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Stochastic Processes: Time Series and Gaussian Processes


Programme (detailed contents) :

Time series

  1. Introduction and Descriptive Analysis: Time series decomposition, Estimation and Elimination of Trend and Seasonal Components
  2. Random Modeling of Time Series: Stochastic process, stationnarity, Autocovariance Function
  3. Statistical Inference of Stationary processes of order 2: Moment Estimation, Best linear predictor, Partial autocorrelation, statistical tests
  4. ARMA and ARIMA Models: AR process, MA process, ARMA et ARIMA processes


Teaching is carried out according to the Progresser En Groupe (PEG) pedagogy. Students work individually on the concepts before lectures, thanks to a handout written in problematized form, then during class they rework on these subjects by groups of 4-5. Tutorials and labworks complete this teaching. These lectures can be given in English if necessary.


Gaussian Processes


Lectures :

  • Part one: Introduction to real-valued Gaussian processes in discrete time; extension to the continuous case; parametric estimation using discrete martingale tools.
  • Part two: On the importance of the covariance function: spectral aspects and link with the regularity of the process.


Exercise classes :

  • Session on calculations for the discretized Brownian motion, expression for the joint density.
  • Session on conditional expectation for Gaussian vectors, as an orthogonal projection.
  • Session on smoothness of the Gaussian process through the smoothness of the covariance function.
  • Session on link between Gaussian processes and RKHS (Reproducing Kernel Hilbert Space)


Lab works :

  • Session simulation of Gaussian processes and its use in modeling

Session application to real data in geostat


At the end of this lecture, the student should have acquired the following skills, as well theoretically than practically with the R statistical Software.


1)    Time series

  • Estimate or eliminate the trend and/or the seasonality of a time series
  • Study the stationnarity of a time series
  • Calculate and estimate the autocorrelogram and the autocorrelograms (total and partial) of a stationary process
  • Study and/or adjust an ARMA (or ARIMA) model on a stationary time series
  • Carry an optimal linear forecast of an ARMA process

2)    Gaussian processes

  • Know the fundamental properties of Gaussian processes
  • Be able to characterize a Gaussian process through its covariance function
  • Be able to use Gaussian Processes for modeling real life situations.

Needed prerequisite

1)    Time series :


Probability and Statistics (MIC2) [I2MIMT31]

Statistics (MIC3) [I3MIMT05]

Probability and Inferential Statistics (I4MMMT21)


2)    Gaussian processes :


Advanced probabilities: martingales, stochastic algorithms and Montecarlo methods [I4MAOPPS21]

Markov chains.

Integration and probabilities.

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...


1] Time series :

  • Aragon, Yves, “Séries temporelles avec R :  Méthodes et cas”, Springer, 2011, ISBN : 978-2-8178-0207-7.
  • Brockwell, Peter J.  and Davis, Richard A.. “Introduction to Time Series and Forecasting”, Springer Texts in Statistics 2nd Edition, 2010. ISBN-13: 978-0387953519, ISBN-10: 0387953515.


2) Gausian Processes :

  • C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, ISBN 026218253X. 2006 Massachusetts Institute of Technology.


  • Solaiman, Bassel. Processus stochastiques pour l'ingénieur. PPUR presses polytechniques, 2006.
  • Jean-Christophe Breton, Processus Gaussiens, Université de Larochelle, 2006.
  • https://perso.univ-rennes1.fr/jean-christophe.breton/Fichiers/gauss_M2.pdf