# Coursebooks 2017-2018

## Linear models

#### Lecturer(s) :

Thibaud Emeric Rolland Georges

English

#### Summary

Regression modelling is a basic tool of statistics, because it describes how one variable may depend on another. The aim of this course is to familiarize students with the basis of regression modelling, and of some related topics.

#### Content

• Properties of the Multivariate Gaussian distribution and related quadratic forms.
• Gaussian linear regression: likelihood, least squares, variable manipulation and transformation, interactions.
• Geometrical interpretation, weighted least squares; distribution theory, Gauss-Markov theorem.
• Analysis of variance: F-statistics; sums of squares; orthogonality; experimental design.
• Linear statistical inference: general linear tests and confidence regions, simultaneous inference
• Model checking and validation: residual diagnostics, outliers and leverage points.
• Model selection: the bias variance effect, stepwise procedures. Information-based criteria.
• Multicollinearity and penalised estimation: ridge regression, the LASSO, relation to model selection, bias and variance revisited, post selection inference.
• Departures from standard assumptions: non-linear least Gaussian regression, robust regression and M-estimation.
• Nonparametric regression: kernel smoothing, roughness penalties, effective degrees of freedom, projection pursuit and additive models.

#### Learning Prerequisites

##### Recommended courses

Analysis, Linear Algebra, Probability, Statistics

#### Learning Outcomes

By the end of the course, the student must be able to:
• Recognize when a linear model is appropriate to model dependence
• Interpret model parameters both geometrically and in applied contexts
• Estimate the parameters determining a linear model from empirical observations
• Test hypotheses related to the structural characteristics of a linear model
• Construct confidence bounds for model parameters and model predictions
• Analyze variation into model components and error components
• Contrast competing linear models in terms of fit and parsimony
• Construct linear models to balance bias, variance and interpretability
• Assess / Evaluate the fit of a linear model to data and the validity of its assumptions.
• Prove basic results related to the statistical theory of linear models

#### Teaching methods

Lectures ex cathedra, exercises in class, take-home projects

#### Assessment methods

Continuous control, final exam.

Second session: from the rulebook of the Section of Mathematics (art. 3 al. 5), the teacher decides of the form of the exam and communicates it to the concerned students.

#### Supervision

 Assistants Yes

No

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16    MAA112
16-17  MAA112
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German