一种修正的马氏距离判别法

马氏距离判别法是一种基于马氏距离的多元统计分析方法，其引入了协方差矩阵的逆矩阵，以排除属性变量的量纲及变量之间的相关性对距离度量的干扰。然而，在属性变量存在严重的多重共线性时，样本协方差矩阵的奇异性会影响其逆矩阵估计的稳定性，从而降低马氏距离判别法的有效性。为此，提出了一种修正的马氏距离判别法，采用了一般交叉验证(GCV)方法，在属性变量间存在高度相关性的情况下，选择预测效果最好的变量维度，同时可以对协方差矩阵的逆矩阵进行稳定的估计。修正的马氏距离判别法可以得到可靠的协方差矩阵的估计，提高模型的判别准确率；也可以抵抗样本外的扰动，提高模型的泛化能力。仿真实验结果验证了在属性变量存在严重的多重共线性情形下，修正的马氏距离判别法的判别效果较经典的马氏距离判别法有明显的提升。 

A modified Mahalanobis distance discriminant method

Mahalanobis distance discriminant method is an effective multivariate statistical analysis method based on the Mahalanobis distance. An important feature of the Mahalanobis distance is its introduction of the inverse of covariance matrix, which avoids the disturbance to the distance measurement from the scales of the attribute variables and the correlations among these variables. However, when there is multicollinearity among the attribute variables, the singularity of the covariance matrix will affect the stability of the inverse matrix estimation, and will greatly damage the effect of the Mahalanobis distance discriminant method. We propose a modified Mahalanobis distance discriminant method, which adopts the general cross-validation (GCV) to choose the dimensions of these variables with the best prediction effect, so that the inverse of the covariance matrix can be well estimated when these attribute variables are highly correlated. The modified Mahalanobis distance discriminant method can provide a reliable estimation of the covariance matrix, resist the disturbances outside the sample set, improve the discriminant accuracy of the model, and enhance the generalization ability of the model. Simulations are conducted to verify the improvement of the discriminant performance of the modified Mahalanobis distance discriminant method compared with the classical one.