STAT 464/864 | Time Series Analysis and Spectrum Estimation

Autocorrelation and autocovariance, stationarity; ARIMA models; model identification and forecasting; spectral analysis. Applications to biological, physical and economic data.

The emphasis in the course will be on understanding ideas rather than speeding through the material. The following outline of topics is therefore subject to change as course progresses.

Expectations for undergraduate students and graduate students will be different, although both groups should expect to complete regular homeworks. In addition, some familiarity with coding will be required by the conclusion of the course (and tutorials will be offered to aid students in their skill development).

Course Reference Materials

Additional Materials

Lectures

Introduction to Time Series
Sept	12	Introduction, examples of time series
	13	Simple models, the autocorrelation function. Trends.
	15	Estimation of trend without seasonality. Slides
	19	Estimation of trend without seasonality, SES and differencing. R Code
	20	Estimation of trend with seasonality. Sample autocorrelation.

Stationary Processes
Sept	22	Autocovariance function, bias, examples.
	26	Properties of ACVF, stationarity.
	27	Proof that stationarity and zero-mean implies existence of MA.
	29	The AR(1) and ARMA(1,1) Models, definition of Causal and Invertible processes
Oct	3	The ARMA(p,q) Model
	4	Existence of stationary solutions, recap of models
	6	Examples. R Code
	10	Class cancelled, Thanksgiving holiday
	11	Examples and the Partial autocorrelation function (PACF)
	13	PACF Example, Introduction to Model Fitting

Prediction and Model Fitting
Oct	17	Best Linear Predictor, Durbin-Levinson Algorithm
	18	The Innovations Algorithm, Example
	20	Prediction of ARMA process
	24	Finish prediction. Yule-Walker.
	25	Yule-Walker and the PACF. Significance test for order p model-fit. R Code
	27	Innovations algorithm for fitting MA models.
	31	Maximum-Likelihood Estimation
Nov	1	Order Selection and the AIC
	3	Diagnostic and Residual Checking

Spectrum Estimation
Nov	7	Examples of ar(). Spectral Density. R Code
	8	Spectral Representation Theorem, example
	10	Class cancelled
	14	Examples, Introduction to the Periodogram
	15	Properties of the Periodogram, Smoothing
	17	Statistical Properties, Examples R Code
	21	More on the Bias/Variance Problem
	22	Direct Spectral Estimators R Code
	24	Zeropadding, the FFT, and other Details R Code

Presentations
Nov	28	Presentations: Day 1
	29	Presentations: Day 2
Dec	1	Presentations: Day 3

Textbook		Introduction to Time Series and Forecasting, Brockwell and Davis, 2nd Edition. Available at Campus Bookstore ($85.41 used) or as a (free) e-book in pdf from the Queen's Library subscription to SpringerLink. You will need to use the Queen's WebProxy service to access the e-book from off-campus.

(additional readings on course reserve)		Time Series: Theory and Methods, P.J. Brockwell and R.A. Davis. Graduate-level version of our textbook, more detail on theory. Also available via SpringerLink Spectral Analysis for Physical Applications, D.B. Percival and A.T. Walden. Arguably the best text available for the second half of the course, and definitely the best for actual algorithm implementation.

(additional readings, e-book)		Time Series Analysis and Its Applications (with R Examples), R.H. Shumway and D.S. Stoffer. Well-written text with good R examples to help in learning to do practical data analysis.

Syllabus		Syllabus for course.
Data		Data Sets for assignments and from in-class examples.
Project		Description and Marking Scheme