Question 1. Consider a p-dimensional response variable j, containing p variables, with n observation vectors 1,..., In. The sample mean vecior of these observation vectors is denoted by j = (1/n) E1 Fi. a) Given that the sample covariance matrix, S, is defined by 1 S = (或一司) (五一)", n-1 1=1 i) Show that 1 S = n - 1 i=1 ii) Using the above result, show that S can alternatively be defined as - ) "צ דה - 1 Y 1 S = Y, where Y is the data matrix, I is the identity matrix and J is a matrix of 1's. [Hint: Consider re-writing the mean vector in terms of the data matrix (see lecture notes).)
Question 1. Consider a p-dimensional response variable j, containing p variables, with n observation vectors 1,..., In. The sample mean vecior of these observation vectors is denoted by j = (1/n) E1 Fi. a) Given that the sample covariance matrix, S, is defined by 1 S = (或一司) (五一)", n-1 1=1 i) Show that 1 S = n - 1 i=1 ii) Using the above result, show that S can alternatively be defined as - ) "צ דה - 1 Y 1 S = Y, where Y is the data matrix, I is the identity matrix and J is a matrix of 1's. [Hint: Consider re-writing the mean vector in terms of the data matrix (see lecture notes).)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Question 1. Consider a p-dimensional response variable j, containing p variables, with n observation
vectors j1, ..., n. The sample mean vector of these observation vectors is denoted by = (1/n)E1 Fi.
a) Given that the sample covariance matrix, S, is defined by
n
1
(5i – 5) (7. – 5)" ,
S
-
1=1
i) Show that
S =
i=1
ii) Using the above result, show that S can alternatively be defined as
s=" (1-) .
yT
Y,
n -
where Y is the data matrix, I is the identity matrix and J is a matrix of 1's.
[Hint: Consider re-writing the mean vector g in terms of the data matrix (see lecture notes).
b) Consider the following (3x3) data matrix, containing three observations of three variables y1, y2 and y3:
5 2 2
1 3 5
6 4 1
Y =
Calculate
i) The sample covariance matrix S (by hand);
ii) The sample correlation matrix R (by hand) using the equation
R= D,-SD,-',
and comment on each of the three elements in the first column of this matrix.
[Hint: Recall how to find the inverse of a diagonal matrix.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8268f00-1e22-49fe-9b99-9cbbb3c8020a%2Fe32724c9-b14f-4739-8c32-5889a9e03c78%2Fsoxzuo2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 1. Consider a p-dimensional response variable j, containing p variables, with n observation
vectors j1, ..., n. The sample mean vector of these observation vectors is denoted by = (1/n)E1 Fi.
a) Given that the sample covariance matrix, S, is defined by
n
1
(5i – 5) (7. – 5)" ,
S
-
1=1
i) Show that
S =
i=1
ii) Using the above result, show that S can alternatively be defined as
s=" (1-) .
yT
Y,
n -
where Y is the data matrix, I is the identity matrix and J is a matrix of 1's.
[Hint: Consider re-writing the mean vector g in terms of the data matrix (see lecture notes).
b) Consider the following (3x3) data matrix, containing three observations of three variables y1, y2 and y3:
5 2 2
1 3 5
6 4 1
Y =
Calculate
i) The sample covariance matrix S (by hand);
ii) The sample correlation matrix R (by hand) using the equation
R= D,-SD,-',
and comment on each of the three elements in the first column of this matrix.
[Hint: Recall how to find the inverse of a diagonal matrix.]
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