MATH5855: Multivariate Analysis

Questions

(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., 

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]

(b) Withou using using R, compute the eigenvalues and the eigenvectors of the matrix [10 Marks]

(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 =

(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]