1. Statistics 349.3(02) Analysis of Time Series

2. Course Information 1.Instructor: W. H. Laverty 235 McLean Hall Tel: 966-6096 Email: laverty@math.usask.ca 2.Class Times MWF 12:30-1:20pm AGRI 1E69 3.Mark break-up Final Exam – 60% Term Tests and Assignments – 40%

3. Course Outline 1. Introduction and Review of Probability Theory –Probability distributions, Expectation, Variance, correlation –Sampling distributions –Estimation Theory –Hypothesis Testing

4. Course Outline -continued 2.Fundamental concepts in Time Series Analysis –Stationarity –Autocovariance function, autocorrelation and partial autocorrelation function 3.Models for Stationary Time series –Autoregressive (AR), Moving average (MA), mixed Autoregressive-Moving average (ARMA) models 4.Models for Non-stationary Time series –ARIMA (Integrated Autoregressive-Moving average models)

5. Course Outline -continued 5.Forecasting ARIMA Processes 6.Model Identification and Estimation 7.Models for seasonal Time series 8.Spectral Analysis of Time series

6. Course Outline -continued 9.State-Space modeling of time series, Hidden Markov Model (HMM) –Kalman filtering 10.Multivariate (Multi-channel) time series analysis 11.Linear filtering

7. Comments The Text Book –Time Series Analysis: Univariate And Multivariate Methods Wei (Paperback - 2006 Ed. 02) Recommended I will present the material in power point slides that will be available on my web site. Hopefully this makes purchasing the text book optional. It is useful for students to purchase text books and build up a library.

8. Some examples of Time series data

9. IBM Stock Price (closing)

10. Time Sequence plot

11. Concentration of residue after production of a chemical

12. Time Sequence plot

13. Yearly sunspot activity (1770 -1869)

14. Time Sequence plot

16. Time Sequence Plots

19. Example Measuring brain activity in an insect as an object is approaching. Time t = 0 is at the point of impact.

20. Simulation To aid in the understanding of time series it is useful to simulate data

21. Generating observations from a distribution

22. Generating a random number from a distribution Let 1. f(x) denote the density function 2. F(x) denote the cumulative distribution function = P[X ≤ x] 3. F -1 (x) denote the inverse cumulative distribution function

23. F(x) denote the cumulative distribution function F(x)F(x) f(x)f(x) x

24. u f(x)f(x) F -1 (u) If u is chosen at random from 0 to 1 then x = F -1 (u) is chosen at random from the density f(x). In EXCEL the following function generates a random observation from a normal distribution. = NORMINV(RAND(), mean, standard deviation)

25. Example Random walk A random walk is a sequence of random variables {x t } satisfying: x t = x t – 1 + u t where {u t } is a sequence of independent random variables having mean 0, standard deviation . (usually normally distributed) The excel functions 1.NORMINV(prob, mean, standard deviation) computes F -1 (prob) for the normal distribution. 2.RAND() computes and random number from the Uniform distribution from 0 to 1. 3.NORMINV(RAND(), mean, standard deviation) computes and random number from the Normal distribution with a given mean and standard deviation.