Bayesian Forecasting and Dynamic Models

1. Bayesian Forecasting and Dynamic Models M.West and J.Harrison Springer, 1997 Presented by Deepak Agarwal

2. Problem Definition • {yt} : 1-d time series to be monitored • E.g. Daily counts of some pattern, e.g., number of emergency room visits to a hospital • Goal: A statistical method which • Forecast accurately (short term, long term behavior), i.e., a good baseline model. • Detects deviations from baseline (detect outliers, gradual changes, structural changes) with good ROC characteristics. • Baseline model adapts to changes over time, e.g., learns gradual changes in day of week effects, learns mean shifts etc.

3. The Approach • Baseline Model learned using a Kalman Filter • Novel and simple way of learning “evolution” covariance using a “discount” concept • Change detection done by cumulating evidence against status quo through residuals. • Procedure adapts to changes in the baseline by using the principle of management by exception • Use the forecasting model unless exceptional circumstances arise wherein one intervenes and corrects the forecasting model.

4. Simple but illustrative model

5. Kalman Filter update at time t: Bayes Rule

6. Asymptotic relation between SNR and EWMA coefficient

7. Estimating Variance components

8. Illustration on Data • Percentages of calls to an automated service at AT&T that ended in Hang ups. • Can’t give you the real numbers, this is what I did • Did an arcsine transform • Generated mean surface using Loess making sure the span was chosen to minimize autocorrelation in residuals. • Generated smooth variances using deviation of observed from mean surface • Simulated observation from this process (See figure on next page).

9. A realization of the simulated process

10. Frequentist property of the procedure.

11. Discount=.8 Red:recovered signal

12. Discount=.95

13. How to detect changes?

14. Detecting changes, continued

15. What to do when a change is detected? • Possibilities: • Ignore points, underestimates variance • Proceed with filtering as usual, introduces bias and overestimates variance • Need something in between. • Intervention: • Management by Exception: Use a forecasting model unless exceptional circumstances arise. • Feed forward: anticipatory in nature, e.g., a new version of the system comes out which is likely to increase hang up rates. • Feed back: Model performance deteriorates, adapt to new conditions, done automatically.

16. How to intervene at time t? • Add additional evolution to state at time t

17. Mild intervention, U_{t}=0

18. Zoomed area, mild intervention

19. Strong intervention, sd of state vector tripled

20. Zoomed in, strong intervention

21. More general models Yt-1 Yt-1 Yt Xt-1 Xt-1 xt Gt

22. Model with Day of week effects on real data.

23. Non-normal models • Observation model is one parameter exponential family. • State equations are same. • Using canonical parametrization, prior on natural parameter \eta_{t}=x^{T}\theta_{t} formed by using prior on \theta_{t} through method of moments • Posterior of \eta_{t} converted to posterior of \theta_{t}. • Details in the book

24. Recent work and possible research questions • Detecting subtle changes that are not outliers • Breakpoints, variance changes, autocorrelated errors (Salvador and Gargallo,JCGS) • Detecting blips might not be important unless it is huge, want to alert only if things persist for a while • Take an EWMA of Bayes factor, similar to Q-chart idea. • Intend to analyse data posted on the AD website using these models. • Comparative analyses with other commonly used methods.