Table 1

Comparison of models using the GCV model selection criterion V c. The model (defined by Eqs. 1and 2) is fitted to the cholera time series after the data are square-root–transformed, which normalizes the residuals and stabilizes the variance. A time lag t of 2 months is used, based on the well-known rule of thumb of choosing the lag for which the autocorrelation function first crosses 0.5. Thus, Eq. 1 takes the general formEmbedded Image Embedded Image(4)where Yt =Embedded Image. When seasonality is incorporated but not the ENSO index, low-dimensional models (d = 1 and d = 2) are selected (that is, they have the smallest values of the cross-validation criterionV c). The importance of seasonality is demonstrated by comparing these models to their autonomous counterparts. Models with an equivalent or larger number of independent variables (for example, d = 4 and d = 6) but no seasonality have larger values ofV c. The importance of ENSO is examined by incorporating the ENSO index into the simplest seasonal model (d = 1) at different time lags (τf between 0 and 12 months). The smallestV c value is obtained for the model with τf = 11. This model is also selected over the seasonal one with an equal number of independent variables (d = 2). It has a higherr 2 value than the seasonal and autonomous models, accounting for a larger fraction of the variance. (For all models, the number of neurons k used in fitting the function f of Eq. 2 varied between 1 and 5. Only the model with the smallest V c is reported here.) A value of k larger than one in the selected ENSO model indicates that a linear time series model is not adequate to fit the causal relationships and would give a less reliable conclusion about the role of ENSO.

Environmental covariateSeasonal clockdkGCV score Vcr2 (%)
+120.93755
+220.93257
431.25951
631.17660
E t−11 +130.81967