1. 数据的矩阵描述

2. 样本
Array of Data

3. Descriptive Statistics
Summary numbers to assess the information contained in data
Basic descriptive statistics
Sample mean
Sample variance
Sample standard deviation
Sample covariance
Sample correlation coefficient

4. Sample Mean and Sample Variance

5. Sample Covariance and Sample Correlation Coefficient

6. Standardized Values (or Standardized Scores)
Centered at zero
Unit standard deviation
Sample correlation coefficient can be regarded as a sample covariance of two standardized variables

7. Properties of Sample Correlation Coefficient
Value is between -1 and 1
Magnitude measure the strength of the linear association
Sign indicates the direction of the association
Value remains unchanged if all xji’s and xjk’s are changed to yji = a xji + b and yjk = c xjk + d, respectively, provided that the constants a and c have the same sign

8. Arrays of Basic Descriptive Statistics

9. Random Vectors and Random Matrices
Vector whose elements are random variables
Random matrix
Matrix whose elements are random variables

10. Expected Value of a Random Matrix

11. Population Mean Vectors

12. Covariance

13. Statistically Independent

14. Population Variance-Covariance Matrices

15. Population Correlation Coefficients

16. Standard Deviation Matrix

17. Correlation Matrix from Covariance Matrix

18. Partitioning Covariance Matrix

19. Partitioning Covariance Matrix

20. Linear Combinations of Random Variables

21. Example of Linear Combinations of Random Variables

22. Linear Combinations of Random Variables

23. Sample Mean Vector and Covariance Matrix

24. Partitioning Sample Mean Vector

25. Partitioning Sample Covariance Matrix

26. Population and Sample 总 体（随机变量或向量）——分布 统计量（随机变量或向量）——分布 数 据 目 标 桥 梁
推 导
统计量（随机变量或向量）——分布
桥 梁
计 算
数 据
出发点

27. Random Matrix

28. Random Sample
Row vectors X1’, X2’, …, Xn’ represent independent observations from a common joint distribution with density function f(x)=f(x1, x2, …, xp)
Mathematically, the joint density function of X1’, X2’, …, Xn’ is

29. Random Sample
Measurements of a single trial, such as Xj’=[Xj1,Xj2,…,Xjp], will usually be correlated
The measurements from different trials must be independent
The independence of measurements from trial to trial may not hold when the variables are likely to drift over time

30. Result 1

31. Proof of Result 1

32. Proof of Result 1

33. Proof of Result 1

34. Some Other Estimators