Partial maximum correntropy regression for robust electrocorticography decoding

Abstract

The Partial Least Square Regression (PLSR) method has shown admirable competence for predicting continuous variables from inter-correlated electrocorticography signals in the brain-computer interface. However, PLSR is essentially formulated with the least square criterion, thus, being considerably prone to the performance deterioration caused by the brain recording noises. To address this problem, this study aims to propose a new robust variant for PLSR. To this end, the maximum correntropy criterion (MCC) is utilized to propose a new robust implementation of PLSR, called Partial Maximum Correntropy Regression (PMCR). The half-quadratic optimization is utilized to calculate the robust projectors for the dimensionality reduction, and the regression coefficients are optimized by a fixed-point optimization method. The proposed PMCR is evaluated with a synthetic example and a public electrocorticography dataset under three performance indicators. For the synthetic example, PMCR realized better prediction results compared with the other existing methods. PMCR could also abstract valid information with a limited number of decomposition factors in a noisy regression scenario. For the electrocorticography dataset, PMCR achieved superior decoding performance in most cases, and also realized the minimal neurophysiological pattern deterioration with the interference of the noises. The experimental results demonstrate that, the proposed PMCR could outperform the existing methods in a noisy, inter-correlated, and high-dimensional decoding task. PMCR could alleviate the performance degradation caused by the adverse noises and ameliorate the electrocorticography decoding robustness for the brain-computer interface.

Keywords: brain-computer interface; electrocorticography decoding; maximum correntropy; partial least square regression; robustness.

Figure 1

Regression performance indicators of the inter-correlated, high-dimensional, and contaminated synthetic dataset under different noise standard deviations with noise levels from 0 to 1.0. (A) Noise standard deviation = 30, (B) noise standard deviation = 100, and (C) noise standard deviation = 300. The performance indicators were acquired from 100 Monte-Carlo repetitive trials and averaged across three dimensions of the output. The proposed PMCR algorithm realized better performance than the existing PLSR algorithms consistently for r, RMSE, and MAE, in particular when the training set was contaminated considerably.

Figure 2

Regression performance indicators of the synthetic dataset with noise standard deviation being 100 under three different noise levels with the number of factors increasing from 1 to 100. (A) Noise level = 0.2, (B) noise level = 0.5, and (C) noise level = 0.8. The performance indicators were obtained from 100 repetitive trials and averaged across three dimensions of the output. The proposed PMCR algorithm not only acquired better prediction results than the other algorithms ultimately with the optimal number of factors, but also achieved admirable regression performance with a small number of factors.

Figure 3

Experimental protocol of the Neurotycho ECoG dataset and decoding paradigm to evaluate the robustness of the different PLSR algorithms. (A) The macaque retrieved foods in a three-dimensional random location, during which the body-centered coordinates of the right wrists and the ECoG signals were recorded simultaneously. (B) Both Monkey B and C were implanted with 64-channel epidural ECoG electrodes on the contralateral (left) hemisphere, overlaying the regions from the prefrontal cortex to the parietal cortex. Ps: principal sulcus, As: arcuate sulcus, Cs: central sulcus, IPs: intraparietal sulcus. (A, B) Were reproduced from Shimoda et al. (2012), which provides the details of this public dataset. (C) Decoding diagram from ECoG signals to three-dimensional trajectories. The training ECoG signals are contaminated to assess the robustness of different algorithms.

Figure 4

Distributions and scalograms of the time-frequency feature noises resulting from the ECoG sampling contamination. (A) Noise level = 10⁻³ (the deteriorated proportion of training set = 0.6645 ± 0.0089), (B) Noise level = 10⁻² (the deteriorated proportion of training set ≈1). The time-frequency feature noises were calculated by subtracting the training datasets which were obtained from acoustic and contaminated ECoG signals, respectively. The distributions were averaged by 20 sessions of Monkey B and C, while the scalograms were averaged across all electrodes. The peaks of distributions are truncated to emphasize the heavy-tailed characteristic.

Figure 5

Spatio-spectro-temporal contributions of the prediction model for Monkey B's Z-position under noise levels 0 and 10⁻³. (A) Spatial patterns, (B) spectral patterns, and (C) temporal patterns. For each domain, the quantitative deterioration is calculated by the absolute value summation of the difference between the original and the deteriorated patterns. The original patterns W_c(ch), W_f(freq), and W_t(temp) were averaged across the 10 acoustic sessions of Monkey B, while the deteriorated patterns $W_c^{\prime }(\mathrm{\text{ch}})$, $W_f^{\prime }(\mathrm{\text{freq}})$, and $W_t^{\prime }(\mathrm{\text{temp}})$ were averaged across 50 trials (10 sessions of Monkey B × 5 repetitive trials). The proposed PMCR achieved the minimal deterioration for each domain.

Figure 6

The connection between PMCR and MCC-PLSR for a univariate response.