Multivariate analysis

The aim of this course is to provide students with a knowledge of the basic concepts underlying the most important multivariate techniques, through the use the statistical methodology and software packages for the analysis of multivariate data.

1. Introduction to multivariate analysis.
Basic concepts in matrix algebra for multivariate statistics. Multivariate data: data matrix, mean vector, covariance matrix and correlation matrix. Proximity measures. Graphical representation.

2. Statistical inference for multivariate normal data.
Inference for the mean and the covariance matrix of a normal population. Confidence regions and simultaneous inference. Comparison of multivariate normal populations. Multivariate Normality Testing.

3. Multivariate analysis of variance.
One-way MANOVA. MANOVA table, hypothesis testing, multiple comparisons. Two-way MANOVA. Interaction effect.

4. Principal components analysis.
Decomposition of a random vector in its principal components. Properties and interpretation. Number of components to choose. Biplot.

5. Correspondence analysis.
Expression of the chi-square statistic by the row and column profiles. Extraction of components. Simultaneous representation of rows and columns. Interpretations.

6. Fundamentals of Discriminant Analysis.
Basic concepts. Disciminant analysis with two populations. Discriminant analysis for normal data. Generalization to more than two populations. Logistic discrimination.

7. Cluster analysis.
Hierarchical clustering. Partitioning methods: k-means method.

Basic pibliography
Everitt, B.S. (2005). An R and S-Plus companion to multivariate analysis. Springer.
Hardle, W.K., Simar, L. (2015). Applied multivariate statistical analysis. Fourth Edition. Springer.
Johnson, R.A., Wichern, D.W. (2007). Applied multivariate statistical analysis. Pearson Education.
Mardia, K.V., Kent, J.T., Bibby, J.M. (1979). Multivariate analysis. Academic Press.

Complementary bibliography
Everitt, B.S., Dunn, G. (2001). Applied multivariate data analysis. Hodder Education.
Hastie, T., Tibshirani, R., Friedman, J. (2009). The elements of statistical learning. Springer.
Koch, I. (2014). Analysis of multivariate and high-dimensional data. Cambridge.
Peña, D. (2002). Análisis de datos multivariantes. McGraw-Hill.
Pérez, C. (2004). Técnicas de análisis multivariante de datos. Pearson Educación, S.A.
Seber, G.A.F. (1984). Multivariate observations. Wiley.

This subject will work on the basic, general, and transversal competences included in the degree's memory. Below are the specific competences that will be enhanced in this subject:
Specific competences:
E1 - Know, identify, model, study, and solve complex problems of statistics and operations research in a scientific, technological, or professional context, arising in real applications.
E2 - Develop autonomy for the practical resolution of complex problems arising in real applications and for the interpretation of results to aid decision-making.
E3 - Acquire advanced knowledge of the theoretical foundations underlying the different methodologies of statistics and operations research, enabling specialized professional development.
E4 - Acquire the necessary skills in the theoretical-practical handling of probability theory and random variables to enable professional development in the scientific/academic, technological, or specialized and multidisciplinary professional field.
E5 - Deepen knowledge in the specialized theoretical-practical foundations of modeling and studying different types of dependency relationships between statistical variables.
E6 - Acquire advanced theoretical-practical knowledge of various mathematical techniques, specifically oriented towards aiding decision-making, and develop the ability to reflect to evaluate and decide between different perspectives in complex contexts.
E8 - Acquire advanced theoretical-practical knowledge of techniques for making inferences and contrasts related to variables and parameters of a statistical model, and know how to apply them autonomously in a scientific, technological, or professional context.
E9 - Know and be able to apply autonomously in scientific, technological, or professional contexts, machine learning techniques and techniques for analyzing high-dimensional data (big data).
E10 - Acquire advanced knowledge of methodologies for obtaining and processing data from different sources, such as surveys, the internet, or "cloud" environments.

The students' face-to-face activity will be 35 hours, including attendance and participation in lectures and interactive sessions. In the lecture part, the faculty will use multimedia presentations, while in the interactive part, students will solve various questions posed on the subject content using the statistical software R.

Students will have access to the teaching materials (presentations, notes, exercises) for the subject. Throughout the course, exercises/tasks will be proposed that students must solve with the guidance of the teachers. This guidance will be provided both through virtual means (communication platforms or email) and in-person in small groups.

The subject assessment is carried out through continuous assessment and the final exam.
Continuous assessment will account for 30% of the final grade and will take into account the exercises/assignments completed throughout the course. Continuous assessment allows for the evaluation of the acquisition of basic competencies CB7-CB9 and general competencies CG1-CG5. The level achieved in cross-cutting competencies CT1-CT5 will be considered. The level achieved in specific competencies CE1, CE2, CE5, CE6, and CE9 will also be evaluated.

The final exam, consisting of problems and questions, will account for 70% of the final grade. The exam will assess specific competencies CE1-CE6.

The grade obtained in continuous assessment will be retained in both assessment opportunities of each course's call. In the second assessment opportunity, an exam will be conducted, and the final grade will be the highest among: i) the grade from the first opportunity, ii) the grade of the new exam, and iii) the weighted average of the new exam (70%) and continuous assessment (30%).

The study time and individual work of the students in orde to pass the course is 125 hours, distributed as follows:

1) Presential work (38 hours): 35 hours (lectures-interactive) + 3 hours (exam)
2) Non presential work (87 hours): 1 hours for each lecture-interactive hour (excluding the exam) and time for continuous assessment assignments.

It is advisable to have basic knowledge of linear algebra and metric geometry, as well as probability calculus and statistics. It is also recommended to have intermediate skills in computer usage, specifically in statistical software. For better learning of the subject, it is helpful to keep in mind the practical sense of the methods being learned, as well as a graphical visualization of procedures dealing with multivariate data.

The development of the course contents will be carried out taking into account that the competencies to be acquired by the students must meet the MECES3 level. Emphasis will be placed on the technical foundations of the multivariate tools studied, and they will be applied in different practical examples, so that students become familiar with both the potentialities and possible limitations of the methods.

In cases of fraudulent completion of exercises or tests, the provisions of the respective regulations of the universities participating in the Master's Degree in Statistical Techniques will apply.

This guide and the criteria and methodologies described herein are subject to modifications resulting from regulations and guidelines of the universities participating in the Master's Degree in Statistical Techniques.