MASTER MATHEMATICS

This course completes the notions developed in the PDE Analysis 1 course. In particular, it is an opportunity to study in depth some linear PDEs posed on an open of Rn, such as the Dirichlet problem, the heat equation, the Schrödinger equation or the wave equation.

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Description: Partial Differential Equations (PDEs) are nowadays an essential mathematical object for the study and understanding of physical or biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, and then to their solution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming exercises will illustrate the scientific computing tools seen in the course with simple examples.

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This course is a continuation of the optimization course of the second semester of L3.

After a reminder of the results and numerical methods for optimization problems of order one and two, without constraint and under equality and inequality constraints, the course focuses on issues of interest today in industrial optimization, and in particular, robust, multicriteria optimization, in the presence of uncertainties.

The course then illustrates the place of optimization in the main machine learning algorithms. These issues are illustrated by examples of classification and regression problems in supervised learning. These examples are the occasion to discuss the issues of metrics and procedures for the evaluation of learning, validation and inference (crossfold, overfitting, etc).

The course presents the different classes of learning: unsupervised, supervised, transfer, reinforcement, incremental, etc.

Issues around database management are addressed: generation, imputation, visualisation, slicing.

The course presents the links between transfer learning and numerical simulation to address issues of synthetic database generation, imputation, non-intrusive prediction, rapid inference, etc.

The course includes an important part of computer projects along the way. All the sessions take place in a computerised environment and allow an immediate implementation of the theoretical elements.

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The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behaviour is essentially based on the use of analytical results and in particular functional analysis. This course presents first important tools for the resolution of PDEs via analytical or geometrical points of view. These tools will be implemented in the study of some examples of PDEs representative of large classes of equations.

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This course develops the classical theory of Banach spaces and is also an introduction to spectral analysis.

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Partial Differential Equations (PDEs) are nowadays an essential mathematical object for the study and understanding of physical or biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, and then to their solution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming exercises will illustrate the scientific computing tools seen in the course with simple examples.

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Finite elements are a widely used numerical method. This course will explain the principles of the method, give the useful equations on various problems and give the keys for the computer implementation of the method.

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This 42 hour course gives the basic elements of the mechanics of continuous media: we study the motions, the deformations and the fields of constraints within media which we consider from a macroscopic point of view, as opposed to a corpuscular description. More precisely, we analyse these physical phenomena by describing them from a mathematical point of view.

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An introductory course in differential geometry, focusing on the notions of subvarieties of Rn, vector fields and flows.

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This course covers the basic aspects of the C++ language as applied to numerical analysis.

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Internship supervised by a researcher or teacher-researcher.

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This course deals with the study of advanced numerical methods for partial differential equations allowing the use of polyhedral meshes. The first part of the course is devoted to analysis tools of general interest. In the second part, the design and analysis of Hybrid High-Order methods, which are an example of the latest generation of numerical methods, are discussed. In the third part, applications of these methods are developed in connection with the research activities present at IMAG: fluid mechanics, solid mechanics and flows in porous media.

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In this course, analytical methods for the solution of partial differential equations (PDEs) -- possibly nonlinear -- and the study of the qualitative behaviour of the solutions will be presented. The class of PDEs and therefore the methods studied may depend on the speaker. They will be related to the applications developed within IMAG: fluid mechanics, solid mechanics, maths-bio.

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This course is a continuation of the optimization course of the second semester of L3 Mathematics and the one on optimization and machine learning of M1 MANU. The course is based on the ingredients given in the other modules of the MANU master in PDE analysis and numerical simulation.

After a reminder of the results and numerical methods for the numerical simulation of PDEs on adaptive mesh, of the results of a posteriori error estimation, and of the supervised learning methods of the M1, the course focuses on the generation of quality databases and their completion and certification thanks to the numerical simulation certified thanks to an error control.

This question is fundamental for the certified use of machine learning in industry. Indeed, the accuracy of mathematical learning during inference is strongly conditioned by the quality of the database.

The course includes an important part of computer projects along the way. All the sessions take place in a computerised environment and allow an immediate implementation of the theoretical elements.

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Inverse problems are studied, emphasising the notion of a 'well-posed problem'.

A presentation is first made of inverse problems in finite dimension (notion of conditioning, singular values, ...), then in infinite dimension (stability of the problem, regularisation, pseudo-inverse...).

In a second step, the properties of the Fourier transform and the convolution operation are recalled in Lp spaces and their usual subspaces. We examine how these two operations lead to well or badly posed problems. For convolution, in particular, the notions of "filter" and "deconvolution" are discussed in the context of signal and image analysis.

Finally, we apply these notions to the study and inversion of the Radon transform, or related transforms, from medical imaging techniques.

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This course covers advanced aspects of the C++ language applied to scientific computing, complemented by a presentation of pre/post-processing tools, modern collaborative work tools (version managers), non-regression tools (test managers).

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This course allows students to become familiar with real problems involving the theoretical or numerical solution of partial differential equations.

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Minimum 4-month internship in a company, EPIC or research laboratory, supervised by a researcher, teacher-researcher or study/research engineer.

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Description: Partial Differential Equations (PDEs) are nowadays an essential mathematical object for the study and understanding of physical or biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, and then to their solution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming exercises will illustrate the scientific computing tools seen in the course with simple examples.

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The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behaviour is essentially based on the use of analytical results and in particular functional analysis. This course presents first important tools for the resolution of PDEs via analytical or geometrical points of view. These tools will be implemented in the study of some examples of PDEs representative of large classes of equations.

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English tutorial for students in the M1 Fundamental Maths programme who are aiming for professional autonomy in the English language.

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In this course we study classical groups (linear, unitary, orthogonal, symplectic), in their algebraic (reduction, conjugation classes...), geometric (actions, exponential application) and topological aspects.

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This course develops the classical theory of Banach spaces and is also an introduction to spectral analysis.

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This course develops the classical theory of modules over a principal ring, and the basics of group representation theory with a focus on finite groups.

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This course develops Cauchy theory for functions of one complex variable, and introduces the notions of conformal representation, fundamental group and coverings.

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An introductory course in differential geometry, focusing on the notions of subvarieties of Rn, vector fields and flows.

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This course develops different aspects of effective calculus in algebra (linear algebra on Euclidean rings, resultants,...) and introduces the first elements of algebraic geometry.

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This course complements the differential geometry course of M1 Fundamental Mathematics. It applies the notions of this course to the study of classical Lie groups.

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An introductory course in the theory of bodies, with the Galois correspondence theorem as the main result.

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Each student works on a project supervised by a researcher or a teacher-researcher. Writing of a thesis of about thirty pages, defended orally before a jury. It is possible to work in pairs.

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Differential geometry course.

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Algebraic geometry course.

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Study and presentation of a mathematical result at M2 level in front of the other students of the course.

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Algebraic topology course.

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Course on a specialised topic in topology or geometry.

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Course on a specialised subject in algebra or geometry.

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Internship supervised by a researcher or teacher-researcher.

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The competition for the external agrégation in mathematics includes two oral tests for eligible candidates. These tests take place in June or July. The first test is entitled "Algebra and Geometry" and the second "Analysis and Probability". In order to prepare for these tests, this UE first presents their specificity and then proposes to prepare the different themes announced by the jury of the competition.

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This course is devoted to the revision of the algebra notions included in the competition programme. It is composed of 36 hours of lectures and 36 hours of tutorials. A large part of the tutorials will be reserved for interventions by the students of the preparation to present either corrections of exercises or important demonstrations of the course.

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The competitive examination for the external agrégation in mathematics contains two 6-hour written tests held in the spring. The first test is entitled "General Mathematics" and the second "Analysis and Probability". In order to prepare for these exams, this UE consists of evaluations and corrections based on subjects at the level of the written exams of the agrégation.

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This course is devoted to the revision of the notions of analysis included in the competition programme. It is composed of 36 hours of lectures and 36 hours of tutorials. A large part of the tutorials will be reserved for interventions by the students of the preparation to present either corrections of exercises or important demonstrations of the course.

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The competition for the external agrégation in mathematics includes two oral tests for eligible candidates. These tests take place in June or July. The first test is entitled "Algebra and Geometry" and the second "Analysis and Probability". In order to prepare for these tests, this course first presents their specificity and then proposes to prepare the different themes announced by the jury of the competition. This course continues the work started in the course "Preparation for the oral exam 1". 

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TD courses in English, for students in the "M2 Préparation au concours de l'agrégation en mathématiques" programme who are aiming for professional autonomy in English.

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The competition for the external agrégation in mathematics contains two 6-hour written tests held in the spring. The first test is entitled "General Mathematics" and the second "Analysis and Probability". In order to prepare for these exams, this UE consists first of all of evaluations and corrections based on subjects at the level of the written exams of the agrégation. This course is designed to help students prepare for these exams by providing assessments and corrections based on subjects at the level of the written exams for the agrégation.

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This course prepares students for the modelling test of the option C of the external agrégation in mathematics. This test, based on the study of texts, is centred on the links between algebra and formal calculation. The formal calculation software SageMaths is used for this preparation.

The problems addressed are those of the option C programme: representation and algorithmic manipulation of the mathematical objects usual in algebra and formal calculus (integers, floats, integers modulo n, polynomials, matrices); limitations posed by the machine (optimisation in space and time, notion of algorithmic complexity), and fields of application of these theories (error-correcting codes, cryptography, information processing and data compression, geometry, etc.).

The classical algorithms (fast exponentiation, extended Euclid, Hörner scheme, Gauss, modular methods, primality tests, etc.) are presented in class, and are then approached on the computer with the help of SageMath software. It is also an opportunity to become familiar with this software.

Each student is required to present one or more oral lessons, in accordance with the agrégation, on texts from previous years.

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The oral exams in algebra/geometry and in analysis/probability of the agrégation competition include a "development" part where the candidate has to present for about fifteen minutes a point of the plan which is particularly interesting to him/her. In this UE, we propose to write a document that will compile the different developments prepared for these oral exams.

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This course is a continuation of the optimization course of the second semester of L3.

After a reminder of the results and numerical methods for optimization problems of order one and two, without constraint and under equality and inequality constraints, the course focuses on issues of interest today in industrial optimization, and in particular, robust, multicriteria optimization, in the presence of uncertainties.

The course then illustrates the place of optimization in the main machine learning algorithms. These issues are illustrated by examples of classification and regression problems in supervised learning. These examples are the occasion to discuss the issues of metrics and procedures for the evaluation of learning, validation and inference (crossfold, overfitting, etc).

The course presents the different classes of learning: unsupervised, supervised, transfer, reinforcement, incremental, etc.

Issues around database management are addressed: generation, imputation, visualisation, slicing.

The course presents the links between transfer learning and numerical simulation to address issues of synthetic database generation, imputation, non-intrusive prediction, rapid inference, etc.

The course includes an important part of computer projects along the way. All the sessions take place in a computerised environment and allow an immediate implementation of the theoretical elements.

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Statistical data are becoming increasingly massive. Before modelling them, it is essential to explore them and to reduce their dimension while losing as little information as possible. This is the objective of this course in multidimensional exploratory statistics. On the methodological level, the tools it uses are essentially those of Euclidean geometry. The statistical problems and notions will thus be translated into the language of Euclidean geometry before being treated in this framework. The two families of exploratory methods that will be seen in this course are:

1) automatic classification methods, which group observations into classes and reduce their disparity to the disparity between these classes;

2) component analysis methods, which look for the main directions of disparity between observations and provide interpretable images of this disparity in reduced dimensions.

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Statistical modelling is based on the two fundamental notions of information (which is to be extracted from the data) and decision (which is to be taken in view of these data). This course introduces the theoretical formalisation of these two notions. It is therefore logically placed at the beginning of the curriculum, as many other courses use its notions and results later on.

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This course focuses on discovering good coding practices for a professional level.
The language used is Python, but some elements of bash and git will also be useful.
There will be a special focus on data processing and visualization at the heart of the course.
We will focus mainly on basic programming concepts, as well as on the discovery of Python's scientific libraries, including "numpy, scipy, pandas, matplotlib, seaborn".
Beyond the knowledge of these fundamental packages, we will introduce modern code practices: (unit) testing, version control (git), automatic documentation generation, etc.

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The importance of statistical science in the process of scientific discovery and industrial advancement is that it allows the formulation of inferences about phenomena of interest to which risks of error or degrees of confidence can be associated. The calculation of these risks of error is based on probability theory, but the principles and methods for associating these risks with inferences constitute a theoretical corpus that serves as a basis for all statistical methodologies.

This module is intended to be a fairly comprehensive presentation of these basic principles and of the mathematical tools, results and theorems used in inferential statistics. It develops the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families and the principle of maximum likelihood and the use of p-value.
For the implementation of certain applications, the adapted tools of the R software are presented.

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The first part of this course concerns complements of probability theory: conditional expectation, Gaussian vectors. The second part presents one of the main families of stochastic processes in discrete time, the Markov chains. These are sequences of dependent random variables, whose dependence relationship is relatively simple since each variable depends only on the previous one. It is also a very powerful modelling tool. We will study the main properties of these processes, as well as their behaviour in long time and the estimation of their parameters.

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This introductory course on time series, i.e. a series of observations made over time, constitutes an indispensable toolbox for the treatment of this type of data frequently encountered in a large number of applications: concentration of a pollutant in the air over time, glucose level in the blood over time, sales of a product in a supermarket, stock market price, etc. This course focuses on both the mathematical presentation of the concepts and the more technical aspects of the implementation of the methods. Numerical illustrations are proposed with the R software.

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Non-parametric methods are important in many statistical applications because they provide an alternative to traditional approaches that require the specification of valid statistical models. Establishing the validity of such a model is a complex undertaking.

Non-parametric methods circumvent this problem by using the transformation of data into ranks and conditioning on certain quantities from the observed pattern of these ranks. The statistics constructed in this way are independent of the law of the raw data, which allows statistical inference procedures to be constructed free of the model underlying the data. Moreover, the loss of statistical efficiency is minimal.

This course is a fairly comprehensive presentation of non-parametric methods. It follows on from a first introductory course on parametric inferential methods, adapting and developing the theory of several advanced concepts such as conditional testing, comparative power of tests (efficiency measures), the notion of "effect size". It focuses on the practical application of these methodologies by providing an overview of the main R commands and their uses.

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This R programming course is aimed at students who will need to know a programming language for advanced data processing in their professional practice. The aim is to learn how to structure, comment and debug code properly. This course is intended for both M1 SSD and M1 Bio-Info students. It is not intended to use the packages as black boxes for the implementation of statistical methods.

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Tutored projects carried out in groups under the guidance of a teacher.

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The linear model is both a simple and very rich tool, which is the basis of many statistical methods. Its mastery and good understanding are very useful both from a practical point of view, to analyze finely certain data sets, and from a conceptual point of view, to understand the theoretical bases of more advanced learning methods, including current ones.

This course proposes an introduction to the linear model of simple and multiple regression, with quantitative or qualitative variables. It presents the formal derivation and theoretical study of least squares and maximum likelihood estimators in the Gaussian case. It also provides validation and variable selection tools to study the limitations of the model. Finally, it introduces the practical use of this tool on simple data sets using the R software.

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This course introduces the mathematical modelling of the behaviour of actors seeking to optimise an individual objective in a competitive situation.

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This course provides an introduction to stochastic control. In this type of
problem, one seeks to modify the natural trajectory of a process to meet a certain
objective. We will place ourselves in the framework of discrete-time decisional Markov processes where one can
choose an action at each time step. We will see how to formalise the stochastic control problems in this framework, and how to solve them theoretically and numerically.

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The aim is to understand the nature of the interrelations between economies and to analyse the conditions for the effectiveness of economic policies in an open economy, taking into account the nature of the exchange rate regime on the one hand and the degree of openness of capital on the other.

In this perspective, the balance of payments will be presented and analysed; issues of competitiveness and attractiveness of economies will be discussed.

It will also look at the impact of financial liberalisation on exchange rate volatility and the possibilities available to different types of economic actors to hedge against exchange rate risk.

Finally, the notions of crisis will be presented (financial, exchange rate). The endogeneity of crises will be highlighted with an analysis of the last two crises: the 2008 financial crisis on the one hand and the health crisis on the other.

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English language tutorials for students in the "M1 Statistics and Data Science" programme, aimed at professional independence in the English language.

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The size of statistical data is constantly increasing, and in particular the richness of the description of the statistical units. However, classical linear statistical modelling becomes invalid in high dimension, i.e. when the number of variables exceeds the number of statistical units. This course presents the most common techniques used to regularize linear models in high dimension.

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This course presents some of the classical and modern methods for the construction of non-parametric density or regression estimators. Both theoretical and practical aspects are covered

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This course proposes an introduction to Bayesian parametric statistics. After the presentation of the Bayesian paradigm, the cases of point and set estimates will be considered and then the methodology of Bayesian model selection will be discussed. Binomial, Gaussian and linear models will be used to illustrate the previous topics.

For complex models, the problems of estimation and model selection in the Bayesian context require the use of advanced tools for approximating integrals. Therefore, the second part of the course will focus on Monte Carlo methods and Markov Chain Monte Carlo algorithms.

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English TD courses, for students in the "M2 Statistics and Data Science" programme, aiming at professional autonomy in the English language.

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This course deals with the framework of machine learning from a statistical point of view.
We will mainly focus on the supervised framework (regression and classification) and will introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond the aspects of modelling and theory, the course will also cover some elements of optimisation and implementation (sklearn, pytorch, etc.) of the methods introduced.

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This course introduces the general framework of linear models where one seeks to express a response variable in terms of a linear combination of predictors. By assuming both a specific relationship between the mean response and the predictors (link function), as well as a specific distribution of the random variation of the response around its mean, it is possible to represent binary data (e.g. presence/absence, mortality/survival) or counts (e.g. number of individuals, number of species). With this general framework, non-normally distributed variables can be modelled. In particular, the use and interpretation of logistic, binomial and Poisson regression models will be detailed.

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The lifetime of an individual in biostatistics, or of a component in reliability analysis, is a quantity whose statistical analysis differs from the usual data. On the one hand, it leads to the consideration of quantities such as the hazard function, the average residual life time, etc., which are not as relevant in other fields of statistics. On the other hand, it often involves a censoring mechanism, because the data are observed incompletely due to the length of the experiments in relation to the time to be allocated to them.

The aim of this module is to introduce the basics of survival analysis. The rationale and main mechanisms for censoring data are discussed. Two main types of statistical approaches are presented: the parametric approach, which despite its limitations, is often favoured by users, because "the parameters speak", and the non-parametric approach, which makes it possible to support and complete the parametric analyses by giving them greater flexibility and depth when the data are numerous. The module also presents various models (Cox model, accelerated failure rate, etc.) that allow survival to be linked to explanatory factors, which makes it possible to determine which factors may have an impact on survival. This information is particularly useful in a health context, as it allows for the personalisation of survival projections for an individual.

These methods will be implemented using R software.

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Many phenomena are only incompletely or indirectly observed, which complicates their analysis. Their statistical modelling must then include unobserved variables, known as latent variables, which are linked in one way or another to the observed variables. This course introduces the various ways of introducing latent variables into a model according to their type (qualitative or quantitative), and of proceeding to the estimation of the model parameters.

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Supplementary courses provide openings into more specialised areas of statistics and stochastic modelling. Their content is likely to change from year to year. Topics may include:
- biological sequence analysis: probabilistic models of biological sequence evolution, phylogeny inference, hidden Markov models for pattern detection, graphical models and inference of gene regulatory networks
- population dynamics: birth and death processes (definitions, properties, asymptotic behaviour, parameter estimation, simulation), deterministic, stochastic or hybrid approximations
- biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects, Likelihood function and applications to bio-medical data, Reminder on survival data, competitive risk models, test based on a U-statistic, Models of analysis of fertility data, Medical diagnosis and ROC curves as an application of a U-statistic, Meta-analyses.
- statistics of extremes and applications to the environment: Univariate and multivariate extreme value theory: law of maxima and high threshold violations for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk studies. Applications to environmental data: rainfall, wave heights, temperatures...
- spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course will be structured around two parts: point processes and geostatistics.
- linear mixed models : Extension of linear models to linear mixed models. Estimation of fixed effect parameters as well as variance parameters within these models. Implementation on different practical cases. Random effects in generalized linear models.

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Supplementary courses provide openings into more specialised areas of statistics and stochastic modelling. Their content is likely to change from year to year. Topics may include:
- biological sequence analysis: probabilistic models of biological sequence evolution, phylogeny inference, hidden Markov models for pattern detection, graphical models and inference of gene regulatory networks
- population dynamics: birth and death processes (definitions, properties, asymptotic behaviour, parameter estimation, simulation), deterministic, stochastic or hybrid approximations
- biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects, Likelihood function and applications to bio-medical data, Reminder on survival data, competitive risk models, test based on a U-statistic, Models of analysis of fertility data, Medical diagnosis and ROC curves as an application of a U-statistic, Meta-analyses.
- statistics of extremes and applications to the environment: Univariate and multivariate extreme value theory: law of maxima and high threshold violations for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk studies. Applications to environmental data: rainfall, wave heights, temperatures...
- spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course will be structured around two parts: point processes and geostatistics.
- linear mixed models : Extension of linear models to linear mixed models. Estimation of fixed effect parameters as well as variance parameters within these models. Implementation on different practical cases. Random effects in generalized linear models.

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4 to 6 month internship in a company, EPIC or research laboratory, supervised by a researcher, teacher-researcher, engineer or research engineer.

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The size of statistical data is constantly increasing, and in particular the richness of the description of the statistical units. However, classical linear statistical modelling becomes invalid in high dimension, i.e. when the number of variables exceeds the number of statistical units. This course presents the most common techniques used to regularize linear models in high dimension.

See full page

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English TD courses, for students in the "M2 Statistics and Data Science" programme, aiming at professional autonomy in the English language.

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This course deals with the framework of machine learning from a statistical point of view.
We will mainly focus on the supervised framework (regression and classification) and will introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond the aspects of modelling and theory, the course will also cover some elements of optimisation and implementation (sklearn, pytorch, etc.) of the methods introduced.

See full page

This course introduces the general framework of linear models where one seeks to express a response variable in terms of a linear combination of predictors. By assuming both a specific relationship between the mean response and the predictors (link function), as well as a specific distribution of the random variation of the response around its mean, it is possible to represent binary data (e.g. presence/absence, mortality/survival) or counts (e.g. number of individuals, number of species). With this general framework, non-normally distributed variables can be modelled. In particular, the use and interpretation of logistic, binomial and Poisson regression models will be detailed.

See full page

The lifetime of an individual in biostatistics, or of a component in reliability analysis, is a quantity whose statistical analysis differs from the usual data. On the one hand, it leads to the consideration of quantities such as the hazard function, the average residual life time, etc., which are not as relevant in other fields of statistics. On the other hand, it often involves a censoring mechanism, because the data are observed incompletely due to the length of the experiments in relation to the time to be allocated to them.

The aim of this module is to introduce the basics of survival analysis. The rationale and main mechanisms for censoring data are discussed. Two main types of statistical approaches are presented: the parametric approach, which despite its limitations, is often favoured by users, because "the parameters speak", and the non-parametric approach, which makes it possible to support and complete the parametric analyses by giving them greater flexibility and depth when the data are numerous. The module also presents various models (Cox model, accelerated failure rate, etc.) that allow survival to be linked to explanatory factors, which makes it possible to determine which factors may have an impact on survival. This information is particularly useful in a health context, as it allows for the personalisation of survival projections for an individual.

These methods will be implemented using R software.

See full page

Many phenomena are only incompletely or indirectly observed, which complicates their analysis. Their statistical modelling must then include unobserved variables, known as latent variables, which are linked in one way or another to the observed variables. This course introduces the various ways of introducing latent variables into a model according to their type (qualitative or quantitative), and of proceeding to the estimation of the model parameters.

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4 to 6 month internship in a company, EPIC or research laboratory, supervised by a researcher, teacher-researcher, engineer or research engineer.

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