自动化模型选择在方差估计与MEG/EEG信号空间白化中的应用

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"这篇文章主要探讨了在脑电图(EEG)和磁脑图(MEG)信号处理中，如何自动选择合适的模型进行协方差估计和信号空间白化的方法。作者Denis A. Engemann等人来自多个研究机构，涵盖了神经科学、精神病学和电信技术等多个领域。文章详细阐述了多种估算方差的策略，并强调了这些方法在精确估计数据方差中的重要性。" 文章的核心关注点在于提高非侵入性脑成像技术，如EEG和MEG的信号分析精度。协方差估计是这一领域的基础，它涉及理解不同传感器记录的信号之间的关系。在EEG和MEG信号中，这种关系反映了大脑神经活动的时空模式。为了更好地解析这些模式，文章介绍了包括主成分分析(PCA)和因子分析(FA)在内的统计学习方法。 PCA是一种常见的数据分析技术，它通过降维来突出显示数据的主要成分，有助于识别信号中的主要变异性。因子分析则试图解释变量间的共变性，通过找出隐藏的因子来简化数据结构。这两种方法在M/EEG信号处理中被用于提取重要的特征向量，以减少数据的复杂性并增强信号的可解释性。文章还讨论了协方差估计和信号白化的概念。协方差估计是确定信号随机变量之间线性关系的过程，对于识别脑活动的空间分布至关重要。而信号白化是一种预处理步骤，其目的是消除信号中的关联性，使各信号分量独立且具有相同的方差，从而优化后续的分析步骤。模型选择是另一关键主题，它涉及到在一系列可能的模型中选择最合适的那一个。在M/EEG信号处理中，这可能意味着找到最佳的滤波器设置或传感器组合，以最大程度地揭示大脑功能连接的细节。文章可能提出了自动化模型选择的算法，以减少人为干预，提高方法的稳健性和效率。关键词中提到的“Statistical learning”表明，作者可能使用了机器学习技术，如贝叶斯方法或支持向量机(SVM)，以自动适应和优化模型选择过程。这种方法可以适应复杂的数据结构，并在大量模型中自动寻找最佳解决方案。这篇文献深入探讨了在EEG和MEG数据处理中，如何通过自动化模型选择和有效的统计学习方法，进行协方差估计和信号空间白化，以提升脑部活动的解析能力。这对于理解和研究大脑功能、疾病诊断以及神经科学研究具有重要意义。

straightforward to verify that C

−

¼ Λ

−1

is a valid inverse square root

of C. An alternative is the symmetric matrix C

−

¼ U

−1

, which we

will use to visualize whitened data.

To reduce redundancy: Eq. (2) reveals that minimum-norm estimates

(MNE) actually implements what is known as Tikhonov regularization

(Tikhonov and Arsenin, 1977) or Ridge regressio n in the ﬁeld of statistical

learning (Hoerl and Kennard, 1970). As a consequence, if the gain matrix

and the data are appropriately whitened, general conditions of statistical

regression models apply to the magnetoencephalography and electroen-

cephalography (M/EEG) inverse problem. Minimum-norm estimates,

therefore, rely on the speciﬁcation of the noise covariance matrix that

needs to be estimated from the data. Or in other words, the quality of

the inverse solution depends on the quality of the covariance estimate.

This holds true for most other source localization in particular

beamformers such as LCMV and DICS (Veen et al., 1997; Gross et al.,

2001). However, this also applies to MNE variants such as dynamical sta-

tistics parametric mapping (dSPM) (Dale et al., 2000) or low resolution

brain electromagnetic tomography (sLORETA) (Pascual-Marqui, 2002),

as well as other distributed models such as minimum-current estimates

(MCE) (Uutela et al., 1999) or mixed-norm estimates (MxNE)

(Gramfort et al., 2012; Gramfort et al., 2013a). It therefore cannot be con-

sidered a local problem.

Covariance estimation

Model selection using cross-validation

The noise covariance estimator is typically applied to segments of

(M/EEG) data that were not used to estimate the noise covariance and

that typically include both, brain signals and noise. Its quality can

hence be assessed by investigating how well the model describes new

data. This idea of model quality assessment on unseen data is put into

practice by aggregating results over random partitions of the data, and

is referred to as cross-validation. Since data are assumed to follow a

multivariate Gaussian distribution, parameterized by a covariance ma-

trix C, the log-likelihood of some data Y reads:

L YjCðÞ¼−

Trace YY

−1



−

log 2πðÞ

det CðÞ



: ð3Þ

The higher this quantity on unseen data, the more appropriate the

estimated noise covariance C and the higher its success at spatiall y

whitening the data. The log-likelihood, hence, allows us to select the

best noise covariance estimators out of a given set of models using

cross-validation with left out data. In the following we will discuss po-

tentially relevant candidate strategies to estimate covariance matrices

on M/EEG data.

Empirical covariance and regularization

The empirical covarian ce matrix can be computed by C ¼

where Y contains the data of size N × T. With a sufﬁcient number of ob-

servations (T large), the sample covariance, which can be derived from

maximum likelihood, is a good estimator of the true covariance. Typi-

cally, a noise covariance is computed on baseline segments preceding

stimulation or for MEG on empty room measurements during which

no subject is present. The latter is however not possible for electroen-

cephalography (EEG) recordings for which the covariance estimation

relies on data segments considered not relevant for the task, typically

during baseline. Biological artifacts often contaminate the data leading

to outlier sample s, and sometimes the data statistics change o ver

time, for example due to changes in environmental noise or changes

in head position. If in such situations only a limited number of samples

is available, the empirical covariance tends to to suffer from high vari-

ance. The estimate then is noisy and unreliable for further analysis.

One typical way to reduce the variance of the covariance estimator is

to apply diagonal loading. It consists of amplifying the diagonal with a

hand-selected constant which attenuates the off-diagonal elements

that correspond to inter-sensor covariance:

¼ C þ αI; α N 0: ð4Þ

The value α is the regularization parameter. This diagonal weighting

of the covariance stabilizes MNE-like estimates by reducing the vari-

ance. However, the introduced bias amounts to assuming a stronger un-

correlated noise level which leads to underestimated amplitudes in the

source estimates. This especially applies to dSPM and sLORETA where

the noise variance is used to rescale MNE estimates and convert

them to statisti cal quantities such as F or T statistics. When used in

beamformers, such a regularization of the data covariance matrices

tends to increase th e point spread function of the spatial ﬁlters and

smear the estimates (Woolrich et al., 2011). In addition, hand-set regu-

larization raises a new problem, which is how to choose the value of α.

Shrinkage models

An improvement of the hand-selected regularization or shrinkage

approach introduced in the section called Covariance estimation is pro-

vided by the Ledoit–Wolf (LW) shrinkage model (Ledoit and Wolf,

2004). This covariance model constitutes an optimal weighted average

of th e invariant identity matrix and the empirical covariance matrix

(Eq. (5)). The LW covariance estimates C

takes the form of:

¼ 1−αðÞC þ αμ I; ð5Þ

where I stands for the identity matrix, μ is the mean of the diagonal el-

ements of C, and α is called the shrinkage parameter. The contribution of

Ledoit and Wolf (2004) is to provide a formula to compute the optimal

value for α. The solution is derived from the values of N, the number of

dimensions, and T, the number of samples. It is provided in closed form

and minimizes the mean squared error between the estimator and the

population covariance. The underlying assumptions of the LW estimator

are that the data are i.i.d. (independent identically distributed) which,

as we will see below, is not a valid assumption for M/EEG data. Howev-

er, Ledoit and Wolf (2004) have shown that the optimal shrinkage pa-

rameter guarantees C

to be well conditioned: matrix inve rsion is

numerically stable, and more stable than with the empirical covariance.

A data-driven extension to the Ledoit–Wolf estimator can be moti-

vated by Eq. (5). Instead of using the Ledoit–Wolf formula to compute

α, cross-validation and likelihood es timation on unse en data can be

compared over a range of α values to select the optimal regularization

parameter. The optimal α can then be determined as the one yielding

a covariance estimator with the maximum likelihood on unseen data.

Throughout the manuscript, models with data-driven shrinkage coefﬁ-

cient as in Eq. (5) will be referred to as SC.

Probabilistic principal component analysis (PPCA)

M/EEG measurements are obtained by sensor recordings at various

locations in space. They include signals from the brain but also artifacts.

Such signals and artifacts yield spatially structured patterns on the sen-

sor array. For example, a source in the brain that would be well modeled

by an equivalent current dipole ECD produces a dipolar pattern on the

sensors. If this dipole does not rotate, due to the physics of the forward

problem, the signal space spanned by this ECD is of dimension one. The

signal space is thus said to be of rank one. Both sources in the brain and

artifacts share this property of generating low rank signals on the sen-

sors. This is for example what justiﬁes the use of signal space projection

SSP (Uusitalo and Ilmoniemi, 1997). The id ea behind SSP is that the

noise subspace includes artifact-related sources of low dimensionality

and that it is approximately orthogonal with the subspace spanned by

the brain signals of interest. Therefore, projecting the data on the or-

thogonal of the noise subspace will remove artif acts and therefore

denoise the data.

Principal component analysis ( PCA) is a statistical method that is

built on this idea of low rank si gnal space. When using cla ssical PCA

330 D.A. Engemann, A. Gramfort / NeuroImage 108 (2015) 328–342

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