Hindriks Rikkert
Department of Mathematics, Faculty of Science, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands.
Neuroimage Rep. 2021 Apr 21;1(1):100007. doi: 10.1016/j.ynirp.2021.100007. eCollection 2021 Mar.
Over the last two decades, a large number of estimators have been proposed to assess brain connectivity from electroencephalography (EEG) and magnetoencephalography (MEG) data. The statistical theory underlying these estimators, however, is relatively underdeveloped. In particular, the theoretical relationships between different estimators were unknown until very recently. In a recent preprint, Nolte et al. derived formulas for such relationships under the assumption that the data are Gaussian. One of the results was that the phase-lag index and the lagged coherence concentrate around identical values for large sample sizes, i.e. have identical asymptotic limits. A proof of this statement, however, was only sketched, the model assumptions were not checked in experimental data, and sampling properties of the estimators were not considered. We derive the probability density of the relative-phase in the Gaussian model and use it to provide an alternative proof of the asymptotic equality of the phase-lag index and the lagged coherence. The proof is based on power series expansions of the Fourier coefficients of the phase-lag index and the lagged coherence, which are demonstrated to be hypergeometric functions. We also assess the sampling properties of the phase-lag index and the lagged coherence through numerical simulations from the Gaussian model. These demonstrate that throughout the entire parameter space of the model and for all sample sizes, the standard error of the phase-lag index is higher than that of the lagged coherence and thus establish that the lagged coherence is a uniformly better estimator than the phase-lag index. We use experimental EEG and MEG data to verify to what extent the Gaussian assumption is appropriate. Deviations from normality were observed precisely at frequencies for which EEG/MEG power spectra had local maxima, i.e. at oscillatory resonances. Depending on the data-set, the resonances were located in the delta, alpha, and beta frequency bands and correspond to the respective brain rhythms. Based on these observations, we propose to model EEG/MEG data with exponential power densities, which include the Gaussian and Laplace densities as special cases. Lastly, we demonstrate that the asymptotic equality of the phase-lag index and the lagged coherence, as well as the large relative standard errors of the phase lag index, also hold in experimental EEG/MEG data. This establishes that the lagged coherence is not only a better estimator for Gaussian data, but for experimental EEG/MEG data as well.
在过去二十年中，已经提出了大量估计器来从脑电图（EEG）和脑磁图（MEG）数据评估脑连接性。然而，这些估计器背后的统计理论相对欠发达。特别是，直到最近，不同估计器之间的理论关系仍不为人知。在最近的一篇预印本中，诺尔特等人在数据为高斯分布的假设下推导出了此类关系的公式。其中一个结果是，对于大样本量，相位滞后指数和滞后相干性集中在相同的值附近，即具有相同的渐近极限。然而，该陈述的证明仅作了概述，模型假设未在实验数据中进行检验，并且未考虑估计器的抽样特性。我们推导了高斯模型中相对相位的概率密度，并使用它来为相位滞后指数和滞后相干性的渐近相等提供另一种证明。该证明基于相位滞后指数和滞后相干性的傅里叶系数的幂级数展开，这些系数被证明是超几何函数。我们还通过高斯模型的数值模拟评估了相位滞后指数和滞后相干性的抽样特性。这些结果表明，在整个模型参数空间和所有样本量下，相位滞后指数的标准误差高于滞后相干性的标准误差，从而确定滞后相干性是比相位滞后指数更好的统一估计器。我们使用实验性EEG和MEG数据来验证高斯假设在何种程度上是合适的。恰好在EEG/MEG功率谱具有局部最大值的频率处，即在振荡共振处，观察到了与正态性的偏差。根据数据集的不同，共振位于δ、α和β频段，并对应于各自的脑节律。基于这些观察结果，我们建议用指数幂密度对EEG/MEG数据进行建模，其中高斯密度和拉普拉斯密度是特殊情况。最后，我们证明了相位滞后指数和滞后相干性的渐近相等，以及相位滞后指数的较大相对标准误差，在实验性EEG/MEG数据中也成立。这表明滞后相干性不仅是高斯数据的更好估计器，也是实验性EEG/MEG数据的更好估计器。
