of this new detector, in comparison with simple SFT, was found to be superior both in simulated and EEG under photic stimulation data.

The present work aims at extending such bivariate local F-test to include more signals/channels. The technique is evaluated with Monte Carlo simulations and applied to the EEG of twelve subjects under photic stimulation.

Section snippets

Local spectral F-test - Fy(fo)

For a single-input, single-output linear model with additive noise, the local spectral F-test applied to a discrete-time output signal y[k] is defined as the ratio between the power in frequency fo and the average power in L neighbouring frequencies [1], i.e. Fy(fo)=|Y(fo)|21Li=oL/2ioo+L/2|Y(fi)|2where Y(fo) is the Discrete Fourier Transform of y[k] evaluated at frequency fo. Y(fi) are the Discrete Fourier Transform values at the L closest neighbouring frequencies to fo. It is assumed that L

New method: additive local spectral F-test – FΣ(fo)

Consider the linear model of EEG in Fig. 1,

where x[k] represents the external stimulus, which is filtered by Hi(f) to drive the evoked responses at different locations on the scalp. The yi[k] represents the EEG signals, which are the addition of evoked responses and the background noise, represented by ni[k].

Assuming that the evoked response spreads out to more than one EEG channel, the Additive Local Spectral F-test is proposed as FΣ(fo)=i=1NFyi(fo)where N is the number of signals.

It is well

Monte Carlo simulation

The critical values as a function of L and for increasing N is shown in Fig. 2. Eqs. (1) and (4) were used for N = 1 and for N > 1, respectively. An increase in the critical values can be noted for increasing N. Additionally, it must be pointed out that the critical values obtained by simulation are in close agreement with the theoretical ones obtained for N = 1 [1] and for N = 2 [6].

The impact in the false positive rate of FΣ(fo) when adding more signals to the estimation is shown in Fig. 3.

Conclusion

In this work the bivariate local F-test proposed by [4] has been extended to accommodate N signals. As its predecessors this new technique has been proposed for assessing if the spectrum at a chosen frequency, normally the frequency related to the response, is statistically different from the neighbour frequencies.

With the addition of more than one signal, it has been shown that the minimum SNR that would result in an improvement in the probability of detection can be determined by continuously

Ethical approval

Work approved by the Local Ethics Committee. (CEP/UFV No. 1.616.098).

Funding

The authors acknowledge FAPEMIG, CAPES and CNPq for financial support.

Conflict of interest

The authors do not have financial and personal relationships with other people or organizations that could inappropriately influence (bias) their work.

References (8)

There are more references available in the full text version of this article.

Cited by (8)

  • Choosing multichannel objective response detectors for multichannel auditory steady-state responses

    2021, Biomedical Signal Processing and Control
    Citation Excerpt :

    Rocha et al. in [19] proposed the Multichannel Local F-test, which led to the consistently higher detection rates in the EEG during photic stimulation in comparison with using a single channel. Other studies [20–23] have also led to an increase in the detection rate of detector with the use of information from more than one EEG channel. In a different scenario, results presented in [24,25] showed the advantages of using more than one signal in the photoelectrochemical analysis.

  • A bayesian approach to the spectral F-Test: Application to auditory steady-state responses

    2020, Computer Methods and Programs in Biomedicine
    Citation Excerpt :

    This characteristic enables the analysis of steady-state evoked potentials by objective response detection (ORD) techniques in the frequency domain, which seek to identify the manifestations of the stimulus in the magnitude and/or phase of specific spectral components present in electroencephalographic (EEG) recordings [7–10]. The most widely used ORD techniques are the spectral F-test (SFT) [11–13], phase synchrony measure (PSM) [14], magnitude squared coherence (MSC) [15–18] and circular T2 test (T2circ) [19]. Comparative studies [20–22] have been performed to determine the most efficient method for application to steady-state potentials; however, the range of available results indicates that no consensus has been reached in the field regarding which technique performs the best.

  • Multi-channel and multi-harmonic analysis of Auditory Steady-State Response detection

    2024, Computer Methods in Biomechanics and Biomedical Engineering
View all citing articles on Scopus
View full text