 # Stochastic Processes: Time Series and Gaussian Processes

## Presentation

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

## Objectives

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...