Combining eigenvalues and variation of eigenvectors for order determination
提出一种新的阶数确定方法，同时利用特征值的递减模式和特征向量方向的可变性，更精确地估计矩阵的秩，并通过模拟和应用验证其一致性。
In applying statistical methods such as principal component analysis, canonical correlation analysis, and sufficient dimension reduction, we need to determine how many eigenvectors of a random matrix are important for estimation. This problem is known as order determination, and amounts to estimating the rank of a matrix. Previous order-determination procedures rely either on the decreasing pattern, or elbow, of the eigenvalues, or on the increasing pattern of the variability in the directions of the eigenvectors. In this paper we propose a new order-determination procedure by exploiting both patterns: when the eigenvalues of a random matrix are close together, their eigenvectors tend to vary greatly; when the eigenvalues are far apart, their variability tends to be small. The combination of both helps to pinpoint the rank of a matrix more precisely than the previous methods. We establish the consistency of the new order-determination procedure, and compare it with other such procedures by simulation and in an applied setting.