1. 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

2. 2 Multivariate Normal Distribution Generalized from univariate normal density Base of many multivariate analysis techniques Useful approximation to “ true ” population distribution Central limit distribution of many multivariate statistics Mathematical 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 Probability Related to Squared Distance

18. 18 Result 4.2

19. 19 Example 4.3 Marginal Distribution

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 Proof of Result 4.6

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 Example 4.8 Linear Combinations

44. 44 Multivariate Normal Likelihood

45. 45 Maximum-likelihood Estimation

46. 46 Trace 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 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 Evaluating Normality of Univariate Marginal Distributions

67. 67 Q-Q Plot

68. 68 Example 4.9

69. 69 Example 4.9

70. Histogram of MidTerm Scores of Students of This Course in 2006 70

71. Q-Q Plot of MidTerm Scores of Students of This Course in 2006 71 n = 33, r Q = 0.946652

72. 72 Example 4.10 Radiation Data of Closed-Door Microwave Oven

73. 73 Measurement of Straightness

74. 74 Table 4.2 Q-Q Plot Correlation Coefficient Test

75. 75 Example 4.11

76. 76 Evaluating Bivariate Normality

77. 77 Example 4.12

78. 78 Example 4.12

79. 79 Chi-Square Plot

80. 80 Example 4.13 Chi-Square Plot for Example 4.12

81. 81 Example 4.13 Chi-Square Plot for Example 4.12

82. 82 Chi-Square Plot for Computer Generated 4-variate Normal Data

83. 83 Steps for Detecting Outliers Make a dot plot for each variable Make a scatter plot for each pair of variables Calculate 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.

84. 84 Helpful Transformation to Near Normality Original Scale Transformed Scale Counts, y Proportions, Correlations, r

85. 85 Box and Cox’s Univariate Transformations

86. 86 Example 4.16 ( ) vs. Example 4.16 ( ) vs.

87. 87 Example 4.16 Q-Q Plot

88. 88 Transforming Multivariate Observations

89. 89 More Elaborate Approach

90. 90 Example 4.17 Original Q-Q Plot for Open-Door Data

91. 91 Example 4.17 Q-Q Plot of Transformed Open-Door Data

92. 92 Example 4.17 Contour Plot of for Both Radiation Data

93. 93 Transform for Data Including Large Negative Values