Multivariate Analysis
GATE ST tests multivariate distributions, dimension reduction, and classification. Questions often require covariance matrices, Mahalanobis distance, principal components, or Hotelling's T² rather than univariate shortcuts.
Core rules
- A multivariate normal vector has mean μ and covariance Σ. Its density involves the Mahalanobis form (x−μ)′Σ⁻¹(x−μ), and linear combinations remain normal.
- Hotelling's T² generalizes the univariate t-test to a mean vector. It is proportional to a scaled F statistic under normality.
- Principal components are ordered eigenvectors of the covariance or correlation matrix. The eigenvalue gives the variance explained by that component.
- Standardize variables before PCA when units differ; otherwise large-variance variables dominate.
- Discriminant analysis uses between-group and within-group covariance. Clustering is unsupervised and does not use known labels.
Worked example
Let Σ=[[4,2],[2,3]]. Solve det(Σ−λI)=(4−λ)(3−λ)−4=λ²−7λ+8=0, giving λ=(7±√17)/2≈5.562 and 1.438. The first principal component explains about 5.562/(5.562+1.438)=79.5% of total variance. For λ=5.562, solve (4−5.562)v1+2v2=0, so v2≈0.781v1. The first component is therefore dominated by the first variable but includes a positive contribution from the second.
Traps
Do not use covariance-based PCA on variables with incompatible scales. Do not confuse the largest eigenvalue with the largest eigenvector entry. Hotelling's T² is not a set of independent univariate t-tests. Correlation matrices change the eigenvectors, so state which matrix you used.