Z-score linear discriminant analysis for EEG based brain-computer interfaces

Abstract

Linear discriminant analysis (LDA) is one of the most popular classification algorithms for brain-computer interfaces (BCI). LDA assumes Gaussian distribution of the data, with equal covariance matrices for the concerned classes, however, the assumption is not usually held in actual BCI applications, where the heteroscedastic class distributions are usually observed. This paper proposes an enhanced version of LDA, namely z-score linear discriminant analysis (Z-LDA), which introduces a new decision boundary definition strategy to handle with the heteroscedastic class distributions. Z-LDA defines decision boundary through z-score utilizing both mean and standard deviation information of the projected data, which can adaptively adjust the decision boundary to fit for heteroscedastic distribution situation. Results derived from both simulation dataset and two actual BCI datasets consistently show that Z-LDA achieves significantly higher average classification accuracies than conventional LDA, indicating the superiority of the new proposed decision boundary definition strategy.

Figure 1. The decision boundaries of Z-LDA and LDA defined from training set.

Blue circles are the weight sum of the first class, blue solid line denotes the Gaussian distribution curve they subject to; red stars are the weight sum of the second class, red solid line denotes the Gaussian distribution curve they subject to; green dashed line denotes the decision boundary of LDA, , and green solid line denotes the decision boundary of Z-LDA, .

Figure 2. The classification performance of Z-LDA and LDA on test samples.

Blue circles are the test samples of the first class; red stars are the test samples of the second class; green dashed line denotes the decision boundary of LDA; green solid line denotes the decision boundary of Z-LDA.

Figure 3. Distribution of the weight sum of subject 1.

Blue dashed line is the Gaussian distribution curve according to the characteristic of weight sum with left hand motor imagery in training set; red dashed line is the Gaussian distribution curve according to the characteristic of weight sum with right hand motor imagery in training set; blue circles denote the weight sum with left hand motor imagery in test set; red stars denote the weight sum with right hand motor imagery in test set; blue solid line is the Gaussian distribution curve derived from the blue circles; red solid line is the Gaussian distribution curve derived from the red stars; green dashed vertical line is the decision boundary defined by LDA from training set; green solid vertical line is the decision boundary defined by Z-LDA from training set; black solid vertical line is the theoretical boundary of test set.