LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2012

ST 3811/3808 – MULTIVARIATE ANALYSIS

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL the following questions:                                                                            (10 x 2 = 20 marks)

1. Define var-cov matrix, correlation matrix and standard deviation matrix. State the relationship among them.
2. State the general expressions for the mean vector and var-cov matrix of linear transformations of a random vector.
3. State the general form of ‘statistical distance’.
4. Explain ‘Bubble Plots’.
5. Define ‘Partial Correlation Coefficient’.
6. State the one-way MANOVA model.
7. State the test for significance of correlation coefficient in a bivariate normal population.
8. Give one reason why in K-means algorithm, the number of clusters ‘K’ is kept an open question.
9. State the postulates on the ‘common factors’ and the ‘specific factors’ in the orthogonal factor model.
10. Explain a situation where the ‘challenge’ of ‘Classification’ arises.

 

SECTION – B

Answer any FIVE questions:                                                                                             (5 x 8 = 40 marks)

11. Briefly explain the terms ‘Data Reduction / Structural Simplification’ and ‘Sorting / Grouping’. Give real-life examples of these two objectives which are addressed by multivariate methods.

 

1. Explain probability plots in general and how it is used for investigation of multivariate normality assumption.

 

1. Derive the moment generating function of multivariate normal distribution.
2. If X =  ~ N_p (μ,Σ) and μ and Σ are accordingly partitioned as  and where  ≠ 0, derive the conditional distribution of X⁽¹⁾ given X⁽²⁾.

 

1. Derive Fisher’s linear discriminant function for discriminating two populations.

 

1. Mention the three linkage methods for hierarchical clustering and present a figurative display of the measure of between-cluster distances in each method.

 

1. Develop the Hotelling’s T² test through the likelihood ratio criterion.         (P.T.O)
2. Give the motivation and the formal definition of Principal Components. State the ‘Maximization Lemma’ (without proof) and hence, obtain the PCs for a random vector

 

SECTION – C

Answer any TWO questions:                                                                                          (2 x 20 = 40 marks)

1. (a) If and S_n­ are the sample mean vector and var-cov matrix from a sample of size ‘n’ from a multivariate population with mean vector μ and var-cov matrix Σ , show that  is an unbiased estimator of μ but S_n is a biased estimator of Σ.

(b) Derive the MLEs of the parameters of multivariate normal distribution.

(10+10)

 

1. (a) Consider the partitions in Q.No. (14). Let =  where= i^th row of  . Show that, for every vector α,

(i)  Var[ X_i – X⁽²⁾ ] ≤ Var [ X_i – α′ X⁽²⁾ ]

(ii) Corr ( X_i, X⁽²⁾ ) ≥ Corr ( X_i­ , α′ X⁽²⁾ ).

Hence, obtain an expression for the multiple correlation coefficient between X_i and X⁽²⁾.

(b) Find the mean vector and the var-cov matrix for the bivariate normal distribution whose p.d.f. is

f(x,y) = exp                 (12 + 8)

1. (a) Exhibit the ‘ambiguity’ in the factor model. Bring out the need for factor  rotation and explain the ‘Varimax’ criterion for rotation.

(b) Explain the ‘Ordinary Least Squares Method’ of estimating the Factor Scores.

(12 + 8)

 

1. (a) Derive an expression for ‘Expected Cost of Misclassification (ECM)’ for classification involving two populations and obtain the optimum allocation regions for the ‘Minimum ECM Rule’.

(b) Consider the following table on three binary variables measured on five subjects with a view to carry out clustering of the five subjects:

variable   Individual	X1              X2               X5
1 2 3 4 5	0                0                   1 1                1                   1 0                0                   1 0                1                   1 1                1                   0

 

 

 

 

 

 

 

 

Obtain the matrices of matches and mismatches, compute the similarity measure

s_ij =  (under usual notations) and carry out the clustering.         (10 + 10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3811 – MULTIVARIATE ANALYSIS

 

 

Date : 01/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL the following questions:                                                                             (10 x 2 = 20 marks)

 

1. Distinguish between ‘Data Exploration’ and ‘Confirmatory Analysis’.
2. Define Variance-Covariance matrix and Generalized Variance of a random vector.
3. Give the motivation for ‘Statistical Distance’ and express its form.
4. Define ‘partial correlation coefficient’ and give the expression for it in a trivariate normal distribution.
5. Explain the objectives of Principal Component Analysis.
6. Explain the ‘Varimax’ criterion for factor rotation.
7. Distinguish between ‘Agglomerative’ and ‘Divisive’ methods in clustering.
8. Present the decomposition of the total sum of squares and cross products in one way MANOVA.
9. State the test for significance of correlation coefficient in a bivariate normal population.
10. Define Wishart distribution.

 

SECTION – B

Answer any FIVE Questions                                                                                              (5 x 8 = 40 marks)

 

11. Find the mean vector and var-cov matrix of X = (X, Y) whose p.d.f. is

f (x , y) =

12. Describe p-p plot and q-q plot and state how the multivariate normality assumption is verified.
13. Let X = ~ N_p (μ,Σ) and μ and Σ be correspondingly partitioned as  and ,  X_i be the i^th component of X⁽¹⁾ and let β = σ_(i)  where σ_(i) is the i^th row of . Derive an expression for multiple correlation coefficient between X_i and X⁽²⁾.
14. Derive the moment generating function of multivariate normal distribution.
15. Derive the T² test for hypothesis concerning the mean vector of a multivariate normal population using the likelihood ratio criterion.
16. Define Principal Components and extract the same for a given random vector, stating the lemma on maximization of quadratic forms (without proof).
17. Present Fisher’s method of discriminating two populations and derive the linear discriminant function. Explain the classification rule based on it.
18. Describe Hierarchical clustering of objects and its algorithm giving figurative depiction of three linkage methods.

 

SECTION – C

Answer any TWO Questions:                                                                                           (2 x 20 = 40 marks)

 

19. (a) Explain the ‘lowess’ curve enhancement of a scatter plot.

(b) Consider the partition in Q. No. (13). If X⁽¹⁾ and X⁽²⁾ are uncorrelated, then show that their distributions are multivariate normal of appropriate dimensions. Also, establish the same result even when X⁽¹⁾ and X⁽²⁾ are correlated.                                                                      (7+13)

 

20. (a) Present the ‘Orthogonal Factor Model’ and develop the ideas of ‘communality’ and ‘specific variance’.

(b) Explain the ‘Principal Factor’ Method of Factor Analysis. Bring out the approach to ‘Reduction of factors’ and ‘Decision on number of factors’.                                   (10+10)

 

21. (a) Carry out the ‘single linkage’ process for clustering six items whose distance matrix is given below (Dendrogram not required):

 

(b) Derive the  expression for ‘Expected Cost of Misclassification’ in  the case  of  two

populations and obtain the Minimum ECM Rule. Discuss the special cases of equal prior

probabilities and equal misclassification costs.                                                        (10+10)

 

22. (a) Derive the MLEs of the parameters of multivariate normal distribution.

(b) Establish the independence of the sample mean vector and sample var-cov matrix of a sample from multivariate normal distribution.                                                         (10+10)

 

 

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