Presentation Transcript

1. Introduction to Time Series Analysis InKwan Yu

2. Time Series? • A set of observations indexed by time t • Discrete and continuous time series

3. Stationary Time Series • (Weakly) stationary • The covariance is independent of t for each h • The mean is independent of t

4. Why Stationary Time Series? • Stationary time series have the best linear predictor. • Nonstationary time series models are usually slower to implement for prediction.

5. Converting Nonstationary Time Series to Stationary Time Series • Remove deterministic factors • Trends • Polynomial regression fitting • Exponential smoothing • Moving average smoothing • Differencing (B is a back shift operator)

6. Converting Nonstationary Time Series to Stationary Time Series • Remove deterministic factors • Seasonality (usually combined with trends removal) • Differencing

7. Converting Nonstationary Time Series to Stationary Time Series • Example

8. Converting Nonstationary Time Series to Stationary Time Series

9. Converting Nonstationary Time Series to Stationary Time Series • After conversion, remaining data points are called residuals • If residuals are IID, then no more analysis is necessary since its mean value will be the best predictor

10. Wold Decomposition • Stationary time series can be represented as the following

11. Pn is a prediction function of Xn+h with forward lag h from Xn. The prediction error is measured in the minimum mean square Stationary Time Series Prediction

12. Stationary Time Series Prediction • Since S is a quadratic function, the minimum value will be obtained when all the partial derivatives are 0.

13. Stationary Time Series Prediction • In another form

14. Stationary Models • AR (AutoRegressive) • AR’s predictor

15. Stationary Models • ARMA • Reduces large autocovariance functions • A transformed linear predictor is used

16. Other Models • Mutivariate Cointegration • ARIMA • SARIMA • FARIMA • GARCH

17. References • Introduction to Time Series and Forecasting 2nd ed., P. Brockwell and R. Davis, Springer Verlag • Adaptive Filter Theory 4th ed., Simon Haykin, Prentice Hall • Time Series Analysis, James Douglas Hamilton, Princeton University Press