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Time Series Analysis and Spatial Statistics, 7.5 Credits

Swedish name: Tidsserieanalys och spatial statistik

This syllabus is valid: 2019-08-19 and until further notice

Course code: 5MS072

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematical 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: VG Pass with distinction, G Pass, U Fail

Established by: Faculty Board of Science and Technology, 2019-09-04


The main purpose of the course is that the student should be well aquainted with the basic notions, theory, models and methods for solutions, in time series analysis and spatial statistics. The course covers models for time dependent or spatially dependent data. Such data frequently occurs  in financial (e.g. the price development of a merchandise) and scientific (e.g. metheorological observations, radar signales) applications.

The course consists of two parts.

Module 1 (6,5 hp) Theory. The module consists of the general theory of time series, stationary and non-stationary models, e.g. ARMA- and ARIMA-models, prediction of time series, spectral theory, parameter estimation, spectrum and filtration. The part also covers methods for measuring spatial dependence (variogram, covariogram), and techniques for spatial interpolation, especially kriging.

Module 2 (1 hp) Lab Assignments. The module consists of analysis of time series and spatial data using suitable software.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding
  • independently define the notions expectation, covariance function and spectral distribution for time series
  • independently define parametric ARMA-type models for the mean
  • thoroughly explain how ARMA-models can be expanded to ARIMA-, FARIMA- and ARCH-models
  • describe Kalman filtration in general terms
  • thoroughly explain how variograms and covariograms are derived
  • thoroughly explain how the (spatial) interpolation methods "The inverse distance method" and "Kriging" work
  • identify trends and seasonal variation in time series
  • derive and estimate expectation, covariance function and spectral distribution for time series, analyse their connections, and derive the uncertainty of the estimates
  • model time series with machine learning methods
  • predict the future of observed time series of different lengths, and critically evaluate the results
  • apply parametric mean models of ARMA-type, analyse the models' properties and fit real data to them
  • interpret spatial dependence using variograma and covariograms
  • predict (spatial) data using the interpolation methods "The inverse distance method" and "Kriging", and critically evaluate the results
Judgement and approach
  • clearly present the reults from time series analyses and analyses of spatial data, and evaluate the resultats from a scientific perspective

Required Knowledge

The course requires 90 ECTS including one of the following options or equivalent knowledge

- minimum 12 ECTS in Mathematical Statistiscs or
- minimum 6 ECTS in Mathematical Statistics and a course in Transform Methods minimum 7,5 ECTS or
- minimum 75 ECTS in Statistics 

In all options we also require a course in Basic Caculus minimum 7,5 ECTS. Proficiency in English equivalent to Swedish upper secondary course English A/5. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The teaching mainly consists of lectures and lessons.

Examination modes

Module 1 is examined  by a written exam. Module 2 is examined by written lab reports (U/G).
On module 1, one of the following judgements is assigned: Fail (U), Pass (G) or Pass with distinction (VG). On module 2, one of the following judgements is assigned: Fail (U) or Pass (G). For the whole course, one of the following grades is assigned: Fail (U), Pass (G) or Pass with distinction (VG). In order to receive a passing grade on the course, all parts must be completed with a passing judgement. The course grade is decided by the grade on Module 1, and is assigned only when all mandatory examination has been completed. A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade.

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 not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be adressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in 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 adressed 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

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.


Valid from: 2019 week 34

Introduction to time series and forecasting
Brockwell Peter J., Davis Richard A.
Third edition. : Cham : Springer : 2016 : xiv, 425 s. :
ISBN: 978-3-319-29852-8
Search the University Library catalogue