Estimating millions of dynamic timing patterns in real time

Ln some business applications, the transaction behavior of each customer is tracked separately with a customer signature. A customer's signature for b uying behavior, for example, may contain information on the likely place of purchase, value of goods purchased, type of goods purchased, and timing of purchases. The signature may be updated whenever the customer makes a tran saction, and, because of storage Limitations, the updating may be able to u se only the new transaction and the summarized information in the customer' s current signature. Standard sequential updating schemes, such as exponent ially weighted moving averaging, can be used to update a characteristic tha t is observed at random, but timing variables Like day of the week are not observed at random, and standard sequential estimates of their distribution s can be badly biased. This article derives a fast, space-efficient sequent ial estimator for timing distributions that is based on a Poisson model tha t has periodic rates that may evolve over time. The sequential estimator is a variant of an exponentially weighted moving average. It approximates the posterior mean under a dynamic Poisson timing model and has good asymptoti c properties. Simulations show that it also has good finite sample properti es. A telecommunications application to a random sample of 2,000 customers shows that the model assumptions are adequate and that the sequential estim ator can be useful in practice.