The sample cross-correlation function between the series PDSI and all ENSO indices available is shown in Figure \ref{455062}. Among all ENSO indices, the series ONI resulted with the highest cross-correlation with the PDSI series at a lag of -1 (\(\hat{\rho}\)=0.43), with the ONI series leading by one time step (one year) the PDSI series. However, since this series is rather short (available from 1950 onwards), we selected the next highly correlated series with PDSI. From all ENSO indices available from 1870 onwards, el Niño3.4 index was selected as having the highest cross-correlation with series PDSI at lag -1 (\(\hat{\rho}\)=0.35), with the Niño3.4 series leading the PDSI series.
Since the PDO times series is available from year 1900, this was considered the initial year for this analysis. The model training period is the interval (1900-1953) and the model validation period is the interval (1954-2014), which coincides with the validation period used for the ES approach in section \ref{995174}.
An ARIMA model was fitted to the PDO series for the training period. An autoregressive model of order 1 (AR(1) ) was adequate for the series. Figures \ref{586715} a2) presents the sample crosscorrelation function (CCF) between the PDO series (X) and the PDSI series (Y) , and the sample CCF between the pre-whitened X series (residuals after fitting an ARIMA model) and the filtered Y series (after applying the AR(1) filter) presented in Figures \ref{586715} a1) .