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

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Problem Definition {y t } : 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.

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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.

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Simple but illustrative model

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Kalman Filter update at time t: Bayes Rule

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Asymptotic relation between SNR and EWMA coefficient

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Estimating Variance components

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Illustration on Data Percentages of calls to an automated service at AT&T that ended in Hang ups. Cant 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).

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A realization of the simulated process

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Frequentist property of the procedure.

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Red:recovered signal Discount=.8

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Discount=.95

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How to detect changes?

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Detecting changes, continued

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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.

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How to intervene at time t? Add additional evolution to state at time t

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Mild intervention, U_{t}=0

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Zoomed area, mild intervention

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Strong intervention, sd of state vector tripled

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Zoomed in, strong intervention

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More general models Y t-1 YtYt xtxt X t-1 GtGt Y t-1 X t-1

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Model with Day of week effects on real data.

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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

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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.

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