MASTER MATHEMATICS

Partial differential equations (PDEs) are nowadays an essential mathematical tool in the study and understanding of physical and biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need for numerical resolution methods.

This course is dedicated to the introduction of PDEs, then to their resolution 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 practical exercises will illustrate the scientific computing tools learnt in class with simple examples.

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Description : Partial differential equations (PDEs) are nowadays an essential mathematical tool for studying and understanding physical and 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, then to their resolution 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 practical exercises will illustrate the scientific computing tools seen in class with 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, and in particular functional analysis. This course introduces the first important tools for solving PDEs from analytical or geometrical points of view. These tools will be put to use in the study of a few examples of PDEs representative of major classes of equations.

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

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

After a review of results and numerical methods for first- and second-order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, in particular robust, multicriteria optimization in the presence of uncertainty.

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

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

Database management issues are addressed: generation, imputation, visualization, slicing.

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

The course includes a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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This course complements the concepts developed in PDE Analysis 1, and provides an opportunity to study in depth certain linear PDEs posed on an open space 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 introduces the basics of continuum mechanics: we study motions, deformations and stress fields in media that we consider from a macroscopic point of view, as opposed to a corpuscular description. More precisely, we analyze these physical phenomena by describing them mathematically.

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Internship supervised by a researcher or teacher-researcher.

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

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This course will present analytical methods for solving partial differential equations (PDEs) -- possibly nonlinear -- and studying the qualitative behavior of solutions. 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 in the second semester of L3 Mathematics and the optimization and machine learning course in M1 MANU. The course builds on the ingredients given in the other MANU master's modules in PDE analysis and numerical simulation.

After a reminder of the results and numerical methods for numerical simulation of PDEs on adaptive meshes, a posteriori error estimation results, and supervised learning methods from M1, the course looks at the generation of quality databases and their completion and certification thanks to numerical simulation certified by error checking.

This question is fundamental to 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 a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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

Inverse problems are first presented in finite dimension (notion of conditioning, singular values, etc.), then in infinite dimension (problem stability, regularization, pseudo-inverse, etc.).

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

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

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This course covers advanced numerical methods for partial differential equations using polyhedral meshes. The first part of the course is devoted to analysis tools of general interest. The second part focuses on the design and analysis of Hybrid High-Order methods, an example of the latest generation of numerical methods. In the third part, applications of these methods are developed in relation to IMAG's research activities in fluid mechanics, solid mechanics and flows in porous media.

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

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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) and non-regression tools (test managers).

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This course introduces students to real-life problems involving the theoretical or numerical solution of partial differential equations.

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In this UE 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 UE develops the classical theory of modules over a principal ring, and the basics of group representation theory, focusing on finite groups.

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Description : Partial differential equations (PDEs) are nowadays an essential mathematical tool for studying and understanding physical and 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, then to their resolution 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 practical exercises will illustrate the scientific computing tools seen in class with simple examples.

See 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, and in particular functional analysis. This course introduces the first important tools for solving PDEs from analytical or geometrical points of view. These tools will be put to use in the study of a few examples of PDEs representative of major classes of equations.

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

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English TD course, for students in the M1 Fundamental Maths program who are aiming for professional autonomy in English.

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

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

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

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

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This course complements the differential geometry course in M1 Fundamental Mathematics. It applies the concepts of this 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 dissertation of around 30 pages, which they defend orally before a jury. Students may 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 an M2-level mathematical result in front of the other students in the course.

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Algebraic geometry course.

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Courses on a specialized subject in algebra or geometry.

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Internship supervised by a researcher or teacher-researcher.

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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 competitive examination program. It comprises 36 hours of lectures and 36 hours of tutorials. A large part of the tutorials will be reserved for students from the preparatory course to present either corrections to exercises or important demonstrations from the course.

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The agrégation extern de mathématiques competitive examination includes two oral tests for eligible candidates. These take place in June or July. The first test is entitled "Algebra and Geometry" and the second "Analysis and Probability". In preparation for these tests, this UE first presents their specific features, and then proposes preparation for the various themes announced by the competition jury.

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This course is devoted to reviewing the analysis concepts included in the competitive examination program. It comprises 36 hours of lectures and 36 hours of tutorials. A large proportion of the tutorials will be devoted to presentations by students of the preparatory course, either to correct exercises or to demonstrate key points of the course.

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

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The oral exams in algebra/geometry and analysis/probability for the agrégation competitive examination include a "development" section, in which the candidate is required to present for around 15 minutes a point in the plan of particular interest to him or her. In this UE, we propose to write a document compiling the various developments prepared for these oral exams.

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This UE prepares students for the modeling test in option C of the agrégation externe in mathematics. This exam, based on the study of texts, focuses on the links between algebra and formal calculus. SageMaths software is used for this preparation.

The issues addressed are those of the Option C program: representation and algorithmic manipulation of mathematical objects common in algebra and formal calculus (integers, floats, integers modulo n, polynomials, matrices); limitations posed by the machine (optimization 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.).

Classical algorithms (rapid exponentiation, extended Euclid, Hörner's scheme, Gauss, modular methods, primality tests, etc.) are introduced in class, and are then worked out on the computer using SageMath software. This is also an opportunity to familiarize yourself with this software.

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

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

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The external agrégation mathematics competitive examination contains two 6-hour written tests, held in the spring. The first is entitled "General Mathematics" and the second "Analysis and Probability". To prepare for these exams, this UE consists firstly of assessments and corrections based on subjects at the level of the agrégation written exams. Complementary work on program concepts is also offered in parallel with the written tests.

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The agrégation extern de mathématiques competitive examination includes two oral tests for eligible candidates. These 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 specific features and then proposes preparation for the various themes announced by the competition jury. This UE continues the work begun in UE "Préparation à l'oral 1". 

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The first part of this course deals with complements to probability theory: conditional expectation, Gaussian vectors. The second part presents 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 preceding one. They are also a very powerful modeling tool. We will study the main properties of these processes, as well as their behavior in long time and the estimation of their parameters.

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Statistical data are becoming ever more massive. Before modeling them, it is essential to explore them and reduce their dimensions, losing as little information as possible. This is the aim of this course in multidimensional exploratory statistics. Methodologically, the tools used are essentially those of Euclidean geometry. Statistical problems and notions will therefore be translated into the language of Euclidean geometry before being treated in this framework. The two families of exploratory methods covered in this course are:

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

2) component analysis methods, which search for the main directions of disparity between observations and provide images of this disparity that can be interpreted in reduced dimensions.

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

After a review of results and numerical methods for first- and second-order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, in particular robust, multicriteria optimization in the presence of uncertainty.

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

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

Database management issues are addressed: generation, imputation, visualization, slicing.

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

The course includes a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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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.
A particular emphasis will be placed on data processing and visualization at the heart of the course.
We will focus mainly on basic programming concepts, as well as the discovery of Python's scientific libraries, including "numpy, scipy, pandas, matplotlib, seaborn".
Beyond knowledge of these fundamental packages, we will introduce modern practices for code: (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 enables the formulation of inferences concerning 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 the basis for all statistical methodologies.

This module is intended to provide a fairly comprehensive presentation of these basic principles and of the mathematical tools, results and theorems used in inferential statistics. It covers the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families, the maximum likelihood principle and the use of p-value.
To implement certain applications, we will present the tools adapted from R software.

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Statistical modeling is based on the two fundamental notions of information (extracted from the data) and decision (made on the basis of the data). This course introduces the theoretical formalization of these two notions. It is therefore logically placed at the beginning of the curriculum, with many other courses using its notions 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 achieve a certain
objective. We will consider discrete-time decisional Markov 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 series of observations made over time, is an indispensable toolbox for processing this type of data, which is frequently encountered in a wide range of applications: the concentration of a pollutant in the air over time, blood glucose levels over time, sales of a product in a supermarket, stock market prices, etc. This course covers both the mathematical presentation of the concepts and the more technical aspects of implementing the methods. The course covers both the mathematical presentation of the concepts and the more technical aspects of implementing the methods. Numerical illustrations are provided using R software.

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Non-parametric methods are important in many statistical applications, as they allow us to move away from conventional 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 transforming 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, enabling the construction of statistical inference procedures free from 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) and the notion of "effect size". It focuses on the practical application of these methodologies, with an overview of the main R commands and their uses.

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The linear model is both a simple and a very rich tool, forming the basis of many statistical methods. Its mastery and understanding are very useful both from a practical point of view, for fine-tuning the analysis of certain data sets, and from a conceptual point of view, for understanding the theoretical foundations of more advanced learning methods, including current ones.

This course provides 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 model limitations. Finally, it introduces the practical use of this tool on simple data sets using R software.

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

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English TD courses for students in the "M1 Statistics and Data Science" program, aimed at professional autonomy in English.

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

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The aim is to understand the nature of the interrelations 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.

From this perspective, the balance of payments will be presented and analyzed, and issues of competitiveness and economic attractiveness will be discussed.

It will also look at the impact of financial liberalization on exchange rate volatility, and the options available to different types of economic player for hedging against exchange rate risk.

Finally, the notions of crisis will be presented (financial, foreign exchange). 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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This R programming course is aimed at students who will need to use a programming language for advanced data processing. The aim is to learn how to structure, comment and debug code properly. This course is aimed at both M1 SSD and M1 Bio-Info students. It is not intended to use packages as black boxes for implementing statistical methods.

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

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This course introduces the general framework of linear models in which we 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), 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 count data (e.g. number of individuals, number of species). Thanks to this general framework, non-normally distributed variables can be modeled. In particular, the use and interpretation of logistic, binomial and Poisson regression models will be detailed.

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TD courses in English for students in the "M2 Statistics and Data Science" program, who are aiming for professional autonomy in English.

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This course provides an introduction to parametric Bayesian statistics. After a presentation of the Bayesian paradigm, the cases of point and ensemble estimation will be considered, followed by the methodology of Bayesian model selection. Binomial, Gaussian and linear models will serve as illustrations for the previous topics.

For complex models, the problems of model estimation and selection in the Bayesian context require the use of advanced integral approximation tools. The second part of the course will therefore focus on Monte Carlo methods and Markov Chain Monte Carlo algorithms.

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The size of statistical data is constantly increasing, and so is 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 dimension.

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

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The lifespan 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 residual mean life time, etc., which are of less interest in other fields of statistics. On the other hand, it often involves a censoring mechanism, as the data are observed incompletely due to the length of the experiments in relation to the time we want to allocate to them.

The aim of this module is to present the basics of survival analysis. The rationale and main mechanisms of data censoring are discussed. Two main types of statistical approach 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 supports and complements parametric analyses, giving them greater flexibility and depth when data are plentiful. The module also presents various models (Cox model, accelerated failure rate, etc.) linking survival to explanatory factors, enabling us to determine which factors may have an impact on survival. This information is particularly useful in a health context, as it can be used to personalize survival projections for an individual.

These methods will be implemented using R software.

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Complementary courses open up more specialized areas of statistics and stochastic modeling. Their content may 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 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, Reminders on survival data, competitive risk models, U-statistic-based testing, Models for fertility data analysis, Medical diagnosis and ROC curves as an application of a U-statistic, Meta-analyses.
- Extreme-value statistics and environmental applications: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, etc.
- spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course can be divided into two parts: point processes and geostatistics.
- linear mixed models : Extension of linear models to linear mixed models. Estimation of both fixed-effect and variance parameters within these models. Implementation on various practical cases. Random effects in generalized linear models.

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Complementary courses open up more specialized areas of statistics and stochastic modeling. Their content may 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 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, Reminders on survival data, competitive risk models, U-statistic-based testing, Models for fertility data analysis, Medical diagnosis and ROC curves as an application of a U-statistic, Meta-analyses.
- Extreme-value statistics and environmental applications: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, etc.
- spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course can be divided into two parts: point processes and geostatistics.
- linear mixed models : Extension of linear models to linear mixed models. Estimation of both fixed-effect and variance parameters within these models. Implementation on various 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 or research engineer.

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Many phenomena are only incompletely or indirectly observed, which complicates their analysis. Their statistical modeling must therefore 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, depending on their type (qualitative or quantitative), and of estimating model parameters.

See full page

This course introduces the general framework of linear models in which we 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), 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 count data (e.g. number of individuals, number of species). Thanks to this general framework, non-normally distributed variables can be modeled. In particular, the use and interpretation of logistic, binomial and Poisson regression models will be detailed.

See full page

TD courses in English for students in the "M2 Statistics and Data Science" program, who are aiming for professional autonomy in English.

See full page

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The size of statistical data is constantly increasing, and so is 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 dimension.

See full page

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

See full page

The lifespan 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 residual mean life time, etc., which are of less interest in other fields of statistics. On the other hand, it often involves a censoring mechanism, as the data are observed incompletely due to the length of the experiments in relation to the time we want to allocate to them.

The aim of this module is to present the basics of survival analysis. The rationale and main mechanisms of data censoring are discussed. Two main types of statistical approach 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 supports and complements parametric analyses, giving them greater flexibility and depth when data are plentiful. The module also presents various models (Cox model, accelerated failure rate, etc.) linking survival to explanatory factors, enabling us to determine which factors may have an impact on survival. This information is particularly useful in a health context, as it can be used to personalize survival projections for an individual.

These methods will be implemented using R software.

See full page

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

See full page

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Many phenomena are only incompletely or indirectly observed, which complicates their analysis. Their statistical modeling must therefore 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, depending on their type (qualitative or quantitative), and of estimating model parameters.

See full page

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The first part of this course deals with complements to probability theory: conditional expectation, Gaussian vectors. The second part presents 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 preceding one. They are also a very powerful modeling tool. We will study the main properties of these processes, as well as their behavior in long time and the estimation of their parameters.

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This EU aims to provide a broad overview of emerging quantitative interdisciplinary fields in the biosciences, ranging from cutting-edge experimental techniques in microscopy and synthetic biology, to systems approaches.

In an innovative way, these methodological aspects will be presented in the context of biological and biophysical concepts such as the robustness and optimality of biological systems, gene regulation and the fundamental principles underlying membrane and genome organization.

The main topics will first be introduced with traditional lectures and will be developed through individual or team projects where students will learn to apply specific techniques through examples, and see how these can be used to explore specific biological questions. These projects will involve bibliographical studies, the use of existing code or the development of new code (depending on the student's experience) and will make up half of the final assessment.

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The course aims to give a general introduction to cellular and molecular biology, and to put into context the use of modern physics, with its quantitative methods and approaches, to describe biological systems and their complexity from the molecular to the cellular and tissue scales.

Quantifying phenomena, their physical interpretation and physico-mathematical modeling are also fundamental aspects of the course. The course opens up to philosophy and to the whole range of themes of this Master's program, centered on the study of the physical principles of the organization and dynamics of living, complex matter.

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The importance of statistical science in the process of scientific discovery and industrial advancement is that it enables the formulation of inferences concerning 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 the basis for all statistical methodologies.

This module is intended to provide a fairly comprehensive presentation of these basic principles and of the mathematical tools, results and theorems used in inferential statistics. It covers the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families, the maximum likelihood principle and the use of p-value.
To implement certain applications, we will present the tools adapted from R software.

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