Statistics and Data Science (SSD)

The first part of this course covers additional topics in probability theory: conditional expectation and 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-run behavior and the estimation of their parameters.

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Statistical data sets are becoming increasingly large. Before modeling them, it is essential to explore them and reduce their dimensionality while losing as little information as possible. This is the objective of this course on multidimensional exploratory statistics. Methodologically, the tools used in this course are primarily those of Euclidean geometry. Statistical problems and concepts will therefore be translated into the language of Euclidean geometry before being addressed within that 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 the variability among the observations to the variability between these classes;

2) principal component analysis methods, which identify the main directions of variation among the observations and enable the generation of interpretable, low-dimensional representations of this variation.

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This course is a continuation of the optimization course offered in the second semester of the third year of the bachelor’s program.

After reviewing the results and numerical methods for first- and second-order optimization problems—both unconstrained and subject to equality and inequality constraints—the course focuses on topics currently of interest in industrial optimization, particularly robust, multi-criteria optimization under uncertainty.

The course then illustrates the role of optimization in the main machine learning algorithms. These topics are illustrated with 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 (cross-validation, overfitting, etc.).

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

Topics related to database management are covered: generation, imputation, visualization, and segmentation.

The course explores the connections between transfer learning in mathematics and numerical simulation to address issues such as the generation of synthetic datasets, imputation, non-intrusive prediction, fast inference, and more.

The course includes a significant number of ongoing computer projects. All sessions take place in a computer-based environment and allow students to immediately apply the theoretical concepts they have learned.

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This course focuses on exploring best practices in professional-level coding.
The primary programming language is Python, but some knowledge of bash and git will also be useful.
A particular emphasis will be placed on data processing and visualization, which are central to the course.
We will focus primarily on the basic concepts of programming, as well as on exploring Python’s scientific libraries, including “numpy, scipy, pandas, matplotlib, and 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 enable the formulation of inferences about phenomena of interest, to which risks of error or levels of confidence can be assigned. The calculation of these error risks is based on probability theory, but the principles and methods for associating these risks with inferences constitute a theoretical framework that serves as the foundation for all statistical methodologies.

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

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Statistical modeling is based on two fundamental concepts: information (which is to be extracted from data) and decision-making (which is to be carried out based on that data). This course provides an introduction to the theoretical formalization of these two concepts. It is therefore logically placed at the beginning of the program, as many other courses subsequently draw on these concepts and results.

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This course provides an introduction to stochastic control. In this type of
problem, the goal is to modify the natural trajectory of a process to achieve a certain
objective. We will focus on discrete-time decision-making Markov processes, in which we can
choose an action at each time step. We will explore how to formalize stochastic control problems within this framework and how to solve them both theoretically and numerically.

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

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Nonparametric methods are important in many statistical applications because they allow us to move beyond 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 distribution of these ranks. The statistics constructed in this way are independent of the distribution of the raw data, which allows for the development of statistical inference procedures that are independent of the underlying model of the data. Furthermore, the loss of statistical efficiency is minimal.
 
This course provides a fairly comprehensive overview of nonparametric methods. It builds upon an introductory course on parametric inferential methods by adapting and expanding the theory behind several advanced concepts, such as conditional tests, the comparative power of tests (measures of effectiveness), and the concept 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 tool that is both simple and very powerful, forming the basis of many statistical methods. Mastering and thoroughly understanding it is very useful both from a practical standpoint—for conducting detailed analyses of certain datasets—and from a conceptual standpoint—for understanding the theoretical foundations of more advanced machine learning methods, including those currently in use.
 
This course offers an introduction to simple and multiple linear regression models, using both quantitative and qualitative variables. It presents the formal derivation and theoretical analysis of least squares and maximum likelihood estimators in the Gaussian case. It also provides tools for validating and selecting variables to explore the model’s limitations. Finally, it introduces the practical application of this tool to simple datasets using the R software.

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Supervised group projects carried out under the guidance of a teacher.

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English tutorial course for students in the “M1 Statistics and Data Science” program who are seeking professional proficiency in English.

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This course provides an introduction to the mathematical modeling of the behavior of agents seeking to optimize an individual objective in a competitive environment.

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This course provides an introduction to the mathematical modeling of the behavior of agents seeking to optimize an individual objective in a competitive environment.

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The goal will be to understand the nature of the interrelationships between economies and to analyze the conditions under which economic policies are effective in an open economy, taking into account both the nature of the exchange rate regime and the degree of capital account openness.

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

It will also involve understanding the impacts of financial liberalization on exchange rate volatility and the options available to various types of economic actors for hedging against exchange rate risk.

Finally, the concept of crises will be introduced (financial and currency crises). The endogeneity of crises will be highlighted through an analysis of the two most recent crises: the 2008 financial crisis, on the one hand, and the health crisis, on the other.

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The goal will be to understand the nature of the interrelationships between economies and to analyze the conditions under which economic policies are effective in an open economy, taking into account both the nature of the exchange rate regime and the degree of capital account openness.

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

It will also involve understanding the impacts of financial liberalization on exchange rate volatility and the options available to various types of economic actors for hedging against exchange rate risk.

Finally, the concept of crises will be introduced (financial and currency crises). The endogeneity of crises will be highlighted through an analysis of the two most recent 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, in their professional practice, will need to know a programming language to perform advanced data processing. The goal is therefore to learn how to properly structure, comment on, and debug code. This course is intended for both M1 SSD and M1 Bio-Info students. It is not intended to treat packages as “black boxes” for implementing statistical methods.

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This course presents some of the classical and modern methods for constructing nonparametric density estimators and regression estimators. It covers both theoretical and practical aspects.

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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) and a specific distribution of the random variation in the response around its mean, it is possible to model binary data (e.g., presence/absence, mortality/survival) or count data (e.g., number of individuals, number of species). Using 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 discussed in detail.

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English tutorial course for students in the “M2 Statistics and Data Science” program who are working toward professional proficiency in English.

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

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

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The volume of statistical data continues to grow, particularly the richness of the descriptions of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions—that is, 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 machine learning framework from a statistical perspective.
We will focus primarily on supervised learning (regression and classification) and introduce some elements of unsupervised learning through clustering methods.
Beyond modeling and theory, the course will also cover some aspects of optimization and implementation (sklearn, PyTorch, etc.) of the methods introduced.

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In biostatistics, the lifespan of an individual—or, in reliability analysis, the lifespan of a component—is a quantity whose statistical analysis differs from that of typical data. On the one hand, it involves considering quantities such as the random function and the mean remaining lifetime, among others, which are not as relevant in other areas of statistics. On the other hand, it often involves a censoring mechanism, since the data are incomplete due to the length of the experiments relative to the time allotted for them.

The purpose of this module is to present the basics of survival analysis. The rationale for survival analysis and the main mechanisms of data censoring are discussed. Two main types of statistical approaches are presented: the parametric approach, which, despite its limitations, is often preferred by users because “the parameters speak for themselves”; and the nonparametric approach, which reinforces and complements parametric analyses by providing greater flexibility and depth when dealing with large datasets. The module also presents various models (the Cox model, the accelerated failure rate model, etc.) that link survival to explanatory factors, thereby identifying those that 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 the 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. Topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogenetic inference, Hidden Markov models for pattern detection, Graph models and inference of gene regulatory networks
     - population dynamics: birth-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, competitive risk models, U-statistic-based tests; fertility data analysis models; medical diagnosis and ROC curves as an application of a U-statistic; meta-analyses.
     - Statistics of Extremes and Applications to the Environment: Univariate and multivariate theory of extreme values: the law of maxima and high-threshold exceedances for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk analysis. Applications to environmental data: rainfall, wave height, temperatures, etc.
    - Spatial Statistics: Introduction to the fundamentals of spatial forecasting and its applications. To cover a broad range of spatial statistics, this course will be structured around 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. Application to 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. Topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogenetic inference, Hidden Markov models for pattern detection, Graph models and inference of gene regulatory networks
     - population dynamics: birth-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, competitive risk models, U-statistic-based tests; fertility data analysis models; medical diagnosis and ROC curves as an application of a U-statistic; meta-analyses.
     - Statistics of Extremes and Applications to the Environment: Univariate and multivariate theory of extreme values: the law of maxima and high-threshold exceedances for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk analysis. Applications to environmental data: rainfall, wave height, temperatures, etc.
    - Spatial Statistics: Introduction to the fundamentals of spatial prediction and its applications. To cover a broad range of spatial statistics, this course will be structured around 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. Application to various practical cases. Random effects in generalized linear models.

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

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Many phenomena are observed only incompletely or indirectly, which complicates their analysis. Their statistical modeling must therefore include unobserved variables—known as latent variables—that are linked in some way to the observed variables. This course introduces the various ways to incorporate latent variables into a model depending on their type (qualitative or quantitative), and to estimate the model’s parameters.

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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) and a specific distribution of the random variation in the response around its mean, it is possible to model binary data (e.g., presence/absence, mortality/survival) or count data (e.g., number of individuals, number of species). Using 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 discussed in detail.

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English tutorial course for students in the “M2 Statistics and Data Science” program who are working toward professional proficiency in English.

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The volume of statistical data continues to grow, particularly the richness of the descriptions of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions—that is, 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.

View the full page

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

View the full page

In biostatistics, the lifespan of an individual—or, in reliability analysis, the lifespan of a component—is a quantity whose statistical analysis differs from that of typical data. On the one hand, it involves considering quantities such as the random function and the mean remaining lifetime, among others, which are not as relevant in other areas of statistics. On the other hand, it often involves a censoring mechanism, since the data are incomplete due to the length of the experiments relative to the time allotted for them.

The purpose of this module is to present the basics of survival analysis. The rationale for survival analysis and the main mechanisms of data censoring are discussed. Two main types of statistical approaches are presented: the parametric approach, which, despite its limitations, is often preferred by users because “the parameters speak for themselves”; and the nonparametric approach, which reinforces and complements parametric analyses by providing greater flexibility and depth when dealing with large datasets. The module also presents various models (the Cox model, the accelerated failure rate model, etc.) that link survival to explanatory factors, thereby identifying those that 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 the R software.

View the full page

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

View the full page

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Many phenomena are observed only incompletely or indirectly, which complicates their analysis. Their statistical modeling must therefore include unobserved variables—known as latent variables—that are linked in some way to the observed variables. This course introduces the various ways to incorporate latent variables into a model depending on their type (qualitative or quantitative), and to estimate the model’s parameters.

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