MMath Mathematics and Statistics

This course unit covers a variety of concepts and models useful for empirical analysis of time series data.

MATH30832 Pre-Requisites: MATH27720 and MATH27711

To develop an understanding of the probabilistic foundations of univariate time series analysis and the ability to select, fit and assess models, construct forecasts and evaluate their uncertainty and accuracy.

On successful completion of this course unit, students will be able to:

•    apply the concepts of stationarity, white noise, autocorrelation, and partial autocorrelation;
•    formulate autoregressive moving-average (ARMA) type models and their seasonal extensions, assess causality and invertibility, and derive autocorrelations and partial autocorrelations.
•    derive best linear predictors and innovations, and apply the Yule-Walker equations, Levinson-Durbin recursion and innovations algorithm;
•    apply differencing to time series data, estimate and interpret sample autocorrelations and partial autocorrelations, and conduct significance tests;
•    apply the Box-Jenkins approach to model identification, estimation and diagnostic checking, using conditional least squares, Gaussian maximum likelihood and information criteria;
•    construct one-step and multi-step forecasts, calculate forecast error variances and prediction intervals under suitable assumptions, and apply single, double and Holt-Winters exponential smoothing;
•    use R to analyse real time series, interpret fitted models and diagnostics, and compare forecast accuracy.
 

•    Time series and stationarity. Strict and weak stationarity; Gaussian processes and white noise; autocovariance and autocorrelation; linear filtering, transformations and decomposition.
•    Linear prediction and innovations. Best linear prediction, orthogonality and innovations; partial autocorrelation; Yule-Walker equations and Levinson-Durbin recursion.
•    AR MA and ARMA models. Stationary solutions, causality and invertibility; characteristic polynomials and the backshift operator; derivation and recursive computation of autocovariances and partial autocorrelations.
•    Estimation of dependence. Sample autocovariance, autocorrelation and partial autocorrelation functions; approximate significance tests and the Ljung-Box test.
•    Seasonality and nonstationarity. Seasonal ARMA models; trend removal, ordinary and seasonal differencing; ARIMA models and the random walk.
•    Model building and assessment. Box-Jenkins identification, estimation and diagnostic checking; conditional least squares; the innovations algorithm and exact Gaussian maximum likelihood; residual diagnostics, parsimony, Akaike information criterion (AIC) and Bayesian information criterion (BIC).
•    Exponential smoothing. Single and double exponential smoothing; additive and multiplicative Holt-Winters methods; connections with ARIMA modelling.
•    Prediction and comparison. Conditional expectation and minimum mean squared error prediction; recursive one-step and multi-step forecasting; forecast error variances and Gaussian prediction intervals; forecast accuracy comparisons and real-data case studies in R.
 

Two lectures and one tutorial each week. In addition students should expect to spend at least six hours each week on private study for this course unit.

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

•    Brockwell, P. J. and Davis, R. A. (2016). Introduction to Time Series and Forecasting. Core reading.
•    Priestley, M. B. (1981). Spectral Analysis and Time Series. Further reading.