Another Look at Low-Order Autoregressive Models in Early Detection of Epidemic Outbreaks and Explosive Behaviors in Economic and Financial Time Series

In our SUGI 2006 presentation, we suggested using low-order autoregressive models, AR(1) and AR(2), in biosurveillance and outbreak detection (PROC ARIMA, SAS/ETS ® ). Our suggestion was based on empirical data. In the NESUG 2007 paper, we proposed strong theoretical grounds for this. Here we provide further development of our approach. Based on a classic susceptibleinfectious-recovered (SIR) model, we arrive at AR(1) models of epidemics where we need to estimate only one parameter, the first-order autoregressive coefficient. Its least squares estimate has a very simple epidemiological meaning. In the vast majority of applications, AR and ARMA are used as purely empirical, stationary models, with no specific substance matter meaning for coefficients. The value of our first-order autoregressive coefficient less than one corresponds to a stationary, no-epidemic regime. If the parameter is greater than one, we have an explosive case (an outbreak of epidemic). When the coefficient is equal to one, we have a unit root case. Based on the observed data in a chosen time window, least squares estimates and confidence intervals allow us to decide which case is more appropriate. The question of bias correction of our estimates is also discussed. After purely temporal analysis, we can proceed to the spatial step with logistic or Poisson regressions as in our SUGI 2006 paper. The approach described above can also be used in describing explosive behaviors of economic and financial time series (e.g., stock market bubbles).