Useful for UGC NET · ASRB NET · ISS
This is the complete study package for Multivariate Analysis (STS-202) together with Section B of STS-205, Estimation Theory and Multivariate Analysis (Conventional). It is the course in which the whole Foundation course is done again with \(p\) variables at once — a mean becomes a vector, a variance becomes a matrix, and the algebra of that matrix turns out to carry nearly all of the statistics.
Mean vectors and dispersion matrices with the three rules used everywhere afterwards; the multinomial, and why every one of its covariances is negative; the multivariate normal density read factor by factor, its moment generating function, and the partition theorem for marginals and conditionals proved in full; maximum likelihood for \(\boldsymbol\mu\) and \(\boldsymbol\Sigma\).
UNIT 2The Wishart as the chi-square in matrix form, with four properties each traced back to its univariate ancestor; why \(m \ge p\) is a hard requirement; the generalized variance as a product of independent chi-squares and the bias that follows; the null distributions of simple, multiple and partial correlations; and inference for regression coefficients.
UNIT 3What goes wrong with \(p\) separate tests; \(T^{2}\) as the likelihood ratio test, with its invariance and its \(F\) transformation; Mahalanobis \(D^{2}\) and the two-sample test; contrasts for equality of components; Wilks' \(\Lambda\) and Bartlett's approximation; and Fisher's discriminant function with its misclassification probability.
UNIT 4Principal components as the solution of a Rayleigh quotient, solved exactly for \(2 \times 2\) and from the characteristic cubic for \(3 \times 3\); canonical correlations; three linkages compared on one data set and \(K\)-means worked to convergence; classical scaling that recovers a triangle from its distances; the orthogonal factor model in closed form; and path, correspondence and conjoint analysis.
PRACTICALAll thirteen prescribed experiments: eleven worked in full in the units on data chosen so that each result checks another, plus the two that are not — writing a \(p\)-variate normal density from its parameters and reading the parameters back out of one, and the one-sample Mahalanobis \(D^{2}\).
REFERENCEThe prescribed unit-wise outline for STS-202 and the Section B practical list for STS-205, as printed, with the objectives, outcomes and the reading list.
| What is built here | Where it is used |
|---|---|
| Mean vectors, dispersion matrices and the rule \(\operatorname{Cov}(\mathbf{AX}) = \mathbf{A}\boldsymbol\Sigma\mathbf{A}'\) | Every derivation in this course, and the variance of any estimator that is a linear function of the data — including the least squares estimator of Linear Algebra & Linear Models, Unit 4 |
| Maximum likelihood for \(\boldsymbol\mu\) and \(\boldsymbol\Sigma\) | The same sufficiency and completeness argument as Estimation Theory, Unit 2, applied to a matrix parameter |
| The Wishart distribution | Pooling covariance matrices; the exact distribution of every statistic in Unit 3; the matrix version of the chi-square results of Distribution Theory, Unit 3 |
| Hotelling's \(T^{2}\) and Wilks' \(\Lambda\) | Multivariate analysis of variance, and therefore the multivariate versions of the designs in Design and Analysis of Experiments |
| Mahalanobis \(D^{2}\) and discriminant analysis | Classification of any kind; outlier detection; the nearest-centroid rules underlying machine learning classifiers |
| Principal components and factor analysis | Dimension reduction before any modelling; the standard remedy for the multicollinearity diagnosed in Linear Algebra & Linear Models, Unit 4 |
| Clustering and scaling | Exploratory analysis of any data set with no response variable — the unsupervised half of the data science material |