STAT8121/STAT7121 Multivariate Analysis Question 1

STAT8121/STAT7121 Multivariate Analysis

Question 1. [4 marks]

Briefly answer the following questions. Please note excessive long answers (usually more than half pages) will incur penalties.  

1. Explain why the maximum likelihood classification rule is a special case of the minimum expected cost of misclassification (ECM) rule. 
2. Explain why the factor analysis problem is difficult to solve.

Question 2 [25 marks]

1. Let  = [ ,  ] ~          ,  be a bivariate standard normal vector.  

           Suppose that  = [ ,  ]       = , where

 .

1. 1. Determine the distribution of .

1. 2. Are             and       independent? Explain.

2. Let  = [ ,  ] ~          ,               be a bivariate normal vector,    where  

= [−1,3]     and           

 .

The eigenvalues of   are = 2.62    and  = 0.38   with the corresponding eigenvectors  #_$ = [−0.53, 0.85]  and  #_& = [0.85, 0.53] .

1. 1. Write down and verify the Singular Value Decomposition (SVD) of     .

1. 2. Hence, using a linear transformation of   = [ ,  ] ~     , ,  

           describe how would you simulate   = [ ,  ] ~           ,             .

           Give your reasoning. 

3. Let  = [ ,  ] ~  ,  be a bivariate normal vector,    where   = [' , ' ]   and  

   

   ^{1 (} .   

           Let   be positive definite and

) ,  =     −          ^*  −  . 

1. 1. Obtain the distribution of   ^*/  −  . Show your working.
   2. Let [             ,             ] , [        ,             ] ,… , [ _-, _-]           be n independent copies of   and  .

      

       

[ / , / ] be the vector of averages  

 Let 6 7 be the cumulative distribution function (CDF) of  8)              ., .   Identify 6 7 including its parameter(s). Give your reasoning.

1. 3. Let the significance level 9 = 0.1  and 6 4.61 = 1 − 9,  where  6 7 is defined in (ii).   Suppose that  is known. The sample of size 8 = 4  is randomly sampled from the distribution of  and the sample mean vector .; = [0.3, 0] . 

Let H₀:  = [0, 0]  and  H₁:  ≠ [0, 0] .  

Determine the interval for all ( for which the hypothesis H₀ will not be rejected at the significance level 9 = 0.1. 

Now suppose that   is unknown.  Describe the changes this would imply in the solutions to (ii) – (iii)? No calculations are required to answer this question.

Question 3 [9 marks] 

Suppose 20 multivariate observations are obtained on

 variables and these observations are denoted by 7 ,… , 7₌, where each 7₁ is a 4 × 1 vector containing 4 values from the 4 variables. Consider the population mean vector ' =   ' ,' , '_?, '_@                        . Suppose we wish to test the following hypotheses involving different contrasts: 

and significance level for the test is 9 = 5%. 

1. Explain how to test this hypothesis. In your answer you must include: (i) assumptions made, (ii) test statistic, and (iii) how to decide if to reject or retain the null hypothesis. 

2. Write down the T² simultaneous confidence intervals for all the contrasts. You are required to simplify the answers to as much as you can.

3. Write down the Bonferroni simultaneous confidence intervals for all the contrasts. You are required to simplify the answers to as much as you can.

Question 4. [17 marks]

1. Suppose there are 3 groups and training sample observations are collected from them. Assume these observations are independent and follow multivariate normal distributions. Summary statistics of the three training samples are given below:

Sample 1:

  Sample 2:

  Sample 1:

The costs of mis-classification are given in the following matrix:

You are required to classify an observation −1, 4    into one of this three groups when: 

1. 1. the three groups variance matrices are assumed equal;

1. 2. the three groups variance matrices are not equal.

2. Suppose

   

    subjects are tested (or measured) repeatedly over the times L = 1, 2, … , M. The observations can be arranged in a matrix form as

,

 where each 7_1O, where is a scalar. This type of data is usually called the repeated-measure. Assume multivariate observations 7₁ = 7₁, … , 7₁ , P = 1, … , 8, are independent and each follows , distribution, where ' = ' , … , ' . 

1. 1. Suppose we wish to compare if the mean measurements at the above times are the same. Explain why ANOVA is generally NOT appropriate for testing the hypothesis 

              A₌:         = … = .

1. 2. Design a test to test the above hypothesis A₌. In your answer you MUST specify clearly the test statistic, the distribution of the test statistic and decision rule.
   3. Design a statistical test for the hypothesis A₌:             =            =         _? = … = ^*                      ,

where  is a known constant. In your answer you MUST specify clearly the test statistic, the distribution of the test statistic and decision rule.