Time-frequency methods

to the functional brain activation vary considerably between subjects. In the classical approach one has to try out a number of possible frequency bands in the search for the ones that display the most pronounced effects—so called reactive frequency bands. As was pointed out by [Pfurtscheller, 1999] it is important to examine event-related changes in signal energy from the broader perspective of the entire time-frequency plane.

In order to accomplish this, one needs to estimate the energy density of the signal in the time-frequency space. In the literature many methods were proposed for this purpose, e.g.:

• Spectrogram [Makeig, 1993],

• Bandpass filtering in overlapping bands [Graimann et al., 2002], • Scalogram [Tallon-Baudry et al., 1996],

• Smoothed pseudo Wigner-Ville transform [Lachaux et al., 2000]),

• Estimate of energy density derived from the matching pursuit (MP) parame-

terization [Durka et al., 2001b],

The mathematical basis, properties, and MATLAB routines to compute the above mentioned estimators of the energy are described in Sect. 2.4.2. Different estimates of the time-frequency distribution of signal energy offer different trade-offs between temporal and spectral resolution. For example, the scalogram has high temporal res- olution and low frequency resolution at high frequencies, low temporal resolution and high frequency resolution at low frequencies. The presence of cross-terms in some of the time-frequency estimators must be taken into account when interpreting the ERD/ERS in the time-frequency space. In contrast, the matching pursuit (MP) parameterization theoretically provides optimal time-frequency resolution through- out the time-frequency plane. Different estimators of energy density were compared in [Zygierewicz et al., 2005]. The results yielded by different estimators gave com- patible results, however those obtained by MP procedure provided more detailed information of the time-frequency structure of the ERD/ERS.

The time frequency estimator of energy density E(t, f ) can be used to evaluate the ERD/ERS in the time-frequency plane in accordance with the general definition (4.17):

ERD/ERS(t, f ) = E(t, f )tr− B( f )

B( f ) (4.23)

where E(t, f )tris the energy density at(t, f ) averaged across trials, and B( f ) is the

mean energy of baseline at frequency f averaged across trials.

The important question to be answered is: which of the ERD/ERS effects are sig- nificantly above the fluctuation level? The statistical problem related to that question is the assessment of significance in the statistical maps (see Sect. 1.5.3.2). In [Durka

et al., 2004, Zygierewicz et al., 2005] a massive univariate approach with the mul- tiple comparisons problem controlled with FDR (Sect.1.5.3.3) was proposed as an efficient solution to that question, and will be described below.

For the null hypothesis of no changes at the given time-frequency coordinates, we have to reduce the resolution to time-frequency boxes(Δt × Δ f ). There are two reasons for decreasing the time-frequency resolution:

• The time and frequency resolution are bounded by the uncertainty principle,

which for the frequencies defined as the inverse of the period (Hz) gives: Δt× Δ f ≥_4π1. The lower bound is only reached by the Gabor functions

• In real data there are big variations of energy estimates due to different kinds of

noise. Increasing the product Δt× Δ f reduces the variance of energy estimate Results from [Durka et al., 2004], suggest that Δt× Δ f = 1/2 gives robust results. This introduces a discretization of the time-frequency plane into resels (resolution elements) r(i, j). In order to obtain En(i, j) — energy in resels rn(i, j) (subscript n

denotes the trial) — we integrate7energy density En(t, f ):

En(i, j) =

Z _(i+1)·Δt

i·Δt

Z _{( j+1)·Δ f}

j·Δ f En(t, f )dtd f (4.24)

At this point, we may proceed to testing the null hypothesis of no significant changes in En(i, j). The distribution of energy density for a given frequency is not

normal. However, in many practical cases it can be transformed to an approximately normal one using an appropriate Box-Cox transformation [Box and Cox, 1964]. The Box-Cox transformations are the family of power transformations:

BC(x,λ) =



xλ₋₁

λ i f λ= 0

log(x) i f λ = 0 (4.25)

For each frequency j the λ parameter is optimized by maximization of the log- likelihood function (LLF) [Hyde, 1999] in the reference period:

λoptj = max λ {LLF(λ)} = maxλ  −m 2log σ 2 BC(x,λ)+ (λ − 1) m

∑

where m is the length of data x, x∈ {En(i, j) : i ∈ tb,n = 1,...,N}. The optimal λoptj is then used to transform all the resels in frequency j. Standard parametric tests can be applied to the normalized data. However, we cannot a priori assume equal variances in the two tested groups. The known solution to the problem of variance

7_{In the spectrogram equation (2.84) and scalogram equation (2.95), the energy is primarily computed on}

the finest possible grid and then the integration is approximated by a discrete summation. In case of MP the integration of equation (2.117) can be strict. The procedure was described in detail in [Durka et al., 2004].

heterogeneity in the t-test is Welch’s [Welch, 1938] correction of the number of de- grees of freedom. The test can be formulated as follows for each of the resels(i, j). The null hypotheses H₀i, jand alternative hypotheses H₁i, j:

H₀i, j: X( j)b,tr = X(i, j)tr (4.27)

H₁i, j: X( j)b,tr = X(i, j)tr (4.28) where: X( j)b,tris the normalized energy in the baseline time averaged across base- line time and trials, and X(i, j)tris the normalized energy in resel(i, j) averaged across trials. The statistics is:

t(i, j) = X( j)b,tr− X(i, j)tr sΔ

(4.29) where sΔis the pooled variance of the reference epoch and the investigated resel. The corrected number of degrees of freedom ν is:

ν= _s2 1 n1+ s2 2 n2 2  s2₁ n1 2 n1−1 +  s2₂ n2 2 n2−1 (4.30)

where s1is the standard deviation in the group of resels from the reference period,

n1= N ·Nbis the size of that group, s2is the standard deviation in the group of resels from the event-related period, and n2= N is the size of that group.

If we cannot assume Xn(i, j) to be distributed normally, we estimate the distribu-

tion of the statistic t from the data using (4.29), separately for each frequency j: 1. From the X(i, j), i ∈ tbdraw with replacement two samples: A of size N and B

of size N· Nb

2. Compute t as in (4.29): tr( j) = XA − XB

sΔ

, where sΔis pooled variance of samples A and B.

. . . and repeat steps 1 and 2 Nrep times. The set of values tr( j) approximates the distribution Tr( j) at frequency j. Then for each resel the actual value of (4.29) is compared to this distribution:

p(i, j) = 2min{P(Tr( j) ≥ t(i, j)),1 − P(Tr( j) ≥ t(i, j))} (4.31) yielding two-sided p(i, j) for the null hypothesis Hi j

0. The relative error of p is (c.f. [Efron and Tibshirani, 1993])

err=σp p = ' (1 − p) pNrep (4.32)

Although the above presented method is computationally intensive, at present it causes no problems in most of the standard applications. However, corrections for multiple comparisons imply very low effective critical values of probabilities needed to reject the null hypothesis. For the analysis presented in [Durka et al., 2004] critical values of the order of 10−4were routinely obtained. If we set p= 10−4in (4.32), we obtain a minimum Nrep= 106resampling repetitions to achieve 10% relative error for the values p(i, j).

In [Durka et al., 2004] either parametric or resampled statistical tests were applied to energies in each resel separately. However, the very notion of the test’s confidence level reflects the possibility of falsely rejecting the null hypothesis. For example, a confidence level of 5% means that it may happen in approximately one in 20 cases. If we evaluate many such tests we are very likely to obtain many such false rejections. This issue is known in statistics as the issue of multiple comparisons, and there are several ways to deal with it properly.

To get a valid overall map of statistically significant changes we suggest the ap- proach chosen in [Durka et al., 2004] that is a procedure assessing the false discovery rate (FDR, proposed in [Benjamini and Hochberg, 1995]). The FDR is the ratio of the number of falsely rejected null hypotheses (m0) to the number of all rejected null hypotheses (m). In our case, if we control the FDR at a level q= 0.05, we know that among resels declared as revealing a significant change of energy, at most 5% of them are declared so falsely. [Benjamini and Yekutieli, 2001] proves that the follow- ing procedure controls the FDR at the level q under positive regression dependency, which can be assumed for the time-frequency energy density maps:

1. Order the achieved significance levels pi, approximated in the previous section

for each of the resels separately, in an ascending series: p1≤ p2≤ ··· ≤ pm

2. Find k= max i pi≤ i mq 1 (4.33) 3. pkis the effective significance level, so reject all hypotheses for which p≤ pk.

Resels r(i, j) are marked significant if the null hypothesis H0i, jcan be rejected using the significance level pk for the probabilities p(i, j) of the null hypothesis (4.27).

An example of ERD/ERS time-frequency maps together with the assessment of their significance obtained with the above described procedure is illustrated inFigure 4.20.

4.1.7.3.4 ERD/ERS in the study of iEEG The ERD/ERS methodology was