Electroencephalography (EEG) signals are resultants of extremely complex
brain activity. Some details of this hidden dynamics might be accessible
through e.g. joint distributions $\rho_{\Delta t}$ of signals of pairs of
electrodes shifted by various time delays (lag $\Delta t$). A standard approach
is monitoring a single evaluation of such joint distributions, like Pearson
correlation (or mutual information), which turns out relatively uninteresting -
as expected, there is usually a small peak for zero delay and nearly symmetric
drop with delay. In contrast, such a complex signal might be composed of
multiple types of statistical dependencies - this article proposes approach to
automatically decompose and extract them. Specifically, we model such joint
distributions as polynomials estimated for all considered lag dependencies,
then with PCA dimensionality reduction find dominant dependency directions
$f_v$. This way we get a few lag dependent features $a_i(\Delta t)$ describing
separate dominating statistical dependencies of known contributions:
$\rho_{\Delta t}(y,z)\approx \sum_{i=1}^r a_i(\Delta t)\, f_{v_i}(y,z)$. Such
features complement Pearson correlation, extracting hidden more complex
behavior, e.g. with asymmetry which might be related with direction of
information transfer, extrema suggesting characteristic delays, or oscillatory
behavior suggesting some periodicity. While this early article is initial
fundamental research, in future it might help e.g. with understanding of cortex
hidden dynamics, diagnosis of pathologies like epilepsy, determination of
precise electrode position, or building brain-computer interface.

该论文提出了一种自动分解和提取脑电信号的方案，将信号的统计依赖作为多项式建模，并使用 PCA 降维找到显著的依赖方向，从而获取描述分离统计依赖的滞后相关特征，此方法提取了更复杂的行为并有助于理解皮层的潜在动力学，诊断癫痫等病症，确定精确的电极位置或构建脑机接口。

时延多特征相关性分析从 EEG 信号中提取微小依赖

Time delay multi-feature correlation analysis to extract subtle  dependencies from EEG signals

Revealing hidden dynamics from the stochastic data is a challenging problem
as randomness takes part in the evolution of the data. The problem becomes
exceedingly complex when the trajectories of the stochastic data are absent in
many scenarios. Here we present an approach to effectively modeling the
dynamics of the stochastic data without trajectories based on the weak form of
the Fokker-Planck (FP) equation, which governs the evolution of the density
function in the Brownian process. Taking the collocations of Gaussian functions
as the test functions in the weak form of the FP equation, we transfer the
derivatives to the Gaussian functions and thus approximate the weak form by the
expectational sum of the data. With a dictionary representation of the unknown
terms, a linear system is built and then solved by the regression, revealing
the unknown dynamics of the data. Hence, we name the method with the Weak
Collocation Regression (WCR) method for its three key components: weak form,
collocation of Gaussian kernels, and regression. The numerical experiments show
that our method is flexible and fast, which reveals the dynamics within seconds
in multi-dimensional problems and can be easily extended to high-dimensional
data such as 20 dimensions. WCR can also correctly identify the hidden dynamics
of the complex tasks with variable-dependent diffusion and coupled drift, and
the performance is robust, achieving high accuracy in the case with noise
added.

本文提出了一种基于 Fokker-Planck 方程的弱形式和高斯核插值的回归方法，对多维随机数据的动力学进行建模，并成功地追踪了隐藏动力学。