样本协方差矩阵和样本相关矩阵及其在样本主成分中的应用

The Sample Covariance Matrix and the Sample Correlation Matrix and Their Applications in the Sample Principal Component

我们给出了样本主成分的性质及证明,分两种情况讨论:从S出发求主成分和从R出发求主成分。在从S出发求主成分中,给出了7个性质(S1)-(S7)及它们的证明,这些性质说明的关系在图1中得到了充分的展现。同样,在从R出发求主成分中,给出了7个性质(R1)-(R7)及它们的证明,这些性质说明的关系在图2中得到了充分的展现。最后我们给出了两个数值模拟的例子来验证性质(S1)-(S7)和(R1)-(R7)的正确性。

We give the properties and proofs of the sample principal component, and discuss them in two different conditions: from S on to calculate principal component and from R on to calculate principal component. From S on to calculate principal component, we give 7 properties (S1)-(S7) and their proofs, and the relationships stated by these properties get full display in Figure 1. Similarly, from R on to calculate principal component, we give 7 properties (R1)-(R7) and their proofs, and the relationships stated by these properties get full display in Figure 2. Finally we give two numerical simulation examples to verify the correctness of properties (S1)-(S7) and (R1)-(R7).