1. 1 Multivariate Normal Distribution 朱永军信息管理学院

2. 2 Generalized from univariate normal densityGeneralized from univariate normal density Base of many multivariate analysis techniquesBase of many multivariate analysis techniques Useful approximation to “true” population distributionUseful approximation to “true” population distribution Central limit distribution of many multivariate statisticsCentral limit distribution of many multivariate statistics Mathematical tractableMathematical tractable

3. 3 Univariate Normal Distribution

4. 4 Table 1, Appendix

5. 5 Square of Distance (Mahalanobis distance)

6. 6 p-dimensional Normal Density

7. 7 Example 4.1 Bivariate Normal

8. 8 Example 4.1 Squared Distance

9. 9 Example 4.1 Density Function

10. 10 Example 4.1 Bivariate Distribution  11 =  22,  12 = 0

11. 11 Example 4.1 Bivariate Distribution  11 =  22,  12 = 0.75

12. 12 Contours

13. 13 Result 4.1

14. 14 Example 4.2 Bivariate Contour

15. 15 Example 4.2 Positive Correlation

16. 16 Probability Related to Squared Distance

17. 17

18. 18 Result 4.2

19. 19 Example 4.3 Marginal Distribution 边际分布

20. 20 Result 4.3

21. 21 Proof of Result 4.3: Part 1

22. 22 Proof of Result 4.3: Part 2

23. 23 Example 4.4 Linear Combinations

24. 24 Result 4.4

25. 25 Example 4.5 Subset Distribution

26. 26 Result 4.5

27. 27 Example 4.6 Independence

28. 28 Result 4.6

29. 29 Proof of Result 4.6

30. 30

31. 31 Example 4.7 Conditional Bivariate

32. 32 Example 4.1 Density Function

33. 33 Example 4.7

34. 34 Result 4.7

35. 35  2 Distribution

36. 36  2 Distribution Curves

37. 37 Table 3, Appendix

38. 38 Proof of Result 4.7 (a)

39. 39 Proof of Result 4.7 (b)

40. 40 Result 4.8

41. 41 Proof of Result 4.8

42. 42 Example 4.8 Linear Combinations

43. 43

44. 44 Multivariate Normal Likelihood

45. 45 Maximum-likelihood Estimation

46. 46 Trace 迹 of a Matrix

47. 47 Result 4.9

48. 48 Proof of Result 4.9 (a)

49. 49 Proof of Result 4.9 (b)

50. 50 Likelihood Function

51. 51 Result 4.10

52. 52 Proof of Result 4.10

53. 53 Result 4.11 Maximum Likelihood Estimators of  and 

54. 54 Proof of Result 4.11

55. 55 Invariance Property

56. 56 Sufficient Statistics

57. 57 Distribution of Sample Mean

58. 58 Sampling Distribution of S

59. 59 Wishart 维希特分布 Distribution

60. 60 Univariate Central Limit Theorem

61. 61 Result 4.12 Law of Large Numbers

62. 62 Result 4.12 Multivariate Cases

63. 63 Result 4.13 Central Limit Theorem

64. 64 Limit Distribution of Statistical Distance

65. 65 Evaluating Normality of Univariate Marginal Distributions

66. 66

67. 67 Q-Q Plot

68. 68 Example 4.9

69. 69

70. 70 Sas prog data sheets; input distance @@; label distance='Hole Distance in cm'; datalines; 9.80 10.20 10.27 9.70 9.76 10.11 10.24 10.20 10.24 9.63 9.99 9.78 10.10 10.21 10.00 9.96 9.79 10.08 9.79 10.06 10.10 9.95 9.84 10.11 9.93 10.56 10.47 9.42 10.44 10.16 10.11 10.36 9.94 9.77 9.36 9.89 9.62 10.05 9.72 9.82 9.99 10.16 10.58 10.70 9.54 10.31 10.07 10.33 9.98 10.15 ; run;

71. Sas prog 续 symbol v=plus; legend2 FRAME CFRAME=ligr CBORDER=black POSITION=center; title 'Normal Quantile-Quantile Plot for Hole Distance'; proc capability data=sheets noprint; spec lsl=9.5 clsl=red usl=10.5 cusl=blue; qqplot distance / cframe = ligr legend = legend2; run; proc capability data=sasuser.finish noprint; qqplot x2 / cframe = ligr legend = legend2; run;

72. Sas prog 续 symbol v=plus; legend2 FRAME CFRAME=ligr CBORDER=black POSITION=center; title 'Normal Quantile-Quantile Plot for Hole Distance'; proc capability data=sheets noprint; spec lsl=9.5 clsl=red usl=10.5 cusl=blue; qqplot distance / cframe = ligr legend = legend2; run; proc capability data=sasuser.finish noprint; qqplot x2 / cframe = ligr legend = legend2; run;

73. Sas prog 另一模块 proc univariate data=Measures; qqplot Width / lognormal(sigma=2 theta=0 zeta=0); qqplot Width / lognormal(sigma=2 theta=0 slope=1); qqplot Width / weibull2(sigma=2 theta=0 c=.25); qqplot Width / weibull2(sigma=2 theta=0 slope=4); Run; data Failures; input Time @@; label Time = 'Time in Months'; datalines; 29.42 32.14 30.58 27.50 26.08 29.06 25.10 31.34 29.14 33.96 30.64 27.32 29.86 26.28 29.68 33.76 29.32 30.82 27.26 27.92 30.92 24.64 32.90 35.46 30.28 28.36 25.86 31.36 25.26 36.32 28.58 28.88 26.72 27.42 29.02 27.54 31.60 33.46 26.78 27.82 29.18 27.94 27.66 26.42 31.00 26.64 31.44 32.52 ; run;

74. Sas prog 续另一模块 symbol v=plus; title 'Three-Parameter Weibull Q-Q Plot for Failure Times'; proc univariate data=Failures noprint; qqplot Time / weibull(c=est theta=est sigma=est noprint) cframe = ligr square href=0.5 1 1.5 2 vref=25 27.5 30 32.5 35 lhref=4 lvref=4 chref=ywh cvref=ywh; run;

75. Q-Q Plot 75

76. 76 Example 4.10 Radiation Data of Closed-Door Microwave Oven

77. 77 Measurement of Straightness

78. 78 Table 4.2 Q-Q Plot Correlation Coefficient Test

79. 79 Example 4.11

80. 80 Evaluating Bivariate Normality

81. 81 Example 4.12

82. 82

83. 83 Chi-Square Plot

84. 84 Example 4.13 Chi-Square Plot for Example 4.12

85. 85

86. 86 Chi-Square Plot for Computer Generated 4-variate Normal Data

87. 87 Steps for Detecting Outliers Make a dot plot for each variableMake a dot plot for each variable Make a scatter plot for each pair of variablesMake a scatter plot for each pair of variables Calculate the standardized values. Examine them for large or small valuesCalculate the standardized values. Examine them for large or small values Calculated the squared statistical distance. Examine for unusually large values. In chi-square plot, these would be points farthest from the origin.Calculated the squared statistical distance. Examine for unusually large values. In chi-square plot, these would be points farthest from the origin.

88. 88 Helpful Transformation to Near Normality Original Scale Transformed Scale Counts, y Proportions, Correlations, r

89. 89 Box and Cox’s Univariate Transformations

90. 90 Example 4.16 ( ) vs. Example 4.16 ( ) vs.

91. 91 Example 4.16 Q-Q Plot

92. 92 Transforming Multivariate Observations

93. 93 More Elaborate Approach

94. 94 Example 4.17 Original Q-Q Plot for Open-Door Data

95. 95 Example 4.17 Q-Q Plot of Transformed Open-Door Data

96. 96 Example 4.17 Contour Plot of for Both Radiation Data

97. 97 Transform for Data Including Large Negative Values