﻿ 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

Statistics and Application
Vol.06 No.01(2017), Article ID:19902,29 pages
10.12677/SA.2017.61005

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

Yingying Zhang, Tengzhong Rong

Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing

Received: Feb. 25th, 2017; accepted: Mar. 14th, 2017; published: Mar. 17th, 2017

ABSTRACT

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).

Keywords:Sample Covariance Matrix, Sample Correlation Matrix, Sample Principal Component, Properties and Proofs, R Software

1. 引言

2. 样本主成分的性质及证明

2.1. 从S出发求主成分

Figure 1. The relationships of the sample covariances (left) and the relationships of the sample correlations (right) among

Figure 2. The relationships of the sample covariances (left) and the relationships of the sample correlations (right) among

Table 1. The covariance matrix and the correlation matrix of the population and the sample

，它是一个正交阵，，写成矩阵形式，就是

(S1).

(S2).

(S3).

(S4).

(S5). 若，则

(S6). 若，则

(S7). 样本总方差

2.2. 从R出发求主成分

，它是一个正交阵，，写成矩阵形式，就是

(R1).

(R2).

(R3).

(R4).

(R5). 若，则

(R6). 若，则

(R7). 样本总方差

3. 数值模拟

(S1). (S2). (S3).

(S4). (S5).

(S6).

(S7).

(R1). (R2). (R3).

(R4). (R5).

(R6). (R7).

.

4. 总结

The Sample Covariance Matrix and the Sample Correlation Matrix and Their Applications in the Sample Principal Component[J]. 统计学与应用, 2017, 06(01): 34-62. http://dx.doi.org/10.12677/SA.2017.61005

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A.1. 准备知识及杂项证明

(A.1)

(A.2)

(A.3)

(A.4)

A.2. 从S出发求主成分

(S1). 由数据资料阵的样本协方差矩阵的定义有

(S2). 易知

(S3). 由(A.1)，得

，得

(A.5)

(S4). 由数据资料阵的样本相关矩阵的定义有

(S5). 由矩阵的样本相关矩阵的定义和(A.1)有

(S6). 由(A.3)有

(S7). 由性质(S1)有

A.3. 从R出发求主成分

(R1). 由(A.1)有

(R2). 由(A.1)有

(R3). 由(A.1)有

(R4). 由(A.3)有

(R5). 由(A.3)有

(R6). 由(A.3)有

，(A.4)和(A.1)有

(R7). 由(R1)有

A.4. 数值模拟

(S1).

(S2).

(S3).

(S4).

(S5).

(S6).

(S7).

(R1).

(R2).

(R3).

(R4).

(R5).

(R6).

(R7).

(S1).

(S2).

(S3).

(S4).

(S5).

(S6).

(S7).

(R1).

(R2).

(R3).

(R4).

(R5).

(R6).

(R7).

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