KTH Mathematics  

Mathematical Statistics

The aim of the course is to introduce basic theories and methods in time series analysis and apply them to real life time series to detect trends, remove seasonal components and estimate statistical characteristics. Techniques developed in time series analysis are important in environmental studies, biology, financial analysis, signal processing, econometrics, etc.


  • SF1901 or equivalent course
  • SF2940 Probability Theory or equivalent course recommended


There will be a written examination on Monday December 13, 2010, 14.00-19.00 hrs. Registration for the written examination via "Mina Sidor"/"My Pages" is required.
Allowed means of assistance for the exam are a calculator (but not the manual for it!) and the "Formulas and survey" from Course litterature below. Each student must bring her/his own calculator to the examination. The department will distribute the "Formulas and survey" and it is not allowed to use your own copy. Grades are set according to the quality of the written examination. Grades are given in the range A-F, where A is the best and F means failed. Fx means that you have the right to a complementary examination (to reach the grade E). The criteria for Fx is a grade F on the exam, and that an isolated part of the course can be identified where you have shown a particular lack of knowledge and that the examination after a complementary examination on this part can be given the grade E.

Hand-in assignments:
There will be mandatory set of hand-in assignments. These are applications to real data of the methods treated during the course and some simulations. They are handed out during one of the lectures at the beginning of the course or available here and are solved in groups of three students. Data files and m-files for the assignments are found here.
There is a deadline for each of the assignments. Students who do not hand in an assignment on time are obliged to solve additional problems.
All home assignments including the additional problems if any, must be handed in no later than the 19th of January 2011, otherwise the whole set of assignments must be redone during the autumn 2011.

Course literature.:
(1) BD P.J.Brockwell and R.A. Davis: Introduction to Time Series and Forecasting. Springer-Verlag. The book can be found at the Kårbokhandeln at the adress Osquars Backe 21 at the campus.
The following sections in the book are planned to be covered: 1.2-1.4, 1.5.1-1.5.2, 2.1-2.6, 3.1-3.3, 4.1-4.4, 5.1.1, 5.1.4, 5.2-5.5, 6.1, 6.3, 7.1-7.2, 7.4-7.5, 7.7, 8.1-8.5, 10.3.5.
(2) Jan Grandell: Lecture notes, Time series analysis. Contains theory that complements the textbook. Not mandatory reading. Can be purchased at "Studentexpeditionen", Lindstedtsvägen 25.
(3) Jan Grandell: Formulas and survey, Time series analysis.

Preliminary plan (TR = Tobias Rydén) 

Day Date Time Hall Topic Lecturer
Tu 26/10 10-12 H1
Sections 1.2-1.4 in BD, stationary models, autocovariance function (ACVF), weak stationarity, AR(1), MA(1). TR
Th 28/10
15-17 D3 Sections 1.5.1-1.5.2 in BD: time series decomposition, trend, seasonal component, shift operator, difference operator. TR
13-15 D3 Section 2.1-2.2: non-negative definiteness of ACVF, strictly stationary models, Gaussian time series, linear process, conditions for convergence in mean square. Causal linear processes, AR(1), MA(1). TR
1/11 13-15 E3 Section 2.3: ARMA(1,1) processes, autocovariance function and invertibility, system polynomials. Section 2.4: Estimation of the mean of a stationary process. Introduction to prediction (Section 2.5). TR
4/11 15-17 D3 Introduction to prediction, the Durbin-Levinson algorithm. (Section 2.5). TR
5/11 13-15 D3 The innovations algorithm for prediction, examples (Section 2.5). TR
10/11 8-10 H1 ARMA(p,q) processes, computing their ACVF (Sections 3.1-3.2) TR
11/11 15-17 D3 The PACF for ARMA(p,q) processes (Section 3.2) TR
12/11 13-15 D3 Forecasting ARMA processes (Section 3.3). TR
17/11 8-12 D3 h-step prediction for ARMA processes (Section 3.3), Spectral density, spectral distribution and spectral representation (Section 4.1) TR
18/11 15-17 D3 The periodogram, spectral estimation (Section 4.2) TR
19/11 15-17 D3 Time-invariant linear filters, transfer functions, spectral density of ARMA processes (Sections 4.3-4.4); Yule-Walker estimation of AR(p) processes (Section 5.1.1) TR
23/11 10-12 H1
The Hannan-Rissanen algorithm for ARMA(p,q) processes (Section 5.1.4), ML estimation of ARMA(p,q) processes (Section 5.2) TR
Th 25/11 15-17 D3
Model diagnosis, forecasting, order selection (Sections 5.3-5.5) TR
26/11 13-15 D3 ARIMA and unit root models (Sections 6.1, 6.3) TR
30/11 10-12 H1
Multivariate time series: second order and spectral properties, multivariate ARMA processes (Sections 7.1-7.2, 7.4) TR
Fr 3/12 13-15 D3
Linear prediction of random vectors (Section 7.5), Cointegration (Section 7.7) TR
7/12 10-12 H1
Stochastic volatility and GARCH processes (Section 10.3.5) TR
Th 9/12 15-17 D3
State-space models, state-space representations of ARIMA models (Sections 8.1-8.3) TR
10/12 13-15 D3 Kalman filtering (Section 8.4), estimation of state-space models (Section 8.5) TR

Welcome, and hope you will enjoy the course!

Tobias Rydén

To course web page

Published by: Mårten Marcus
Updated: 16/10-2009