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STAT 530/J530 B.Habing Univ. of S.C. (^1)
STAT 530/J
September 1 st, 2005
Instructor: Brian Habing
Department of Statistics
LeConte 203
Telephone: 803-777-
STAT 530/J530 B.Habing Univ. of S.C. (^2)
Today
- The Multivariate Normal Distribution
STAT 530/J530 B.Habing Univ. of S.C. (^3)
Data!
Multivariate Data is typically presented in
a matrix, with each of then rows being a different observation, and each of theq columns being a different variable.
n n nj nq
i ij iq
q
j q
x x x x
x x x
x x x
x x x x
X
L L
M O M
M O M
L L
1 2
1
21 22 2
11 12 1 1
STAT 530/J530 B.Habing Univ. of S.C. (^4)
Summarizing Data
Like in univariate
statistics the Mean and Variance are commonly used as summary measures.
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
μ q
μ
μ M
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
= 2
σ q
σ
σ M
STAT 530/J530 B.Habing Univ. of S.C. (^5)
Covariances
It is also necessary to summarize the
relationship between the variables, and this can be done with the covariance.
( )( ) 1
1 ( , )
( , ) [( )( )]
1
∑
∧ − − −
=
= − − n
i
j k ij j il l
j k j j k k
x x x x n
Covx x
Covx x E x μ x μ
STAT 530/J530 B.Habing Univ. of S.C. (^6)
Covariance Matrix
The covariance is thus a matrix:
q qq
q
L L
M O M
M
L
1
21 11
11 12 1
STAT 530/J530 B.Habing Univ. of S.C. (^10)
Sample Correlation Matrix
The Sample Correlation Matrix can be
written as:
Where D -1/2^ is the matrix with 1/s i on the
diagonals.
R D SD
STAT 530/J530 B.Habing Univ. of S.C. (^11)
What if Data is Missing?
• Missing Completely at Random?
• Imputation
• Multiple Imputation
STAT 530/J530 B.Habing Univ. of S.C. (^12)
(Multivariate) Normal Distribution
− −
f x e x
x 2
2 2
( )
( )^ σ
μ
( (x ) (x ))
1
(x) | 2 |
μ μ
T
f e
