Multivariate Statistics


It is intended to familiarize the student with inference techniques for multivariate mean values ​​and covariance matrices, as well as multivariate linear models in Gaussian populations, dimensionality reduction methods, discrimination and data classification methods.

General characterization





Responsible teacher

Carlos Manuel Agra Coelho


Weekly - 4

Total - 56

Teaching language



Basic concepts of analysis and intermediate level knowledge in Linear Algebra, Probabilities and Statistical Inference


Anderson, T. W. (2003), An Introduction to Multivariate Statistical Analysis, 3rd ed., J. Wiley & Sons, New York

Flury, B. (1997), A First Course in Multivariate Statistics, Springer. New York

Johnson, R. and Wichern, D. W. (2007), Applied Multivariate Statistical Analysis, 6th Edition, Prentice Hall, New Jersey

Morrison, D. F. (2004), Multivariate Statistical Methods, 4th Edition, Duxbury Press

Rencher, A. C. (1998), Multivariate Statistical Inference and Applications, John Wiley & Sons

Rencher, A. C. and Christensen, W. F. (2012). Methods of Multivariate Analysis, Third Edition, John Wiley & Sons

Zelterman, D. (2015). Applied Multivariate Statistics with R. Springer

Teaching method

Lectures will be presential (if possible, to be confirmed). 
It is intended that the classes work according to an essentially interactive teaching model, 
using practical exercise resolution and data analysis in R environment.

Evaluation method

Continuous evaluation:

It includes the following components:

  • 1st test (MT1) - Test with a weight of 30%. The test will have a time limit of 2 hours. The test is rated on a scale of 0 to 20 points (with a minimum rating of 7).
  • 2nd test (MT2) - Test with a weight of 30%. The test will have a time limit of 2 hours. The test is rated on a scale of 0 to 20 points (with a minimum rating of 7).
  • Work/Report (T) - Individual work, with a weight of 40%, rated on a scale of 0 to 20 values. This work has to be turned in between November 27 and December 4.

Final grade calculation formula (NF): NF = 0.3 x MT1+ 0.3 x MT2 + 0.4 x T (partial grades rounded to the nearest unit)

Resource (Improvement)/Special: In-person written test, to be held on a single date, within the period provided for in the academic calendar, with a weight of 100%. The exam will have the duration of 3 hours. Rated on a scale from 0 to 20 points.

Subject matter

  1. Succint revisions (linear algebra, random variables and random vectors, confidence intervals and statistical inference)
  2. Introduction to Multivariate Statistics: the multivariate Normal distribution and the Wishart distribution
  3. Graphical representations and an outlier test 
  4. Inference for vectors of expected values
    1. Inference for a single vector of expected values
    2. Test of equality for two vectors of expected values
    3. Test of equality for several vectors of expected values
  5. Inference on covariance matrices
    1. Tests and Confidence Intervals for eigenvalues and the sphericity test
    2. The test of equality of several covariance matrices
  6. Analysis of covariance structures
    1. Principal Components Analysis (tests and confidence intervals for eigenvalues and the test for outliers revisited)
    2. Canonical Correlations Analysis and the Multivariate Linear Model (the test of independence of 2 sets of variables and tests for canonical correlations and for models and submodels)
  7. Classificatory Analysis and Clustering 
    1. Discriminant Analyis
    2. Cluster Analysis