Course Syllabus for

# Mathematical Statistics, Time Series Analysis Matematisk statistik, tidsserieanalys

## FMSN45F, 7.5 credits

Valid from: Autumn 2020
Decided by: Professor Thomas Johansson
Date of establishment: 2020-05-19

## 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: FMSN45
Teaching language: English

## Aim

Practical and theoretical knowledge in modelling, estimation, validation, prediction, and interpolation of time discrete dynamical stochastic systems, mainly linear systems. The course also gives a basis for further studies of time series systems, e.g. Financial statistics and Non-linear systems.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to construct a model based on data for a concrete practical time series problem,
• be able to perform simple transformations of a non-stationary time series into a stationary time series,
• be able to predict and interpolate in linear time series models,
• be able to estimate parameters in linear time series models and validate a resulting model,
• be able to construct a Kalman-filter based on a linear state model,
• be able to estimate in time varying stochastic systems using recursive and adaptive techniques.

Competences and Skills

For a passing grade the doctoral student must be able to present the analysis of a practical problem in a written report and present it orally.

## Course Contents

Time series analysis concerns the mathematical modelling of time varying phenomena, e.g., ocean waves, water levels in lakes and rivers, demand for electrical power, radar signals, muscular reactions, ECG-signals, or option prices at the stock market. The structure of the model is chosen both with regard to the physical knowledge of the process, as well as using observed data. Central problems are the properties of different models and their prediction ability, estimation of the model parameters, and the model's ability to accurately describe the data. Consideration must be given to both the need for fast calculations and to the presence of measurement errors. The course gives a comprehensive presentation of stochastic models and methods in time series analysis. Time series problems appear in many subjects and knowledge from the course is used in, i.a., automatic control, signal processing, and econometrics. Further studies of ARMA-processes. Non-stationary models, slowly decreasing dependence. Transformations. Optimal prediction and reconstruction of processes. State representation, principle of orthogonality, and Kalman filtering. Parameter estimation: Least squares and Maximum likelihood methods as well as recursive and adaptive variants. Non-parametric methods,covariance estimation, spectral estimation. An orientation on robust methods and detection of outliers.

## Course Literature

Jakobsson, A.: An Introduction to Time Series Modeling. Studentlitteratur, 2019.

## Instruction Details

Types of instruction: Lectures, laboratory exercises, exercises

## Examination Details

Examination formats: Written exam, written report, seminars given by participants