The development of accurate electroencephalography (EEG) and magnetoencephalography (MEG) source imaging algorithm is of great importance for functional brain research and non-invasive presurgical evaluation of epilepsy. In practice, the challenge arises from the fact that the number of measurement channels is far less than the number of candidate source locations, rendering the inverse problem ill-posed. A widely used approach is to introduce a regularization term into the objective function, which inevitably biased the estimated amplitudes towards zero, leading to an inaccurate estimation of the estimator's variance. This study proposes a novel debiased EEG/MEG source imaging (DeESI) algorithm for detecting sparse brain activities, which corrects the estimation bias in signal amplitude, dipole orientation and depth. The DeESI extends the idea of group Lasso by incorporating both the matrix Frobenius norm and the L1-norm, which guarantees the estimators are only sparse over sources while maintains smoothness in time and orientation. We also derived variance of the debiased estimators for standardization and hypothesis testing. A fast alternating direction method of multipliers (ADMM) algorithm is proposed for solving the matrix form optimization problem directly without the need for vectorization. The proposed algorithm is compared with eleven existing ESI methods using simulations and an open source EEG dataset whose stimulation locations are known precisely. The DeESI exhibits the best performance in peak localization and amplitude reconstruction.

Accurate prediction of the out-of-distribution data is desired for a learning algorithm. In domain generalization, training data from source domains tend to have different distributions from that of the target domain, while the target data are absence in the training process. We propose a Distribution Free Domain Generalization (DFDG) procedure for classification by conducting standardization to avoid the dominance of a few domains in the training process. The essence of the DFDG is its reformulating the cross domain/class discrepancy by pairwise two sample test statistics, and equally weights their importance or the covariance structures to avoid dominant domain/class. A theoretical generalization bound is established for the multi-class classification problem. The DFDG is shown to offer a superior performance in empirical studies with fewer hyperparameters, which means faster and easier implementation.

We consider distributed statistical optimization and inference in the presence of heterogeneity among distributed data blocks. A weighted distributed estimator is proposed to improve the statistical efficiency of the standard ”split-and-conquer" estimator for the common parameter shared by all the data blocks. The weighted distributed estimator is at least as efficient as the would-be full sample and the generalized method of moment estimators with the latter two estimators requiring full data access. A bias reduction is formulated for the weighted distributed estimator to accommodate much larger numbers of data blocks (relaxing the constraint from $K = o(N^{1/2})$ to $K = o(N^{2/3})$, where $K$ is the number of blocks and $N$ is the total sample size) than the existing methods without sacrificing the statistical efficiency at the same time. The mean squared error bounds, the asymptotic distributions, and the corresponding statistical inference procedures of the weighted distributed and the debiased estimators are derived, which show an advantageous performance of the debiased weighted estimators when the number of data blocks is large.

This paper proposes a partitioning structure learning method for segmented linear regression trees (SLRT), which assigns linear predictors over the terminal nodes. The recursive partitioning process is driven by an adaptive split selection algorithm that maximizes, at each node, a criterion function based on a conditional Kendall’s τ statistic that measures the rank dependence between the regressors and the fit- ted linear residuals. Theoretical analysis shows that the split selection algorithm permits consistent identification and estimation of the unknown segments. A suffi- ciently large tree is induced by applying the split selection algorithm recursively. Then the minimal cost-complexity tree pruning procedure is applied to attain the right-sized tree, that ensures (i) the nested structure of pruned subtrees and (ii) consistent estimation to the number of segments. Implanting the SLRT as the built-in base predictor, we obtain the ensemble predictors by random forests (RF) and the proposed weighted random forests (WRF). The practical performance of the SLRT and its ensemble versions are evaluated via numerical simulations and empirical studies. The latter shows their advantageous predictive performance over a set of state-of-the-art tree-based models on well-studied public datasets.