1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia
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 Univariate Normal Distribution
4 Table 1, Appendix
5 Square of Distance (Mahalanobis distance)
6 p-dimensional Normal Density
7 Example 4.1 Bivariate Normal
8 Example 4.1 Squared Distance
9 Example 4.1 Density Function
10 Example 4.1 Bivariate Distribution 11 = 22, 12 = 0
11 Example 4.1 Bivariate Distribution 11 = 22, 12 = 0.75
12 Contours
13 Result 4.1
14 Example 4.2 Bivariate Contour
15 Example 4.2 Positive Correlation
16 Probability Related to Squared Distance
17 Probability Related to Squared Distance
18 Result 4.2
19 Example 4.3 Marginal Distribution
20 Result 4.3
21 Proof of Result 4.3: Part 1
22 Proof of Result 4.3: Part 2
23 Example 4.4 Linear Combinations
24 Result 4.4
25 Example 4.5 Subset Distribution
26 Result 4.5
27 Example 4.6 Independence
28 Result 4.6
29 Proof of Result 4.6
30 Proof of Result 4.6
31 Example 4.7 Conditional Bivariate
32 Example 4.1 Density Function
33 Example 4.7
34 Result 4.7
35 2 Distribution
36 2 Distribution Curves
37 Table 3, Appendix
38 Proof of Result 4.7 (a)
39 Proof of Result 4.7 (b)
40 Result 4.8
41 Proof of Result 4.8
42 Example 4.8 Linear Combinations
43 Example 4.8 Linear Combinations
44 Multivariate Normal Likelihood
45 Maximum-likelihood Estimation
46 Trace of a Matrix
47 Result 4.9
48 Proof of Result 4.9 (a)
49 Proof of Result 4.9 (b)
50 Likelihood Function
51 Result 4.10
52 Proof of Result 4.10
53 Result 4.11 Maximum Likelihood Estimators of and
54 Proof of Result 4.11
55 Invariance Property
56 Sufficient Statistics
57 Distribution of Sample Mean
58 Sampling Distribution of S
59 Wishart Distribution
60 Univariate Central Limit Theorem
61 Result 4.12 Law of Large Numbers
62 Result 4.12 Multivariate Cases
63 Result 4.13 Central Limit Theorem
64 Limit Distribution of Statistical Distance
65 Evaluating Normality of Univariate Marginal Distributions
66 Evaluating Normality of Univariate Marginal Distributions
67 Q-Q Plot
68 Example 4.9
69 Example 4.9
Histogram of MidTerm Scores of Students of This Course in
Q-Q Plot of MidTerm Scores of Students of This Course in n = 33, r Q =
72 Example 4.10 Radiation Data of Closed-Door Microwave Oven
73 Measurement of Straightness
74 Table 4.2 Q-Q Plot Correlation Coefficient Test
75 Example 4.11
76 Evaluating Bivariate Normality
77 Example 4.12
78 Example 4.12
79 Chi-Square Plot
80 Example 4.13 Chi-Square Plot for Example 4.12
81 Example 4.13 Chi-Square Plot for Example 4.12
82 Chi-Square Plot for Computer Generated 4-variate Normal Data
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 Helpful Transformation to Near Normality Original Scale Transformed Scale Counts, y Proportions, Correlations, r
85 Box and Cox’s Univariate Transformations
86 Example 4.16 ( ) vs. Example 4.16 ( ) vs.
87 Example 4.16 Q-Q Plot
88 Transforming Multivariate Observations
89 More Elaborate Approach
90 Example 4.17 Original Q-Q Plot for Open-Door Data
91 Example 4.17 Q-Q Plot of Transformed Open-Door Data
92 Example 4.17 Contour Plot of for Both Radiation Data
93 Transform for Data Including Large Negative Values