Motivation: The objective of EEG data analysis is to extract meaningful insights, enhancing our understanding of brain function. However, the high dimensionality and temporal dependency of EEG data present significant challenges to the effective application of statistical methods. This study systematically addresses these challenges by introducing a high-dimensional statistical framework that includes testing changes in the mean vector and precision matrix, as well as conducting relevant analyses.Specifically, the Ridgelized Hotelling's $T^2$ test (RIHT) is introduced to test changes in the mean vector of EEG data over time while relaxing traditional distributional and moment assumptions. Secondly, a multiple population de-biased estimation and testing method (MPDe) is developed to estimate and simultaneously test differences in the precision matrix before and after stimulation. This approach extends the joint Gaussian graphical model to multiple populations while incorporating the temporal dependency of EEG data.Meanwhile, a novel data-driven fine-tuning method is applied to automatically search for optimal hyperparameters.

Results: Through comprehensive simulation studies and applications, we have obtained substantial evidence to validate that the RIHT has relatively high power, and it can test for changes when the distribution is unknown. Similarly, the MPDe can infer the precision matrix under time-dependent conditions. Additionally, the conducted analysis of channel selection and dominant channel can identify significant channels which play a crucial role in human cognitive ability, such as PO3, PO4, Pz, P4, P8, FT7 and FT8. All findings confirm that the proposed methods outperform existing ones, demonstrating the effectiveness of the framework in EEG data analysis.

The RIHT and MPDe programs used in our studies can be executed in MATLAB. We have provided separate examples for each program. The file 'Theta_generate_p150_low_sparsity.csv' in MPDe contains simulated data. The initial step for testing precision matrices involves calling the R program from the R package 'JGL' to estimate the precision matrix. After this, you can run our program 'MPDe_codes/example.m'. Note: We have run the R program to estimate the precision matrix. The precision matrix saved in 'MPDe_codes/Opt_thetahat_X1_p150_nk200_sparsity1.0.csv' corresponds to the sample of 'MPDe_codes/Opt_sample_X1_p150_nk200_sparsity1.0.csv'. Similarly, the precision matrix saved in 'MPDe_codes/Opt_thetahat_X2_p150_nk200_sparsity1.0.csv' corresponds to the sample of 'MPDe_codes/Opt_sample_X2_p150_nk200_sparsity1.0.csv'. Additionally, the EEG data is included in the accompanying zip file.