Data Science | MSc in Industrial and Applied Mathematics - Grenoble

An introduction to shape and topology optimization

Mon, 01 Jan 0001 00:00:00 +0000

Credits

3 ECTS, C. 18h

Instructors

Eric Bonnetier and Charles Dapogny

Objectives

In a very broad acceptation, shape and topology optimization is about finding the best domain (which may represent, depending on applications, a mechanical structure, a fluid channel,…) with respect to a given performance criterion (e.g. robustness, weight, etc.), under some constraints (e.g. of a geometric nature). Fostered by its impressive technological and industrial achievements, this discipline has aroused a growing enthusiasm among mathematicians, physicists and engineers since the seventies. Nowadays, problems pertaining to fields so diverse as mechanical engineering, fluid mechanics or biology, to name a few, are currently tackled with optimal design techniques, and constantly raise new, challenging issues.

The purpose of this course is to discuss the main aspects related to the numerical resolution and the practical implementation of shape and topology optimization problems, and to present state-of-the-art elements of response. It focuses as well on the needed theoretical ingredients as on the related numerical considerations. More specifically, the following issues will be addressed:

How to define a `good’ notion of derivative for a ``cost’’ function depending on the domain; How to calculate the shape derivative of a function which depends on the domain via the solution of a Partial Differential Equation posed on it;

How to devise efficient first-order algorithms (e.g. steepest-descent algorithms) based on the notion of shape derivative; How to numerically represent shapes so that it is at the same time convenient to perform Finite Element computations on them, and to deal with their evolution in the course of the optimization process.

Prerequisites

Only a basic knowledge of functional analysis and scientific computing will be assumed: differential calculus, Finite Element method, etc.

Computational biology

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Public

M2 MSIAM and 3A MMIS

Instructor

Clovis Galiez et Antoine Frenoy (18h CM + 18h CM)

Description

This interdisciplinary course is designed for students with a computational or mathematical background, providing them with the skills necessary to move into bioinformatics and computational biology. The objective is to provide an introduction to the modeling of biological phenomena, and to present advanced software and mathematical tools for the analysis of sequencing data.Through a realistic bacterial outbreak scenario, the first part of the lecture will deal with omics biology from sequence to protein structure. Through a challenging project, you will need to leverage fundamental computational biology concepts and knowledge (standard tools like blast and standard biological databases) in order to implement your own algorithmic and statistical solution.The second part of the course focuses on evolutionary biology and population biology. The central concepts of the field are presented in a computational form, with emphasis on methods for modeling and simulating the phenomena studied. Through data analyses projects and critical reading of research articles, the focus is put on modern research questions (microbiome, drug discovery, etc) involving large datasets and tools from artificial intelligence.

Requirements

Basic statistics (Poisson distribution), algorithmic (complexity), programming (python required, and R or Matlab).

Evaluation

Session one: Evaluation on project report and source code.Session two: oral examination.

Keywords

population biology, evolution, sequence analysis, algorithms for genomics, protein structures, machine learning, stochastic simulations

Data Science Seminars and Challenge

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Public

M2MSIAM (DS) et M2(TSI / SIGMA)

Instructor

Ronaldo Phlypo and Sana Louhichi

Description

This course contains two parts.

Part I concerns Data challenge.

This part consists in a real problem that is given to the students for which data are readily available. The goal is to have teams of five to six students compete in solving (at least partially) the problem.

The work is spread over the Autumn semester and consists of: building a prediction model or a methodology to solve the problem based on a set of training data, blind evaluation of the model or methodology on a test bench (unseen data, withheld from the students), using an appropriate performance measure.

At the end, the teams will present their solution path in a formal presentation and a short report.

Part II concerns Data Science seminars.

This is a cycle of seminars or presentations with a common factor that is the project of the data challenge. A first seminar will settle the context and the problem for that year’s data challenge.

The other seminars will propose different industrial or academic approaches and problems that are (loosely) related to the objective of the data challenge. Presentations have a time slot of one hour and students will have to read up front some ressources to orient their questions about the subject after the seminar.

Course Outline

Prerequisites

basic concepts on applied mathematics, probability, statistics

Evaluation:

written report (10–20 pages) 40% (discusses the problem, details the developed method(s) with a bibliography covering the state-of-the-art and situates the problem or one of the proposed approaches with respect to one of the seminars in 1–2 pages)
oral presentation 20% (discusses the problem, the proposed technical solution, and perspectives)
project utility 20% (covers a utility vote from the customer/company and⋅or a ranking score of the proposed solution)

Differential Calculus, Wavelets and Applications

Mon, 01 Jan 0001 00:00:00 +0000

Public

M2 MSIAM (DS, MSCI)

Credits

6 ECTS

Instructors

Sylvain Meignen and Kevin Polisano

Description

The course is structured in two parts, treated respectively and independently by Sylvein Meignen and Kévin Polisano. The first part is devoted to differential calculus and its applications in image restoration and edge detection. The second part is dedicated to the construction and practical use of the wavelet transform. Wavelets are basis functions widely used in a large variety of fields: signal and image processing, data compression, smoothing/denoising data, numerical schemes for partial differential equations, scientific visualization, etc. Connections between the two parts will be made on the aspects of denoising, edge detection and graph analysis.

Course outline

Part I: Differential Calculus

Differentiability on normed vector spaces
Image restoration
Edge detection

Part II: Wavelets and Applications

From Fourier to the 1D Continuous Wavelet Transform
Wavelet zoom, a local characterization of functions
The 2D Continuous Wavelet Transform
The 1D and 2D Discrete Wavelet Transform
Linear and nonlinear approximations in wavelet bases
The graph Fourier and wavelets transforms

Assessments

First session

Part I – Differential calculus: a written exam (3h) [N1]
Part II – Wavelets and application: one project and two lab sessions [N2]

Second session

The student will have the choice of retaking only one or both parts:

Part I: a new written exam (3h)
Part II: continuation of the project after feedbacks from the teacher

Details for N1:

The lab sessions are each one graded out of 2,5 points
The project is graded out of 15 points
- Choice of the article (difficulty, length, …): 1 point
- Summary and outline (in line with the targets announced): 2 points
- Report redaction (including statement of the method, novelty of the paper, …): 3 points
- Codes (from scratch or existing librairies): 4 points
- Numerical results (replication results or extended): 3 points
Interpretations of the results: 2 points

From Basic Machine Learning models to Advanced Kernel Learning

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Instructor

Julien Mairal

Syllabus

Statistical learning is about the construction and study of systems that can automatically learn from data. With the emergence of massive datasets commonly encountered today, the need for powerful machine learning is of acute importance. Examples of successful applications include effective web search, anti-spam software, computer vision, robotics, practical speech recognition, and a deeper understanding of the human genome. This course gives an introduction to this exciting field. In the first part, we will introduce basic techniques such as logistic regression, multilayer perceptrons, nearest neighbor approaches, both from a theoretical and methodological point of views. In the second part, we will focus on more advanced techniques such as kernel methods, which is a versatile tool to represent data, in combination with (un)supervised learning techniques that are agnostic to the type of data that is learned from. The learning techniques that will be covered include regression, classification, clustering and dimension reduction. We will cover both the theoretical underpinnings of kernels, as well as a series of kernels that are important in practical applications. Finally we will touch upon topics of active research, such as large-scale kernel methods and the use of kernel methods to develop theoretical foundations of deep learning models.

Assessment

Evaluation: project (1/2) + final exam (1/2)

Generative, Multimodal AI

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS - 36h

Instructors

Karteek Alahari, Xavier Alameda-Pineda, Ahlame Douzal, Eric Gaussier, Georges Quénot and Didier Schwab

Description

The course is split into two parts. During the first part, a wide range of machine learning algorithms will be discussed. The second part will focus on deep learning, and presentations more applied to the three data modalities and their combinations. The following is a non-exhaustive list of topics discussed:

Computing dot products in high dimension & Page Rank
Matrix completion/factorization (Stochastic Gradient Descent, SVD)
Monte-carlo, MCMC methods: Metropolis-Hastings and Gibbs Sampling
Unsupervised classification: Partitionning, Hierarchical, Kernel and Spectral clustering
Alignment and matching algorithms (local/global, pairwise/multiple), dynamic programming, Hungarian algorithm,…
Introduction to Deep Learning concepts, including CNN, RNN, Metric learning
Attention models: Self-attention, Transformers
Auditory data: Representation, sound source localisation and separation.
Natural language data: Representation, Seq2Seq, Word2Vec, Machine Translation, Pre-training strategies, Benchmarks and evaluation
Visual data: image and video representation, recap of traditional features, state-of-the-art neural architectures for feature extraction
Object detection and recognition, action recognition.
Multimodal learning: audio-visual data representation, multimedia retrieval.
Generative Adversarial Networks: Image-image translation, conditional generation

Assessment

Final exam

GPU Computing

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, Lectures 18h, Labs 18h

Instructor

Christophe Picard

Syllabus

In this course, we will introduce parallel programming paradigms to the students in the context of applied mathematics. The students will learn to identify the parallel pattern in numerical algorithm. The key components that the course will focus on are : efficiency, scalability, parallel pattern, comparison of parallel algorithms, operational intensity and emerging programming paradigm. Trough different lab assignments, the students will apply the concepts of efficient parallel programming using Graphic Processing Unit. In the final project, the students will have the possibility to parallelize one of their own numerical application developed in a previous course.

Introduction to parallelism
Introduction to general context of parallelism
Models of parallel programming
Description of various model of parallelism
Paradigm of parallelism
Templates of parallelism
Parallel architectures
Programming tools: Cuda

Prerequisite

C or C++, Compiling, Data structures, Architecture, Concurrency

Assessment

Project

Handling uncertainties in (large-scale) numerical models

Mon, 01 Jan 0001 00:00:00 +0000

Public

M2 MSIAM (DS, MSCI)

Credits

6 ECTS

Instructors

Elise Arnaud, Eric Blayo, Arthur Vidard, Olivier Zahm

Description

Numerical simulation is ubiquitous in today’s world. Initially confined to well-mastered physical problems, it has spread to all fields (oceanography, biology, ecology, etc.), the aim being to make forecasts of the systems under study. This has been possible thanks to the combination of numerical models and access to a considerable amount of data. However, there are many sources of uncertainty in these modelling systems. They can come from poorly known processes, approximations in the model equations and/or in their discretization, partial and uncertain data, … The objective of this course is to explore in depth the mathematical methods that have allowed these two worlds to meet. Firstly, we will focus on sensitivity analysis approaches that allow us to study the behavior of the system and its response to perturbations. In particular, this permits to study the way in which uncertainties are propagated. Next, we will look at data assimilation methods that aim at reducing said uncertainties by combining numerical models and observation data. Finally, the notions of model reduction will be discussed, which allow the implementation of the previous methods on high dimensional problems.

This course is intended for DS and MSCI students and will start with a differentiated refresher course on the necessary basic mathematical notions.

Course outline

General introduction and reminder of the basic concepts
Sensitivity analysis
- Local sensitivity analysis
- Global sensitivity analysis
Data assimilation
- Variational methods
- Stochastic methods
Model reduction
- Gaussian processes
- Polynomial Chaos

Prerequisites

Keywords

numerical model, uncertainty quantification, sensitivity analysis, data assimilation, inverse problem, meta modelisation, dimension reduction

Learning, Probabilities and Causality

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Track

M2 MSIAM (DS)

Instructors

Xavier Alameda-Pineda, Karim Assaad, Emilie Devijver, Eric Gaussier, Thomas Hueber

Objectives

The main aim of this course is to provide the principles and tools to understand and master learning models based on probabilities and causality.

Description

Causality is at the core of our vision of the world and of the way we reason. It has long been recognized as an important concept and was already mentioned in the ancient Hindu scriptures: “Cause is the effect concealed, effect is the cause revealed”. Even Democritus famously proclaimed that he would rather discover a causal relation than be the king of presumably the wealthiest empire of his time. Nowadays, causality is seen as an ideal way to explain observed phenomena and to provide tools to reason on possible outcomes of interventions and what-if experiments, which are central to counterfactual reasoning, as ‘‘what if this patient had been given this particular treatment?’’

Course Outline

Probabilistic Learning

In this part of the course we will study various probabilistic models assuming that the causality relationships between random variables are given. We will focus on unsupervised probabilistic models, from classical Gaussian mixtures to more recent variational techniques including diffusion models. A non-exhaustive list of models discussed in class is:

Gaussian mixture models
Hidden Markov models
Probabilistic principal component analysis
Linear dynamical systems (i.e. Kalman filter)
Variational autoencoders, and their dynamical counterpart.
Normalising flows and diffusion models.

Causal Learning

How to represent causal relations through structural causal graphs?
How to infer causal relations from purely observational data, from purely interventional data and from a mixture of them?
How to exploit and reason upon causal knowledge? In particular, can one quantify the relation between a cause and its effect? Can one compute the effect of an intervention? Can one use causal knowledge for counterfactual reasoning or mediation analysis? Theoretical and practical work The course will be divided into lectures and practical sessions aiming to better understand the different notions introduced. The concepts behind causality are not too difficult to grasp but nevertheless differ from traditional probability concepts.

Prerequisites

Probability and statistics background.

Keywords

Objective

Description

Selected references

Pattern Recognition and Machine Learning, by C. Bishop, 2005. [link].

An introduction to variational autoencoders, by D. P. Kingma and M. Welling [link].

The matrix cook book.

Dynamical Variational Autoencoders: A Comprehensive Review, by Laurent Girin et. al. 2021

The Book of Why: The New Science of Cause and Effect, by Pearl and Mackenzie, 2018

Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, by Pearl, 1988

Causation, Prediction, and Search, by Spirtes, Glamour and Scheines, 2000

Elements of Causal Inference: Foundations and Learning Algorithms, by Peters, Janzing and Scholkopf, 2017

Causality: Models, Reasoning and Inference, by Pearl, 2009

Mathematical Foundations of Machine Learning

Mon, 01 Jan 0001 00:00:00 +0000

Program

The program is composed of two parts of offline learning and online learning presented below.

Part I: Offline learning (taught by Massih-Reza Amini)

Supervised Learning

This part gives an overview of foundations of supervised learning. We will see that learning is an inductive process where a general rule is to be found from a finite set of labeled observations by minimizing the empirical risk of the rule over that set. The study of consistency gives conditions that, in the limit of infinite sample sizes, the minimizer of the empirical risk will lead to a value of the risk that is as good as the best attainable risk. The direct minimization of the empirical risk is not tractable as the latter is not derivative, hence learning algorithms find the parameters of the learning rule by minimizing a convex upper-bound (or surrogate) of the empirical risk. We present, classical strategies for unconstrained convex optimization: gradient descente, Quasi-Newton approach, and conjugate gradient descente. We present classical learning algorithms for binary classification: the perceptron, logistic regression and boosting by linking the development of these models to the Empirical Risk Minimization framework as well as the Multi-class classification paradigm. Particularly, we present Multi-Layer Perceptron as well as the back-propagation algorithm that is in use in deep learning.

Unsupervised and semi-supervised learning

In this part, we will present generative models for clustering as well as two powerful tools for parameter estimation namely Expectation-Maximization (EM) and Classification Expectation-Maximization (CEM) algorithms. In the context of Big Data, labeling observations for learning is a tedious task. Semiu-supervised paradigm aims at learining with few labeled and a huge amount of unlabeled data. In this part we review the three families of techniques proposed in semi-supervised learning, that is Graphical, Generative and Discriminant models.

Part II: Online learning (taught by Pierre Gaillard and Nicolas Gast)

Adversarial bandits and online learning

This part present different key paradigms that address sequential decision-making under uncertainty. Online prediction with expert advice which focuses on leveraging the wisdom of multiple experts to make predictions, where the goal is to perform nearly as well as the best expert in hindsight. This approach is crucial in scenarios where the true underlying model is unknown, and the learner must adapt to the advice of various experts over time. Online convex optimization extends this concept to more general settings, allowing for the optimization of convex functions in an online manner. Here, the learner makes decisions in a convex set and receives feedback in the form of convex loss functions, aiming to minimize regret over time. And finally, Adversarial bandits introduce a more challenging setting where the learner must explore and exploit in an environment where the rewards are chosen by an adversary. Unlike stochastic bandits, adversarial bandits do not assume any probabilistic structure in the reward generation process, making the learning task significantly more complex. The learner must employ strategies that balance exploration and exploitation effectively, even in the face of potentially malicious reward assignments. These three paradigms collectively contribute to the robustness and versatility of online learning algorithms, enabling them to tackle a wide range of real-world problems characterized by uncertainty and adversity.

Reinforcement learning

This part presents Reinforcement Learning (RL) which is a type of machine learning where an agent learns to make decisions by interacting with an environment, often modeled as a Markov Decision Process (MDP), which provides a mathematical framework describing how the agent’s actions affect the environment’s states and rewards. Classical RL algorithms, such as Q-Learning and SARSA, focus on estimating value functions or direct policy optimization to maximize cumulative rewards. These algorithms have laid the foundation for understanding and solving sequential decision-making problems. Modern RL has seen significant advancements with the integration of deep learning, giving rise to Deep Reinforcement Learning (Deep RL) algorithms like Deep Q-Networks (DQN) and Proximal Policy Optimization (PPO), which can handle high-dimensional state and action spaces. Additionally, techniques like Monte Carlo Tree Search (MCTS), popularized by AlphaGo, have further enhanced RL by enabling more sophisticated planning and decision-making. These modern approaches have expanded the applicability of RL to complex real-world problems, including game playing, robotics, and resource management, where traditional methods may struggle.

Materials

Slides, homework and passed exams of Part I

In brief

Period : Semester 9
Credits : 6 ECTS
Number of hours : 36h

Pedagogical team

References

[1] Massih-Reza Amini - Machine Learning, de la théorie à la pratique, Eyrolles (2nd edition), 2020.

[2] Christopher Bishop - Neural Networks for Pattern Recognition, Oxford University Press, 1995.

[3] Richard Duda, Peter Hart & David Strok - Pattern Classification, John Wiley & Sons, 1997.

[4] John Shawe-Taylor & Nello Cristianini - Kernel Methods for Pattern Analysis, Cambridge University Press, 2004.

[5] Colin McDiarmid - On the method of bounded differences, Surveys in Combinatorics, 141:148-188, 1989.

[6] Mehryar Mohri, Afshin Rostamzadeh & Ameet Talwalker - Foundations of Machine Learning, MIT Press, 2012.

[7] Bernhard Schölkopf & Alexander J. Smola - Learning with Kernels, MIT Press, 2002.

[8] Vladimir Kolchinskii - Rademacher penalties and structural risk minimization, IEEE Transactions on Information Theory, 47(5):1902–1914, 2001.

Mathematical optimization.

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, C. 36h

Instructor

Anatoli Iouditski

Syllabus

This course deals with

Topic 1: convex analysis
Topic 2: convex programming
Basic notions: vector space, affine space, metric, topology, symmetry groups, linear and affine hulls, interior and closure, boundary, relative interior
Convex sets: definition, invariance properties, polyhedral sets and polytopes, simplices, convex hull, inner and outer description, algebraic properties, separation, supporting hyperplanes, extreme and exposed points, recession cone, Carathéodory number, convex cones, conic hull
Convex functions: level sets, support functions, sub-gradients, quasi-convex functions, self-concordant functions
Duality: dual vector space, conic duality, polar set, Legendre transform
Optimization problems: classification, convex programs, constraints, objective, feasibility, optimality, boundedness, duality
Linear programming: Farkas lemma, alternative, duality, simplex method
Algorithms: 1-dimensional minimization, Ellipsoid method, gradient descent methods, 2nd order methods
Conic programming: barriers, Hessian metric, duality, interior-point methods, universal barriers, homogeneous cones, symmetric cones, semi-definite programming
Relaxations: rank 1 relaxations for quadratically constrained quadratic programs, Nesterovs π/2 theorem, S-lemma, Dines theorem Polynomial optimization: matrix-valued polynomials in one variable, Toeplitz and Hankel matrices, moments, SOS relaxations

Assessment

A two-hours written exam (E1) in December. For those who do not pass there will be another two-hours exam (E2) in session 2 in spring.

Modeling Seminar

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, Tut. 36h

Instructors

Christophe Picard

Objectives

This lecture proposes modelling problems. The problems can be industrial or academic. Students are faced to an industrial problem or an academic problem (research oriented). They are in charge of this project. An teacher/tutor may guide them to find solutions to the problem. For industrial project, they have to understand the user needs, to analyze and model the problem, to derive specifications, to implement a solution and to develop the communication and the presentation of the proposed solution. More academic projects are linked to the courses. They are constructed such that the students can go deeper into a subject.

This lecture introduces basic communication methods in industry. This part is in french and optional.

Rules: the students have to choose TWO subjects (either academic or industrial). They work in small groups on both projects with tutor (analysis of the problem, bibliography, construction of a solution, numerical simulations, etc.). At the end, they defend their results in front of a jury and provide a short report.

Prerequisites

No specific prerequisites.

Natural Language Processing & Information Retrieval

Mon, 01 Jan 0001 00:00:00 +0000

Overview

The automatic processing of languages, whether written or spoken, has always been an essential part of artificial intelligence. This domain has encouraged the emergence of new uses thanks to the arrival in the industrial field of many technologies from research (spell-checkers, speech synthesis, speech recognition, machine translation, …). In this course, we present the most recent advances and challenges for research. We will discuss discourse analysis whether written or spoken, text clarification, automatic speech transcription and automatic translation, in particular recent advances with multimodal large Language models.

Information access and retrieval is now ubiquitous in everyday life through search engines, recommendation systems, or technological and commercial surveillance, in many application domains either general or specific like health for instance. In this course, we will cover Information retrieval basics, information retrieval evaluation, models for information retrieval, medical information retrieval, and deep learning for multimedia indexing and retrieval.

In brief

Period: semester 9
Credits: 6 ECTS
Number of hours: 36h
Apogée: GBX9MO75

Recommended prerequisites

Basic knowledge in linear algebra, differential calculus and probabilities.

Pedagogical team

Responsibles:

Jean-Pierre Chevallet,
Didier Schwab.

Lecturers:

Laurent Besacier,
Jean-Pierre Chevallet,
Marco Dinarelli,
Emmanuelle Esperança-Rodier,
Lorraine Goeuriot,
Philippe Mulhem,
Didier Schwab
Romain Xu-Darme.

Evaluation

Final written exam, 3 hours.

Statistical learning: from parametric to nonparametric models

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Public

M2 MSIAM (DS)

Instructors

Sana Louhichi and Anatoli Juditsky

Description

This course is related to mathematical and statistical methods which are very used in supervised learning.

It contains two parts.

In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.

In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.

This course will be followed by practical sessions with the R software.

Course Outline

Introduction. Penalized linear methods for regression and classification. Non linear methods for regression. Cross Validation.

Prerequisites

basic probability statistical inference, linear model.

Keywords

High-dimension, Lasso, Ridge, Information Criteria, Mallows criterion, Cross validation, Nonparametric trend estimation, Kernel nonparametric models, Smoothing parameter selection, Average squared error, Mean average squared error, Generalized cross validation, Dependent random variables, Martingale difference sequences, Stochastic Volatility, Moment inequalities, Maximal inequalities. Supervised classification.

Stochastic Calculus and Applications to finance

Mon, 01 Jan 0001 00:00:00 +0000

Credits

3 ECTS, 18h

Instructors

Pierre Etoré

Objectives

This MSc course aims at presenting the fundamental concepts of Stochastic Calculus, and the way this concepts have been used in order to build models for applications to finance. Stochastic calculus is a theory that uses Brownian motion and Itô’s integral as basic building blocks, and Itô’s formula as a multipurpose tool, in order to describe and manipulate a rather large variety of continuous time Stochastic processes , called « continuous semimartingales » (Stochastic calculus for processes with jumps is out of the scope of this course). The theory of Stochastic calculus is largely due to the seminal work by K. Itô, that goes back to the 1940s and 1950s. This work has been rediscovered by economists (among them Myron Scholes) in the 1970s, giving rise to the famous Black-Scholes model. Since the late 1980s the link between Stochastic calculus and economics has been more and more formalized, giving rise to the field of « Mathematical Finance ».

This course requires knowledge of probability and integration theory. Some previous knowledge of Stochastic processes is welcomed. No previous knowledge of Brownian motion or Stochastic Calculus is required. The content is planned to be:

Continuous time stochastic processes, Brownian motion (definition and properties)
Continuous time martingales
Itô’s integral
Itô’s formula, Theorem of Lévy, Theorem of Girsanov
Black-Scholes model; notion of pricing and hedging
Pricing and hedging formulas, illustration of the link between Stochastic Differential Equations and Partial Differential Equations inside Black-Scholes type models.

Resources

Text course of the course: https://membres-ljk.imag.fr/Pierre.Etore/fichiers/poly_SCAF.pdf

Bibliography

“Continuous martingales and brownian motion”, D. Revuz, M. Yor
“Brownian motion and stochastic calculus” I.K. Karatzas S.E. Shreve
“Stochastic Calculus for Finance”, S.E. Shreve

Prerequisites

Statistics (Master 1 level), Probability (Master 1 level)

Temporal, spatial and extreme event analysis

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Public

M2 MSIAM (DS)

Instructors

Julien Chevallier (part I), Jean-François Coeurjolly (II) and Stéphane Girard (part III)

Description

Modelling extreme temperatures, extreme river flows, earthquakes intensities, neuronal activity, map diseases, lightning strikes, forest fires, for example is a risk modelling and assessment task, which is tackled in statistics using point processes and extreme value theory.

On the one hand, point processes are a class of stochastic processes modelling random events in interaction. By event we can think of the time a neuron activates, an earthquake occurs, the time a tweet has been retweeted, etc or the location of a tree in a forest, the impact of a lightning strike, etc. The first two parts provide an introduction to stochastic models and statistical inference which could cover such applications. Main characteristics of such processes, standard models (properties, simulation) and statistical procedures to infer them will be presented.

On the other hand, taking into account extreme events such as heavy rainfalls, floods, extreme temperatures is often crucial in the statistical approach to risk modeling. In this context, the behavior of the distribution tail is then more important than the shape of the central part of the distribution. Extreme-value theory offers a wide range of tools for modeling and estimating the probability of extreme events.

Course Outline

Part I (9 hours) - Temporal point processes

Definition and simulation of one-dimensional point processes (conditional/stochastic intensity);
Likelihood and goodness-of-fit tests (illustration on the Poisson point process);
Hawkes processes (estimation, goodness-of-fit, stationarity, ergodicity).

Part II (12 hours) - Spatial point processes

Definition and characterization of a spatial point process, intensity functions and conditional intensity functions; Poisson point process;
Intensity estimation and summary statistics;
Models for spatial point processes (Cox, determinantal and Gibbs point processes): characterization, simulation, statistical inference and validation.

Part III (15 hours) - Extreme-value analysis

Asymptotic behavior of the largest value of a sample. Extreme-value Distribution (EVD). Maximum domains of attraction (Fréchet, Weibull and Gumbel). Asymptotic behavior of excesses over a threshold. Generalized Pareto Distribution (GPD). Regularly varying functions.
Estimation of the parameters of the EVD and GPD. Hill estimator. Application to the estimation of extreme quantiles. Illustration on simulated and real data.

Prerequisites

Background on statistics and probability (master 1 level)

Keywords

stochastic processes; dependence modelling; simulation and statistical inference; Poisson point process; quantiles; excess process.