MASTER MATHEMATICS

Partial Differential Equations (PDE) 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, 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 in simple examples the tools of scientific calculation seen in the course.

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Description : Partial Differential Equations (PDE) 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, 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 behavior is essentially based on the use of analytical results and in particular of functional analysis. This course presents the first important tools for the solution 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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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 first and second order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, and in particular, robust, multi-criteria 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 used to discuss issues of metrics and procedures for learning evaluation, validation and inference (crossfold, overfitting, etc).

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

The issues surrounding database management are addressed: generation, imputation, visualization, 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 computerized environment and allow an immediate implementation of the theoretical elements.

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This course completes the notions developed in the course Analysis of PDEs 1. 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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This 42-hour course gives the basic elements of continuum mechanics: we study the motions, deformations and stress fields within 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 from a mathematical point of view.

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Internship supervised by a researcher or teacher-researcher.

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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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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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In this course, we will present analytical methods for solving partial differential equations (PDEs) -- possibly nonlinear -- and studying the qualitative behavior of the solutions. The class of PDEs and therefore the methods studied will depend on the speaker. They will be related to the applications developed within IMAG: fluid mechanics, solid mechanics, math-bio.

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This course is a continuation of the optimization course of the second semester of L3 Mathematics and the optimization and machine learning course 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 by an error control.

This question is fundamental for a 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 computerized environment and allow an immediate implementation of the theoretical elements.

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Inverse problems are studied, emphasizing 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, regularization, 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 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, we focus on the design and analysis of Hybrid High-Order methods, which are an example of the latest generation of numerical methods. In the third part, we develop applications of these methods in connection with the research activities present at IMAG: 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 study/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), 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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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 UE develops the classical theory of modules on a principal ring, and the basics of the theory of representations of groups by concentrating on finite groups.

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Description : Partial Differential Equations (PDE) 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, 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.

See full page

The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results and in particular of functional analysis. This course presents the first important tools for the solution 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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English tutorial for students in the M1 Fundamental Math program who are aiming for professional autonomy in the English language.

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

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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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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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An 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 of M1 Fundamental Mathematics. We apply the notions 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 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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Course in algebraic topology.

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

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Course on a specialized topic in topology or geometry.

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

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The competition for the external agrégation in mathematics includes two oral exams 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 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 UE is devoted to the revision of the notions of analysis included in the competition program. 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 course 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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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 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 UE prepares students for the modelling test of the option C of the external agrégation of mathematics. This test, based on the study of texts, is centered 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 program: representation and algorithmic manipulation of mathematical objects usual 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 (fast exponentiation, extended Euclid, Hörner scheme, Gauss, modular methods, primality tests, etc.) are presented in class, and are then discussed 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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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 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 first of all of evaluations and corrections based on subjects at the level of the written exams of the agrégation. Supplements on the notions of the program are also proposed in parallel with the written tests.

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The competition for the external agrégation in mathematics includes two oral exams 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 exams, this UE first presents their specificity and then proposes to prepare the different themes announced by the jury of the competition. This UE continues the work started in the UE " Preparation to the oral 1 ". 

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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 : Markov chains. These are sequences of dependent random variables, whose dependence relation is relatively simple since each variable depends only on the previous one. It is 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 more and more massive. Before modeling 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 on multidimensional exploratory statistics. From a methodological point of view, the tools used are essentially those of Euclidean geometry. The statistical problems and notions will 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 the disparity of the observations to the disparity between these classes;

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

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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 first and second order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, and in particular, robust, multi-criteria 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 used to discuss issues of metrics and procedures for learning evaluation, validation and inference (crossfold, overfitting, etc).

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

The issues surrounding database management are addressed: generation, imputation, visualization, 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 computerized environment and allow an immediate implementation of the 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 special 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 discovering 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 one can associate risks of error or degrees of confidence. 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 complete presentation of these basic principles and of the tools, results and mathematical 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 will be presented.

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Statistical modeling is based on the two fundamental notions of information (which must be extracted from the data) and decision (which must be taken in view of these data). This course introduces the theoretical formalization 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 is an introduction to stochastic control. In this type of
problem, we try 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 we can
choose an action at each time step. We will see how to formalize the 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 large number of applications: concentration of a pollutant in the air over time, blood glucose levels over time, sales of a product in a department store, stock market prices, 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 allow to get away from classical approaches which require the specification of valid statistical models. However, 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 by 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 allows the construction of statistical inference procedures 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 builds on a first course on parametric inferential methods by 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 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 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 basis 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 gives validation and variable selection tools to study the limits of the model. 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 direction of a teacher.

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English TD courses, for students in the "M1 Statistics and Data Science" program, aiming 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 openness of capital on the other.

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

It will also be a question of understanding the impacts of financial liberalization on the volatility of foreign exchange and the possibilities offered 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 financial crisis of 2008 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 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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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 introduces the general framework of linear models where we try 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 counts (e.g. number of individuals, number of species). Thanks to this general framework, we can then model non-normally distributed variables. The use and interpretation of logistic, binomial and Poisson regression models will be discussed in detail.

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English tutorials 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 Bayesian parametric statistics. After the presentation of the Bayesian paradigm, the cases of point and set estimates will be considered and 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 integral approximation tools. 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 is constantly increasing, and in particular the richness of the description of the statistical units. However, classical linear statistical modeling 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 deals with the machine learning framework from a statistical point of view.
We will mainly focus on the supervised framework (regression and classification) and introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond the modeling and theory aspects, the course will also cover some elements of optimization and implementation (sklearn, pytorch, etc.) of the introduced methods.

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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 interesting 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 compared to the time one wants to allocate to them.

The purpose of this module is to introduce the basics of survival analysis. The rationale and main mechanisms of data censoring 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" and the non-parametric approach which allows to support and complete the parametric analyses by giving them more flexibility and depth when the data are numerous. The module also presents different models (Cox model, accelerated failure rate, etc.) allowing to link survival to explanatory factors, which makes it possible to determine those which can impact this survival. This information is particularly useful in a health context, as it allows the personalization of survival projections for an individual.

The implementation of these methods will be done on the R software.

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Complementary courses present openings to more specialized areas of statistics and stochastic modeling. Their content may change from year to year. The topics covered may be the following
- 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 bio-medical data, Reminders on survival data, competitive risk models, test based on a U-statistic, 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...
- 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.
- 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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Complementary courses present openings to more specialized areas of statistics and stochastic modeling. Their content may change from year to year. The topics covered may be the following
- 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 bio-medical data, Reminders on survival data, competitive risk models, test based on a U-statistic, 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...
- 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.
- 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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Many phenomena are only incompletely or indirectly observed, which complicates their analysis. Their statistical modeling must then include unobserved variables, called 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 the parameters of the model.

See full page

This course introduces the general framework of linear models where we try 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 counts (e.g. number of individuals, number of species). Thanks to this general framework, we can then model non-normally distributed variables. The use and interpretation of logistic, binomial and Poisson regression models will be discussed in detail.

See full page

English tutorials 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 in particular the richness of the description of the statistical units. However, classical linear statistical modeling 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

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

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 interesting 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 compared to the time one wants to allocate to them.

The purpose of this module is to introduce the basics of survival analysis. The rationale and main mechanisms of data censoring 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" and the non-parametric approach which allows to support and complete the parametric analyses by giving them more flexibility and depth when the data are numerous. The module also presents different models (Cox model, accelerated failure rate, etc.) allowing to link survival to explanatory factors, which makes it possible to determine those which can impact this survival. This information is particularly useful in a health context, as it allows the personalization of survival projections for an individual.

The implementation of these methods will be done on the R software.

See full page

4 to 6 month internship in a company, EPIC or research laboratory, supervised by a researcher, teacher-researcher, engineer 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 then include unobserved variables, called 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 the parameters of the model.

See full page

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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 : Markov chains. These are sequences of dependent random variables, whose dependence relation is relatively simple since each variable depends only on the previous one. It is 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 bioscience, ranging from advanced 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 robustness and optimality of biological systems, gene regulation and the fundamental principles underlying membrane and genome organization.

The main topics will be introduced first 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 literature reviews, use of existing code, or development of new code (depending on the student's experience) and will constitute half of the final assessment.

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

A fundamental point addressed is also the quantification of phenomena, their physical interpretation and their physico-mathematical modeling. The course opens to the philosophy and to the set of themes of this master's course centered on the study of the physical principles of the organization and the dynamics of the living and complex matter.

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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 one can associate risks of error or degrees of confidence. 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 complete presentation of these basic principles and of the tools, results and mathematical 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 will be presented.

See full page

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