Modeling North Pacific Climate Time Series

We present a case study in modeling the North Pacific (NP) index, which is a time series related to atmospheric pressure variations at sea level. We consider three statistical models, namely, a Gaussian stationary autoregressive process, a Gaussian stationary fractionally differenced (FD) process, and a ‘signal plus noise’ process consisting of a square wave oscillation with a pentadecadal period embedded in Gaussian white noise. Each model depends upon three parameters, so all three models are equally simple. Statistically each model fits the NP index equally well. The fact that this index consists of just a hundred observations makes it unrealistic to expect to be able to clearly prefer one model over the other. Although the models fit equally well, their implications for the long term behavior of the NP index can be quite different in terms of, e.g., generating regimes of characteristic lengths (i.e., stretches of years over which the NP index is predominantly either above or below its long term average value). Because we cannot determine a preferred model statistically, we are faced with either entertaining multiple models when considering what the long term behavior of the NP index is likely to be or using physical arguments to select one model. The latter approach would arguably favor the FD process because it has an interpretation as the synthesis of first order differential equations involving many different damping constants.