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Stationary stochastic Processes, 7.5 Credits

Swedish name: Stationära stokastiska processer

This syllabus is valid: 2022-07-25 and until further notice

Course code: 5MS083

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematical Statistics: Second cycle, has second-cycle course/s as entry requirements
Statistics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements

Grading scale: TH teknisk betygsskala

Established by: Faculty Board of Science and Technology, 2022-03-02


The aim of this course is that the student shall acquire 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.

The course covers models for stochastic dependence, concepts for description of stationary stochastic processes in the time domain such as expectation, covariance, and cross-covariance functions, and concepts of description of stationary stochastic processes in the frequency domain such as effect spectrum and cross spectrum. Some important types of processes are introduced: Gaussian processes, Wiener processes, white noise and Gaussian fields in time and space. The course also covers stochastic processes in linear filters: relationships between in- and out-signals, auto regression and moving average (AR, MA, ARMA), and differentiation and integration of stochastic processes. Finally, the basics in statistical signal processing are introduced, including estimation of expectations, covariance function, spectrum, and applications of linear filters: frequency analysis and optimal filters.

The course consists of two modules:
Module 1 (6 ECTS) Theory and
Module 2 (1.5 ECTS) Computer labs.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding
  • describe the structure and conclusions of the models treated
  • present the possibilities and limitations of stochastic models.
Skills and abilities
  • identify natural situations where a stationary process is a suitable mathematical model, e.g., within at least one engineering, science, or economics application
  • formulate a stationary stochastic process model using a concrete problem within the chosen application
  • determine suitable model parameters, with the help of data
  • interpret models and translate model concepts to a conclusion regarding the original problem
  • perform calculations using expectations, variance, covariance, and cross-covariance within and between different stationary processes
  • calculate the relationship between covariance properties in the time domain and spectral properties in the frequency domain for one and several processes
  • formulate linear filters using covariance and spectral properties
  • estimate covariance functions, spectra, and other parameters in stationary processes using data
  • model measurement data from nature as a simple stationary process

Judgement and approach
  • critically read and interpret technical literature with elements of stationary processes within the chosen application

Required Knowledge

The course requires a total of 90 ECTS including a course in Probability Theory on advanced level minimum 7,5 ECTS. Proficiency in English equivalent to the level required for basic eligibility for higher studies.

Form of instruction

The teaching is mainly through lectures, problem solving sessions and computer labs.

Examination modes

Module 1 is assessed by a written exam. Module 2 is assessed through written presentations of computer lab reports. On Module 1, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). On Module 2, one of the following grades is assigned: Fail (U) or Pass (G). For the whole course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). In order to receive a passing grade on the course, both modules must be completed with a passing grade.The grade is decided by the grade on Module 1. The course grade is assigned once all mandatory modules have been assessed.

Deviations from the syllabus examination form can be made for a student who has a decision on pedagogical support due to disability. Individual adaptation of the examination form shall be considered based on the student's needs. The examination form is adapted within the framework of the expected learning outcomes of the course syllabus. At the request of the student, the course coordinator, in consultation with the examiner, must promptly decide on the adapted examination form. The decision shall then be communicated to the student.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Credit transfer
All students have the right to have their previous education or equivalent, and their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be addressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university,, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.


Valid from: 2022 week 30

Stationary stochastic processes for scientists and engineers
Lindgren Georg, Rootzén Holger, Sandsten Maria
Boca Raton : CRC Press : cop. 2014 : xv, 314 s. :
ISBN: 9781466586185
Search the University Library catalogue