MASTER'S DEGREE IN MATHEMATICS

Partial differential equations (PDEs) are now an essential mathematical tool for studying and understanding physical and biological phenomena. Their extreme complexity often makes them impossible to solve analytically, hence the need to use numerical solution methods.

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

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Description: Partial differential equations (PDEs) are now an essential mathematical tool for studying and understanding physical and biological phenomena. Their extreme complexity often makes them impossible to solve analytically, hence the need to use numerical solution methods.

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

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The construction of solutions to partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results, particularly functional analysis. This course presents important initial tools for solving PDEs from analytical or geometric perspectives. These tools will be applied in the study of several examples of PDEs representative of large classes of equations.

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

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

After reviewing the results and numerical methods for first- and second-order optimization problems, without constraints and under equality and inequality constraints, the course focuses on issues of current interest in industrial optimization, in particular robust, multi-criteria optimization in the presence of uncertainty.

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

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

Issues surrounding database management are addressed: generation, allocation, visualization, and segmentation.

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

The course includes a significant amount of ongoing IT projects. All sessions take place in a computerized environment and allow for immediate implementation of theoretical concepts.

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This course complements the concepts developed in the EDP Analysis 1 course. In particular, it provides an opportunity to study in depth certain linear EDPs posed on an open set of Rn, such as the Dirichlet problem, the heat equation, the Schrödinger equation, and the wave equation.

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This 42-hour course provides the basics of continuum mechanics: we study movements, deformations, and stress fields within media that are considered from a macroscopic point of view, as opposed to a corpuscular description. More specifically, we analyze these physical phenomena by describing them from a mathematical point of view.

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Internship supervised by a researcher or teacher-researcher.

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Introductory course in differential geometry, focusing on the concepts of submanifolds of^Rn, vector fields, and flow.

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

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Finite elements are a widely used numerical method. This course will explain the principles of the method, provide useful equations for various problems, and give the keys to implementing the method in computer science.

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This course will present analytical methods for solving partial differential equations (PDEs) — including nonlinear ones — and studying the qualitative behavior of solutions. The class of PDEs and therefore the methods studied may depend on the instructor. They will be related to the applications developed within IMAG: fluid mechanics, solid mechanics, mathematical biology.

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This course is a continuation of the optimization course from the second semester of the third year of mathematics and the course on optimization and machine learning from the first year of the MANU master's program. The course builds on the content covered in other modules of the MANU master's program in PDE analysis and numerical simulation.

After reviewing the results and numerical methods for numerical simulation of PDEs on adaptive meshes, a posteriori error estimation results, and supervised learning methods from the M1, the course focuses on the generation of high-quality databases and their completion and certification through certified numerical simulation using error control.

This question is fundamental to the certified use of machine learning in industry. Indeed, the accuracy of mathematical learning during inference is heavily dependent on the quality of the database.

The course includes a significant amount of ongoing IT projects. All sessions take place in a computerized environment and allow for immediate implementation of theoretical concepts.

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We study inverse problems, emphasizing the concept of a "well-posed problem."

First, inverse problems are presented in finite dimensions (concepts of conditioning, singular values, etc.), then in infinite dimensions (stability of the problem, regularization, pseudo-inverse, etc.).

Next, the properties of the Fourier transform and the convolution operation are reviewed in Lp spaces and their usual subspaces. We examine how these two operations lead to well-posed or ill-posed problems. For convolution in particular, the concepts of "filter" and "deconvolution" are discussed in the context of signal and image analysis.

Finally, these concepts are applied to the study and inversion of Radon transforms, or related transforms, derived from medical imaging techniques.

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

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Minimum 4-month internship in a company, EPIC (public industrial and commercial establishment), or research laboratory, supervised by a researcher, teacher-researcher, or research/development engineer.

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

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This course allows students to familiarize themselves with real-world problems involving the theoretical or numerical solution of partial differential equations.

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In this EU, we study classical groups (linear, unitary, orthogonal, symplectic) in their algebraic aspects (reduction, conjugacy classes, etc.), geometric aspects (actions, exponential map), and topological aspects.

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

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Description: Partial differential equations (PDEs) are now an essential mathematical tool for studying and understanding physical and biological phenomena. Their extreme complexity often makes them impossible to solve analytically, hence the need to use numerical solution methods.

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

See the full page

The construction of solutions to partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results, particularly functional analysis. This course presents important initial tools for solving PDEs from analytical or geometric perspectives. These tools will be applied in the study of several examples of PDEs representative of large classes of equations.

See the full page

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

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English tutorial course for students in the M1 Fundamental Mathematics program who wish to achieve professional autonomy in English.

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Introductory course on field theory, with Galois correspondence theorem as the main result.

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This course develops Cauchy's theory for functions of a complex variable and introduces the concepts of conformal representation, fundamental group, and coverings.

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

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Introductory course in differential geometry, focusing on the concepts of submanifolds of^Rn, vector fields, and flow.

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This course complements the differential geometry course in the M1 Fundamental Mathematics program. It applies the concepts covered in that course to the study of classical Lie groups.

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Each student works on a project supervised by a researcher or teacher-researcher. Students write a thesis of approximately 30 pages, which they defend orally before a jury. It is possible to work in pairs.

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Differential geometry course.

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Algebraic topology course.

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Study and presentation of a Master's level mathematics result in front of other students on the course.

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Algebraic geometry course.

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Course on a specialized topic in algebra or geometry.

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Internship supervised by a researcher or professor.

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Course on a specialized topic in topology or geometry.

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This course is devoted to reviewing the algebra concepts included in the exam syllabus. It consists of 36 hours of lectures and 36 hours of tutorials. A large part of the tutorials will be reserved for presentations by students preparing for the exam, or for corrections of exercises or important demonstrations from the course.

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The external mathematics aggregation exam consists of two oral exams for eligible candidates. These exams take place in June or July. The first exam is entitled "Algebra and Geometry" and the second "Analysis and Probability." In order to prepare for these exams, this course first presents their specific characteristics and then offers preparation for the various topics announced by the exam jury.

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This course is devoted to reviewing the analytical concepts included in the exam syllabus. It consists of 36 hours of lectures and 36 hours of tutorials. A large part of the tutorials will be reserved for presentations by students preparing for the exam, corrections of exercises, or important demonstrations from the course.

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The external mathematics aggregation exam consists of two six-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 tests, this course consists of assessments and corrections based on topics at the level of the written aggregation tests.

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The oral exams in algebra/geometry and analysis/probability for the agrégation competitive examination include an "essay" section in which candidates must present a topic of particular interest to them for approximately 15 minutes. In this course, students are asked to write a document compiling the various essays prepared in preparation for these oral exams.

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This course prepares students for the modeling exam in Option C of the external mathematics aggregation. This exam, based on the study of texts, focuses on the links between algebra and computer algebra. The SageMaths computer algebra software is used for this preparation.

The topics covered are those of Option C: representation and algorithmic manipulation of common mathematical objects in algebra and formal calculation (integers, floats, integers modulo n, polynomials, matrices); limitations imposed by the machine (space and time optimization, the concept of algorithmic complexity), and areas of application for these theories (error-correcting codes, cryptography, information processing and data compression, geometry, etc.).

Classic algorithms (fast exponentiation, extended Euclidean algorithm, Hörner's scheme, Gauss, modular methods, primality tests, etc.) are presented in class and then explored on a computer using SageMath software. This 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 terms of the competitive examination, based on texts from previous years.

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English tutorial course for students enrolled in the "M2 Preparation for the competitive examination for the agrégation in mathematics" program who wish to achieve professional autonomy in the English language.

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The external mathematics aggregation exam consists of two six-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 tests, this course consists primarily of assessments and corrections based on topics from the written tests of the aggregation. Additional information on the concepts covered in the program is also provided alongside the written tests.

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The external mathematics aggregation exam consists of two oral exams for eligible candidates. These exams take place in June or July. The first exam is entitled "Algebra and Geometry" and the second "Analysis and Probability." In order to prepare for these exams, this course first presents their specific characteristics and then offers preparation for the various topics announced by the exam jury. This course continues the work begun in the course "Preparation for Oral Exam 1." 

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The first part of this course covers additional probability theory topics: conditional expectation, Gaussian vectors. The second part introduces one of the main families of discrete-time stochastic processes: Markov chains. These are sequences of dependent random variables, whose dependency relationship is relatively simple since each variable depends only on the previous one. They are also a very powerful modeling tool. We will study the main properties of these processes, as well as their long-term behavior and the estimation of their parameters.

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Statistical data continues to grow in volume. Before modeling it, it is essential to explore it and reduce its size while losing as little information as possible. This is the objective of this course in multidimensional exploratory statistics. Methodologically, the tools used are essentially those of Euclidean geometry. Statistical problems and concepts will therefore be translated into the language of Euclidean geometry before being addressed within this framework. The two families of exploratory methods that will be covered 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 seek out the main directions of disparity between observations and provide interpretable images of this disparity in reduced dimensions.

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

After reviewing the results and numerical methods for first- and second-order optimization problems, without constraints and under equality and inequality constraints, the course focuses on issues of current interest in industrial optimization, in particular robust, multi-criteria optimization in the presence of uncertainty.

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

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

Issues surrounding database management are addressed: generation, allocation, visualization, and segmentation.

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

The course includes a significant amount of ongoing IT projects. All sessions take place in a computerized environment and allow for immediate implementation of theoretical concepts.

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This course focuses on discovering best practices for professional-level coding.
The language used is Python, but some elements of bash and git will also be useful.
A particular emphasis will be placed on data processing and visualization at the heart of the course.
We will focus mainly on the basic concepts of programming, as well as discovering Python's scientific libraries, including "numpy, scipy, pandas, matplotlib, seaborn."
Beyond knowledge of these fundamental packages, we will introduce modern coding 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 lies in its ability to formulate inferences about phenomena of interest that can be associated with risks of error or degrees of confidence. The calculation of these risks of error is based on probability theory, but the principles and methods used to associate these risks with inferences constitute a theoretical corpus that serves as the basis for all statistical methodologies.

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

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Statistical modeling is based on two fundamental concepts: information (which must be extracted from data) and decision-making (which must be based on this data). This course introduces the theoretical formalization of these two concepts. It is therefore logically placed at the beginning of the curriculum, as many other courses use these concepts and results later on.

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This course is an introduction to stochastic control. In this type of problem
, we seek to modify the natural trajectory of a process in order to fulfill a certain objective
. We will focus on discrete-time Markov decision processes, where we can
choose an action at each time step. We will see how to formalize stochastic control problems in this framework, and how to solve them theoretically and numerically.

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This introductory course on time series, i.e., a sequence of observations made over time, provides an essential toolkit for processing this type of data, which is frequently encountered in a wide range of applications: pollutant concentration in the air over time, blood glucose levels over time, sales of a product in a supermarket, stock market prices, etc. This course focuses both on the mathematical presentation of concepts and on the more technical aspects of implementing methods. Numerical illustrations are provided using R software.

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Nonparametric methods are important in many statistical applications because they allow us to move away from traditional approaches that require the specification of valid statistical models. However, establishing the validity of such a model is a complex undertaking. 

Nonparametric methods circumvent this problem by transforming the data into ranks and conditioning on certain quantities derived from the observed configuration of these ranks. The statistics thus constructed are independent of the distribution of the raw data, which makes it possible to construct statistical inference procedures that are free of the underlying model of the data. In addition, the loss of statistical efficiency is minimal.
 
This course provides a fairly comprehensive overview of non-parametric methods. It follows on from an introductory course on parametric inferential methods, adapting and developing the theory behind several advanced concepts such as conditional tests, comparative test power (efficiency measures), and the notion of "effect size." It emphasizes the practical application of these methodologies by providing an overview of the main R commands and their uses.

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The linear model is a simple yet powerful tool that forms the basis of many statistical methods. Mastering and understanding it is very useful both from a practical standpoint, for analyzing certain data sets in detail, and from a conceptual standpoint, for understanding the theoretical foundations of more advanced learning methods, including current ones.
 
This course offers an introduction to simple and multiple linear regression models, 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 tools for validating and selecting variables in order to study the model's limitations. Finally, it introduces the practical use of this tool on simple data sets using the R software.

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

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English tutorial course for students enrolled in the "M1 Statistics and Data Science" program who wish to achieve professional autonomy in English.

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This course introduces mathematical modeling of the behavior of actors seeking to optimize an individual objective in a competitive situation.

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This course introduces mathematical modeling of the behavior of actors seeking to optimize an individual objective in a competitive situation.

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The aim will be to understand the nature of the interrelationships between economies and to analyze 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 capital openness on the other.

With this in mind, the balance of payments will be presented and analyzed; issues of competitiveness and attractiveness of economies will be discussed.

It will also be necessary to understand the impact of financial liberalization on exchange rate volatility and the options available to different types of economic actors to hedge against exchange rate risk.

Finally, the concepts of crisis will be presented (financial, currency). The endogenous nature 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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The aim will be to understand the nature of the interrelationships between economies and to analyze 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 capital openness on the other.

With this in mind, the balance of payments will be presented and analyzed; issues of competitiveness and attractiveness of economies will be discussed.

It will also be necessary to understand the impact of financial liberalization on exchange rate volatility and the options available to different types of economic actors to hedge against exchange rate risk.

Finally, the concepts of crisis will be presented (financial, currency). The endogenous nature 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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This R programming course is intended for students who will need to know a programming language for advanced data processing in their professional practice. The aim is therefore 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 for the use of packages as black boxes for the implementation of statistical methods.

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This course presents some of the classical and modern methods for constructing nonparametric density or regression estimators. Both theoretical and practical aspects are covered.

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This course introduces the general framework of linear models, which seek to express a response variable as a function of a linear combination of predictors. By assuming both a specific relationship between the mean response and the predictors (link function) and 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 count data (e.g., number of individuals, number of species). Thanks to this general framework, it is then possible to model non-normally distributed variables. The use and interpretation of logistic, binomial, and Poisson regression models will be detailed in particular.

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English tutorial course for students in the "M2 Statistics and Data Science" program who wish to achieve professional autonomy in English.

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This course offers an introduction to parametric Bayesian statistics. After presenting the Bayesian paradigm, point and ensemble estimators will be considered, followed by a discussion of Bayesian model selection methodology. Binomial, Gaussian, and linear models will be used to illustrate the above topics.

For complex models, the issues 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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The size of statistical data continues to grow, particularly in terms of the richness of the description of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions, 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 dimensions.

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This course covers the framework of machine learning from a statistical perspective.
We will focus mainly on the supervised framework (regression and classification) and introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond modeling and theory, the course will also cover some elements of optimization and implementation (sklearn, pytorch, etc.) of the methods introduced.

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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 that of usual data. On the one hand, it leads to considering quantities such as the hazard function, the mean residual lifetime, etc., which are not as interesting in other areas 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 we want to allocate to them.

The purpose of this module is to present the basics of survival analysis. The reasons for censoring data and the main mechanisms involved are discussed. Two main types of statistical approaches are presented: the parametric approach, which despite its limitations is often favored by users because "the parameters speak for themselves," and the non-parametric approach, which can reinforce and complement parametric analyses by giving them greater flexibility and depth when there is a large amount of data. The module also presents different models (Cox model, accelerated failure rate model, etc.) that link survival to explanatory factors, making it possible to determine which factors may impact survival. This information is particularly useful in a healthcare context, as it allows for the personalization of an individual's survival projections.

These methods will be implemented using R software.

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The supplementary courses provide an introduction to more specialized areas of statistics and stochastic modeling. Their content is subject to change from year to year. The topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogeny inference, Hidden Markov models for pattern detection, Graphical models and inference of gene regulation networks
     - Population dynamics: birth and death processes (definitions, properties, asymptotic behavior, parameter estimation, simulation), deterministic, stochastic, or hybrid approximations
- Biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects, likelihood function and applications to biomedical data, review of survival data, competing risks models, U-statistic-based testing, fertility data analysis models, medical diagnosis and ROC curves as an application of a U-statistic, meta-analyses.
     - Extreme statistics and applications to the environment: Theory of univariate and multivariate extreme values: law of maxima and high threshold exceedances for random variables and vectors, extremal dependencies, estimation of extreme quantiles, risk analysis. Applications for environmental data: rainfall, wave height, temperatures, etc.
    - 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 divided into two parts: point processes and geostatistics.
- Mixed linear models: Extension of linear models to mixed linear models. Estimation of fixed effect parameters such as variance within these models. Implementation in various practical cases. Random effects in generalized linear models.

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The supplementary courses provide an introduction to more specialized areas of statistics and stochastic modeling. Their content is subject to change from year to year. The topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogeny inference, Hidden Markov models for pattern detection, Graphical models and inference of gene regulation networks
     - Population dynamics: birth and death processes (definitions, properties, asymptotic behavior, parameter estimation, simulation), deterministic, stochastic, or hybrid approximations
- Biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects, likelihood function and applications to biomedical data, review of survival data, competing risks models, U-statistic-based testing, fertility data analysis models, medical diagnosis and ROC curves as an application of a U-statistic, meta-analyses.
     - Extreme statistics and applications to the environment: Theory of univariate and multivariate extreme values: law of maxima and high threshold exceedances for random variables and vectors, extremal dependencies, estimation of extreme quantiles, risk analysis. Applications for environmental data: rainfall, wave height, temperatures, etc.
    - 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 divided into two parts: point processes and geostatistics.
- Mixed linear models: Extension of linear models to mixed linear models. Estimation of fixed effect parameters such as variance within these models. Implementation in various practical cases. Random effects in generalized linear models.

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4- to 6-month internship in a company, EPIC (public industrial and commercial establishment), or research laboratory, supervised by a researcher, teacher-researcher, or research engineer.

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Many phenomena are only partially or indirectly observed, which complicates their analysis. Their statistical modeling must therefore include unobserved variables, known as latent variables, which are linked in some way 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 estimating the model parameters.

See the full page

This course introduces the general framework of linear models, which seek to express a response variable as a function of a linear combination of predictors. By assuming both a specific relationship between the mean response and the predictors (link function) and 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 count data (e.g., number of individuals, number of species). Thanks to this general framework, it is then possible to model non-normally distributed variables. The use and interpretation of logistic, binomial, and Poisson regression models will be detailed in particular.

See the full page

English tutorial course for students in the "M2 Statistics and Data Science" program who wish to achieve professional autonomy in English.

See the full page

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The size of statistical data continues to grow, particularly in terms of the richness of the description of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions, 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 dimensions.

See the full page

This course covers the framework of machine learning from a statistical perspective.
We will focus mainly on the supervised framework (regression and classification) and introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond modeling and theory, the course will also cover some elements of optimization and implementation (sklearn, pytorch, etc.) of the methods introduced.

See the full page

The lifetime of an individual in biostatistics, or of a component in reliability analysis, is a quantity whose statistical analysis differs from that of usual data. On the one hand, it leads to considering quantities such as the hazard function, the mean residual lifetime, etc., which are not as interesting in other areas 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 we want to allocate to them.

The purpose of this module is to present the basics of survival analysis. The reasons for censoring data and the main mechanisms involved are discussed. Two main types of statistical approaches are presented: the parametric approach, which despite its limitations is often favored by users because "the parameters speak for themselves," and the non-parametric approach, which can reinforce and complement parametric analyses by giving them greater flexibility and depth when there is a large amount of data. The module also presents different models (Cox model, accelerated failure rate model, etc.) that link survival to explanatory factors, making it possible to determine which factors may impact survival. This information is particularly useful in a healthcare context, as it allows for the personalization of an individual's survival projections.

These methods will be implemented using R software.

See the full page

4- to 6-month internship in a company, EPIC (public industrial and commercial establishment), or research laboratory, supervised by a researcher, teacher-researcher, or research engineer.

See the full page

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Many phenomena are only partially or indirectly observed, which complicates their analysis. Their statistical modeling must therefore include unobserved variables, known as latent variables, which are linked in some way 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 estimating the model parameters.

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