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Programmes → I.M.Sc. in Mathematical Sciences and M.Sc. in Statistics-OR

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Multivariate Analysis

Course Code: IM 401; ST 501  |  Credits: 4  |  Prerequisite: Linear Algebra and Matrix Theory

Syllabus Details

  • Unit I:(Multivariate Distributions and the Multivariate Normal) Concept of random vector and random matrix; multivariate distribution function and marginal and conditional distribution; bivariate distributions: Binomial, Poisson, Exponential distributions, Multinomial, Dirichlet distributions; conditional and marginal distributions; multivariate normal distribution (MVN): singular and non-singular MVN, properties of MVN, marginal distributions, conditional distributions, linear transformations, characteristic function, estimation of MVN parameters, sampling distributions of MLEs, hypothesis testing for MVN mean, distribution of sample mean vector and its independence.
  • Unit II:(Correlation, Hotelling’s T2 and Wishart Distribution) Multiple linear equations; multiple correlation, partial correlation in multiple setup; Hotelling’s T2 and its applications; Hotelling’s T2 statistic as a generalization of square of Student’s statistic; distance between two populations, Mahalanobis D2 statistic and its relation with Hotelling’s T2 statistic; Wishart distribution, its properties, usage in confidence region construction.
  • Unit III:(Multivariate Regression and Techniques) Multivariate regression, MANOVA, MANCOVA; multivariate techniques: principal component analysis, discriminant analysis, multidimensional scaling, factor analysis, canonical correlation.
  • Unit IV:(Classification and Clustering) Classification problem, cluster analysis.

Course Learning Objective

  1. Explain and discuss the different multivariate distributions, their properties and characterizations.
  2. Explain the inadequacy of Euclidean distance between two data points of a multivariate random vector, explain why measures based on the dispersion matrix are more suitable, and test whether data come from a multivariate normal population.
  3. Estimate parameters of the MVN distribution and confidence regions, and carry out hypothesis tests for the multivariate normal distribution on both the mean vector and the dispersion matrix.
  4. Explain multivariate techniques like PCA, factor analysis and discriminant analysis and use them on multivariate data for analysis.
  5. Explain different ways of classification of variables and observations: hierarchical, clustering.
  1. Johnson, R. A. and Wichern, D. W. Applied Multivariate Analysis. 5th ed. Prentice-Hall, 2002
  2. Mardia, K. V., Kent, J. T. and Bibby, J. M. Multivariate Analysis. Academic Press, 1979
  3. Anderson, T. W. Introduction to Multivariate Analysis. John Wiley, 1984
  4. Kshirsagar, A. M. Multivariate Analysis. Marcel Dekker, 1979
  5. Kotz, S., Balakrishnan, N. and Johnson, N. L. Continuous Multivariate Distributions, Volume 1, Models and Applications. John Wiley & Sons, 2000
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