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MATH5855: Multivariate Analysis
Assignment 1
Due data: 5 pm on Tuesday October 4, 2022
Instructions:
The assignment 1 contains 3 questions and worth a total of 70 points which will count towards
15% of the final mark for the subject.
Use tables, graphs and concise text explanations to support your answers. Unclear answers
may not be marked at your own cost. All tables and graphs must be clearly commented and
identified.
You need to submitTWO files, the pdf file of the answers and the R markdown file, containing
the R codes.
Questions
Question 1. [20 Marks] Let X =
X1
X2
X3
∼ N3
1
2
3
,
11 −6 2
−6 10 −4
2 −4 6
.
(a) Find the best linear approximation of X3 by a linear function of X1 and X2. [5 Marks]
(b) Using R, simulate n = 1000 samples from X. Transform the data to Z =
(
Z1 Z2 Z3
)
,
where Z1 = X2 − X3, Z2 = X2 + X1 and Z3 = Z1 + Z2. Plot the scatter-plots of pairs of
observations and do the test to confirm the multivariate normality of Z. To generate the data,
set the seed equal to the last 3 digits of your student zID; i.e., if your student zID is 1234567,
you need to use ”‘set.seed(567)”’. Interpret the result of the test. [15 Marks]
Question 2. [25 Marks]
(a) State, explicitly, all possible values that a and b can take in order for the following matrix to
be a covariance matrix. Give arguments that justify your answer [5 Marks]
Σ =
1 2
a b
.
1
(b) Withou using using R, compute the eigenvalues and the eigenvectors of the matrix [10 Marks]
Σ =
13 −4
−4 7
(c) Using R, confirm the eigenvalues and the eigenvectors of Σ obtained in (b) and define the
matrix P and the diagonal matrix Λ such that Σ = PΛP T . [10 Marks]
Question 3. [25 Marks] Consider X =
X1
X2
∼ N3
2
2
,
1 0
0 1
. Let a = 1
1
and b =
1
−1
.
(a) Show that aTX and bTX are independent. [5 Marks]
(b) Using R, simulate n = 1000 samples from X and use the appropriate plot to confirm inde-
pendence of aTX and bTX, visually. Make sure to set the seed equal to the last 3 digits of
your student zID, as in Question 1. [10 Marks]
(c) Assume that the mean and covariance matrix of the simulated data is unknown. Find the
ML estimators for µ and Σ based on the sample. [10 Marks]
