Stationary Stochastic Processes
Stationära stokastiska processer

FMSF10F, 7.5 credits

Valid from: Autumn 2020
Decided by: Anders Gustafsson / FUN
Date of establishment: 2020-05-18

General Information

Division: Mathematical Statistics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMSF10
Teaching language: English

Aim

The student shall aquire a toolbox containing concepts and models for description and handling of stationary stochastic processes within many different areas, such as, signal processing, automatic control, information theory, economics, biology, chemistry, and medicine. The mathematical and statistical elements are therefore illustrated using a wide variety of examples from different areas of application. The course shall also give the student the ability to identify the presence of stationary processes in other courses in the education, use the knowledge of stationary processes in other courses, and translate the concepts and tools between different courses, building on stationary processes.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to perform calculations using expectations, variance, covariance, and cross-covariance within and between different stationary processes,
• be able to calculate the relationship between covariance properties in the timedomain and spectral properties in the frequency domain for one and several processes,
• be able to formulate linear filters using covariance and spectral properties,
• be able to estimate covariance function, spectrum, and other parameters in stationary processes using data.

Competences and Skills

For a passing grade the doctoral student must

• be able to identify natural situations where a stationary process is a suitable mathematical model, e.g., within at least one engineering, science, or economics application,
• be able to formulate a stationary stochastic process model using a concrete problem within the chosen application,
• be able to suggest model parameters, with the help of data,
• be able to interpret the model and translate model concepts to a conclusion regarding the original problem.

Judgement and Approach

For a passing grade the doctoral student must

• be able to read and interpret technical literature with elements of stationary processes within the chosen application,
• be able to describe the model structure and the conclusions,
• be able to describe the possibilities and limitations of stochastic models.

Course Contents

Models for stochastic dependence. Concepts of description of stationary stochastic processes in the time domain: expectation, covariance, and cross-covariance functions. Concepts of description of stationary stochastic processes in the frequency domain: power spectrum, cross spectrum. Special processes: Gaussian process, Wiener process, white noise, Gaussian fields in time and space. Stochastic processes in linear filters: relationships between in- and out-signals, auto regression and moving average (AR, MA, ARMA), derivation and integration of stochastic processes. The basics in statistical signal processing: estimation of expectations, covariance function, and spectrum. Application of linear filters: frequency analysis and optimal filters.

Course Literature

Lindgren, G., Rootzén, H., Sandsten & M.: Introduction to Stationary Stochastic Processes: Applications in Science and Engineering.. Chapman & Hall, 2013. ISBN 9781466586185.

Instruction Details

Types of instruction: Lectures, laboratory exercises, exercises

Examination Details

Examination format: Written exam
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: A basic course in mathematical statistics and knowledge in complex and linear analysis.

Course Occasion Information

Contact and Other Information

Course coordinator: Maria Sandsten <maria.sandsten@matstat.lu.se>
Web page: http://www.maths.lu.se/kurshemsida/fmsf10masc04/