LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3801/S916 – MULTIVARIATE ANALYSIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

1. Let X ~ N . Obtain the conditional distribution of X₁ given X₂ = x₂.
2. Write the characteristic function of bivariate normal distribution.
3. Explain how the collinearity problem can be solved in the multiple regression

Y = Xb + Î.

4. If X and Y are two independent standard normal variables, obtain the distribution of two times of the mean of these two variables.
5. Let X be trinormal with

m =  and   compute  .

6. Define Fisher’s Z – transformation.
7. Explain classification problem into two classes.
8. Write down any four similarity measures used in cluster analysis.
9. Distinguish between principal component and factor analysis.
10. What is meant by residual plot?

SECTION-B

Answer any FIVE  questions.                                                                          (5×8=40 marks)

11. Define multiple correlation coefficient between X₁ and X₂, …., X_p. Show that the multiple correlation coefficient between X₁ and X₂, …., X_p has the expression

.

12. Let Y ~ N_p ( 0, å) . Show that Y¹ has – distribution.
13. Test at level 0.05 whether m = in a bivariate normal population with s₁₁ = s₂₂ = 5 and s₁₂ = -2 ,   by using the sample mean vector  based on a sample of size 10.
14. How will you test the equality of covariance matrices of two multivariate normal distributions on the basis of independent samples drawn from two populations?
15. Derive the characteristic function of Wishart distribution.
16. In Principal component analysis derive the first principal component.
17. Obtain the rule to assign an observation of unknown origin to one of two p-variate normal populations having the same dispersion matrix.
18. Outline single linkage and complete linkage clustering procedures with an example.

SECTION-C

Answer any TWO  questions.                                                                          (2×20=40 marks)

19. a) Derive the MLE of å when the sample is from N_p (m, å).
20. b) Define Hottelling’s T² – statistic.
21. c) Using the likelihood ratio test procedure, show that the rejection region for testing m =

m_o against m ¹ m_o is given by

T² = n( S^-1 (³  T.                                                             (10+3+7)

20. a) Prove that under some assumptions (to be stated), Variance- covariance matrix can be

written as å = LL¹ +  in the factor analysis model.  Also discuss the effect of an

orthogonal transformation.

1. b) Let X₁,X₂,…., X_p have covariance matrix å with eigen value vector pairs (l₁, e₁),…,

(l_p, e_p),  l₁ l₂ ….. ≥ l_p 0,   then prove that

s₁₁ + s₂₂ + …..+ s_pp = ,

Where ,  Y_i  represents the i – th principal component.

1. c) Explain the principal component (principal factor) method of estimating L in the factor

analysis model.                                                                                                  (10+5+5)

21. a) Explain the method of extracting canonical correlations and canonical variables. Also

explain how the theory of canonical correlation is helpful in the analysis of multivariate

data.

1. b) State an establish the additive property of Wishart distribution. (10+10)
2. Write shot notes on:-
3. Roy’s Union – Intersection principle
4. Step – Wise regression
5. Mohalanobis Squared distance. (5+10+5)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3801/ 3805 / 3808 – MULTIVARIATE ANALYSIS

Date & Time: 24/04/2007 / 9:00 – 12:00      Dept. No.                                       Max. : 100 Marks

SECTION A

10 X 2 =20

ANSWER ALL QUESTIONS. EACH CARRIES TWO MARKS

1. Write the likelihood function corresponding to a sample of size drawn from .
2. Mention any two applications of F-distribution in Multivariate Statistical Analysis.
3. Find the distribution of if.
4. Mention any two properties of variance-covariance matrix.
5. How will you obtain the MLE of population correlation coefficient between two variables having bivariate normal distribution?
6. What do mean by agglomerative algorithms?
7. Define : Communalities
8. Define Non-central Chi-square statistic.
9. What is meant by “Expected Cost of Misclassification”?
10. Give an unbiased estimator of variance-covariance matrix in

SECTION B

5 X 8 = 40

ANSWER ANY FIVE. EACH CARRIES EIGHT MARKS

1. Derive the distribution of where and is a matrix of rank

1. State and prove a characterization of multivariate normal distribution

1. Obtain an expression of under usual notations.

1. Obtain the rule which minimizes Expected Cost of Misclassification(ECM) in the case of two multivariate normal populations assuming priori probabilities are known to be equal and the populations have equal variance-covariance matrices.
2. Show that where B,C,D and E are matrices of suitable order and mention any two places the above identity is used in multivariate analysis.

1. State and prove any two properties of Wishart’s Distribution.

1. Define Generalized variance. Show it is equal to the product of eigen roots of

1. Obtain expressions for the first r-principal components in a p-variate normal distribution.

SECTION C

2 X 20 = 40

ANSWER ANY TWO. EACH CARRIES TWENTY MARKS

1. Derive the conditional distribution of a sub vector of
2. Derive the sampling distribution sample correlation coefficient assuming the population multiple correlation coefficient is zero.

1. Derive the likelihood ratio test for assigned mean based on a sample of size N drawn from assuming is unknown.

1. Write short notes of the following :

• Similarity and Distance Measures.
• Discriminant Analysis
• Cluster Analysis
• Orthogonal Factor Model

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2008

    ST 3808 / 3801 – MULTIVARIATE ANALYSIS

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer all the questions.                                                                         (10 X 2 = 20)

1. If X and Y are two independent standard normal variables then obtain the distribution of two times of the mean of these two variables.

• If X = ( X₁, X₂ )’~N₂ then write the c.f. of the marginal distribution

of X₂.

3. Mention any two properties of multivariate normal distribution.

4 . What is meant by residual plot?.

5. Explain the use of partial and multiple correlation coefficients.
6. Define Hotelling’s T² – statistics.
7. Define Fisher’s Z-transformation
8. Write a short on discriminant analysis.
9. Explain how canonical correlation is used in multivariate data analysis
10. Explain classification problem into two classes.

PART B

Answer any FIVE questions.                                                                     (5 X 8 = 40)

11.Find the multiple correlation coefficient between X₁ and X ₂ , X₃, …     , X _p.

Prove that the conditional variance of X₁ given the rest of the variables can not be

greater than unconditional variance of X₁.

12. Derive the c.f. of multivariate normal distribution.
13. Let Y ~ N_p ( 0 , S ). Show that Y’S ^-1 Y   has     distribution.
14. Obtain a linear function to allocate an object of unknown origin to one of the two

normal populations.

15. Giving a suitable example describe how objects are grouped by complete linkage

method.

16. Let X ~ N_p (m, S). If X⁽¹⁾ and X⁽²⁾ are two subvectors of X, obtain the conditional

distribution of X⁽¹⁾ given X⁽²⁾.

17. Prove that the extraction of principal components from a dispersion matrix is the

study of characteristic roots and vectors of the same matrix  .

18. Explain step-wise regression.

PART C

Answer any two questions.                                                                 (2 X 20 = 40)

               19 .  Derive the MLE of  m and S when the sample is from N ( m ,S ).

20. a) Derive the procedure to test the equality of mean vectors of two p-variate

normal populations when the dispersion matrices are equal.

1. b) Test at level 0.05 ,whether µ = ( 0 0  )’ in a bivariate normal population with

σ₁₁ = σ₂₂= 10 and  σ₁₂= -4 , by using the sample mean vector= (7  -3)^‘

based on a sample size 10.

(15 + 5)

21. a) Define i) Common factor ii) Communality  iii) Total variation
22. b) Explain the principal component ( principal factor )method of estimating L

in the factor analysis method.

1. c) Discuss the effect of an orthogonal transformation in factor analysis method.

( 6 + 6 + 8 )

22. What are canonical correlations and canonical variables? .Describe the extraction

of canonical correlations and their variables from the dispersion matrix. Also

show that there will be p canonical correlations if the dispersion matrix is of size p.

( 2+10+8 )

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2010

    ST 3811  – MULTIVARIATE ANALYSIS

Date : 29-10-10                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

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

1. State and briefly explain any one broad objective for which multivariate analysis techniques are used.
2. Define Variance-Covariance matrix of a random vector.
3. Explain ‘Bubble Plot’ for three-dimensional data display.
4. Define ‘Partial Correlation Coefficient’.
5. Explain the dimensionality reduction role of Principal components.
6. Distinguish between ‘Clustering’ and ‘Classification’.
7. State any two similarity measures for pairs of items when variables are binary.
8. State any one of the commonly employed scalings on the coefficient vector of the Fisher’s linear discriminant function and the interpretative use of the scaling.
9. State the Hotelling’s T² statistic for testing of hypothesis about the mean vector of a multivariate normal population.
10. State the test for significance of the correlation coefficient in a bivariate normal population.

SECTION – B

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

1. If is the sample mean vector and S_n is the sample var-cov matrix based on a random sample of size ‘n’ from a p-variate distribution with mean vector μ and Var-cov matrix Σ, show that Var-Cov() = Σ /n and S_n is a biased estimator of Σ.
2. Explain the construction of ‘probability plots’ in general and the investigation of multivariate normality assumption in particular.
3. If X ~ N_p(μ, Σ ) and C is a non-singular matrix of order p x p, show that                CX ~ N_p ( Cμ, CΣC^T ). Hence, deduce the distribution of DX where D is a q x p matrix with rank q (≤ p).
4. Under usual notations of partitions of a random vector and its var-cov matrix, with  = , show that ρ( X_i ,X⁽²⁾ ) ≥ ρ( X_i , X⁽²⁾ ) for every vector.
5. Develop the MANOVA for comparing mean vectors of a number of normal populations and explain the test procedure for the same.
6. Establish the relationship of principal components to the eigen values and eigen vectors of the var-cov matrix of the underlying random vector.
7. State the three linkage methods for hierarchical clustering. Present a figurative display of the measure of between-cluster distances in each method.
8. Derive the ‘Minimum ECM Rule’ for classification involving two multivariate normal populations with a common var-cov matrix.

(P.T.O)

SECTION – C

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

1. (a) Describe the enhancement of scatter plots with ‘lowess’ curves.

(b) Develop the multivariate normal density function.                                                              (8 +12)

1. (a) Derive the MLE’s of the parameters μ, Σ of multivariate normal distribution.

(b) Show that these MLEs are independently distributed.                                                         (12 + 8)

1. Present the motivation, definition and derivation of Fisher’s (multiple) discriminant functions.

1. (a) Present the ‘Orthogonal Factor Model’ in detail and develop the notions of ‘Communality’ and ‘Specific Variance’.

(b) Bring out the ambiguity in a factor model, the rationale for ‘factor rotation’ and explain the varimax criterion for factor rotation.                                                                                          (12 + 8)

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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:

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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