Class 3 (Module III)

25 Maio 2022, 15:00 • Pedro Cristiano Santos Martins da Silva

This class was lectured on the 1 June, 2022

Hierarchical minimum variance aggregation criterion (Ward's method). Inversions and ultra-metrics. Cophenetic distances, Sheppard diagrams and Pearson and Spearman cophenetic correlation coefficients. Partitional methods: K-means and K-medoids. Internal quality indices and estimation of the best number of clusters. Comparing partitions using the Rand and adjusted Rand indices. Clusters interpreatition: comparing the cluster mean of each quantitative (active and supplementary) variable with the population mean (v-test). 

Class 4 (Module III)

23 Maio 2022, 09:30 • Pedro Cristiano Santos Martins da Silva

This class was lectured on the  30 may, 2022

Clustering analysis: introduction, motivation and general definitions and concepts. 

Dissimilarity and distances for quantitative data (Minkowski distance, Canberra distance, generalized euclidean distances including Mahalanobis), binary data (simple matching and Jacard and  Legendre and Gower distance) and categorical data (chi-square distances). Generic hierarchical clustering algorithm and dendrogram. Single-linkage, complete-linkage, average and centroid methods: properties and examples. Hierarchical methods via Lance-Williams updating formula table. 

Class 2 (Module III)

18 Maio 2022, 15:00 • Jorge Filipe Campinos Landerset Cadima

[Slides PCA+LDA 51-65 + 69-117] Interpreting PCA in the space of variables. Some additional remarks about PCA. The Singular Value Decomposition of a generic matrix. PCA as the SVD of the centred data matrix. SVDs in R. Biplots: concept, construction and interpretation. Mahalanobis distances. Biplots in R. An example of biplots: the crayfish data (both Covariance and Correlation Matrix PCA). Linear Discriminant Analysis: introduction, the classification matrix, Fisher's formulation of the problem, the generalized eigenvalue problem and the solution to the problem. Properties of discriminant axes. Discriminant axes as the linear combinations of the original variables which maximize the F-test statistic in a one-way ANOVA over the grouping factor. Alternative formulations of LDA, all equivalent to minimizing the angle between the column-space of the centred data and the column-space of the classification matrix. Classification of new individuals using the discriminant axes. LDA in R: the lda command in the MASS package; its usage and characteristics.

Nota: Esta aula foi leccionada na sexta-feira, dia 27.5.22, das 14h30 às 17h10, na Sala S1, simultaneamente presencial e por zoom.

Class 1 (Module III)

16 Maio 2022, 09:30 • Jorge Filipe Campinos Landerset Cadima

[Slides PCA+LDA 1-50] Introduction to Multivariate Analysis: programme, bibliography, a few introductory remarks. Some essential matrix concepts: orthogonal matrices; trace of a square matrix and its properties, eigenvalues and eigenvectors. A fundamental result: the Spectral Decomposition of symmetric matrices. A statistical approach to Principal Component Analysis. The goal, the formulation of the problem and its solution via the Rayleigh-Ritz Theorem. Properties of principal components. PCA with R: the prcomp command. An example with the crayfish (lavagantes) dataset. The dependence of PCA on the units of measurement. PCA on the standardised data (correlation matrix PCA): concept and discussion. Again the crayfish example.
Nota: Esta aula foi leccionada na quarta-feira, dia 25.5.22, simultaneamente presencial e por zoom. Devido a um problema de conexão à rede, acabou por ser leccionada na Sala S1, das 15h30 às 18h00,

Class 15 (Module II) – Linear Mixed Models

11 Maio 2022, 15:00 • Elsa Maria Félix Gonçalves

Linear mixed model with one factor of fixed effects and one factor of random effects, without interaction, balanced and diagonal matrices G e R (exercise 3).
Linear mixed model with one factor of fixed effects and one factor of random effects, with interaction, balanced and diagonal matrices G e R (exercise 4).
Some examples of other linear mixed models (exercises 6 and 8).

Nota: Esta aula foi leccionada na segunda-feira, dia 23.5.22, das 15h às 17h30, na Sala 39, simultaneamente presencial e por zoom.