Sumários

Module II, Class 4

15 Março 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

Online class on Tuesday, March 16, from 14h30 to 17h00
[Slides 116-155] The Linear Model (in the regression context): the additional assumptions, the vector notation. Tools for random vectors: the  expected vector and its properties; the (co-)variance matrix and its properties; the MultiNormal distribution and its properties. The Linear Model in vector notation. The consequences of the Linear Model for the observations of the response vector Y (Normal distribution, independence). The vector of parameter estimators Beta-hat: definition; its probability distribution under the Linear Model and its interpretation. The final hurdle to using the distribution of each beta-hat_j for inference: the unknown variance of random errors, sigma^2. Estimating sigma^2 with the Residual Mean Square (QMRE). The effect of replacing sigma^2 with QMRE on the distribution: the appropriate Student's-t distribution. A confidence interval for any parameter beta_j: deduction and interpretation. Hypothesis tests for any beta_j.

Module II, Class 3

9 Março 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

Online class on Monday, March 15, as on Tuesday March 9 there was the Module I test
Revision of matrix operations and properties. Exercise 11. [Slides 87-115] Again the geometry of the space of variables. An alternative right triangle beginning with the centred vector y^c: the fundamental formula of linear regressions as a direct application of the Pythagorean Theorem; the geometric interpretation of the Coefficient of Determination. Properties of linear regressions (with the intercept). Interpretation and units of measurment of the fitted coefficients b_j and of residuals. The vector of residuals as an orthogonal projection on the orthogonal complement of C(X). An example: a multiple regression on the iris dataset. Multiple regressions in R. Submodels: definition, properties and warning. Example: a simple linear regression with the iris data, as a submodel of the previous example. Polynomial regression: the concept; the trick to fit them as multiple linear regressions; an example (videira dataset). The transition to an inferential context. Initial assumptions and the model equation.

Module II - Class 2

8 Março 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

[Slides 48-86] More non-linear relations that can be linearized by suitable transformations: the (2-parameter) logistic relation; the power law; hyperbolic relations; Michaelis-Menten relation (for each, we discussed the linearizing transformations and solved differential equations that gave rise to them). Multiple linear regression: the problem, a motivating example; the difficulties associated with more than one predictor; the least-squares criterion. An alternative representation of the data: the space of variables. The geometric problem in the space of variables that corresponds to a least-squares solution. The model matrix and its column-space, C(X). Orthogonal projections onto C(X) and the matrix of orthogonal projections, the hat-matrix H. The projected vector y-hat and the formula for the model parameters, b. The three sums of squares and the fundamental formula as an application of the Pythagorean Theorem.

Module II - Class 1

2 Março 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

[Slides 1 to 47] Introduction (Programme, Bibliography, webpage). Motivating examples. A few basic ideas about Statistical Modelling. Reviewing simple linear regression: least squares criterion, formulas, properties, drawbacks. Commands for linear regression in R. Examples. Nonlinear relations and linearizing transformations: the exponential relation: its equation, its linearizing transformation, its differential equation, its use in population models and drawbacks.

Lesson 5 -Module I

1 Março 2021, 14:30 • Manuela Neves

Principais métodos de estimação - o método dos momentos e  o método da máxima verosimilhança (definição, resolução de exemplo de aplicação). Propriedades.

A utilização do R. Resolução de exercícios. 

A construção e a interpretação de intervalos de confiança. Intervalos de confiança para os parâmetros usuais em uma e duas populações. Interpretação no R.

Testes de hipóteses paramétricos. Utilização do R. Testes de ajustamento - o teste de Shapiro-Wilk