(Created 2011-09-01.)

MATHEMATICAL STATISTICS, TIME SERIES ANALYSIS | FMS051 |

**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.

*Knowledge and understanding*

For a passing grade the 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.

*Skills and abilities*

For a passing grade the student must

- be able to present the analysis of a practical problem in a written report and present it orally.

**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.

**Literature**

Madsen, H.: Time Series Analysis. Chapman & Hall (CRC texts in Statistical Sciences Series) 2007. ISBN 987-1-4200-5967-0