This paper shows theoretically and with examples that climatological means derived from spectral methods predict independent data with less error than climatological means derived from simple averaging. Herein, “spectral methods” indicates a least squares fit to a sum of a small number of sines and cosines that are periodic on annual or diurnal periods, and “simple averaging” refers to mean averages computed while holding the phase of the annual or diurnal cycle constant. The fact that spectral methods are superior to simple averaging can be understood as a straightforward consequence of overfitting, provided that one recognizes that simple averaging is a special case of the spectral method. To illustrate these results, the two methods are compared in the context of estimating the climatological mean of sea surface temperature (SST). Cross-validation experiments indicate that about four harmonics of the annual cycle are adequate, which requires estimation of nine independent parameters. In contrast, simple averaging of daily SST requires estimation of 366 parameters—one for each day of the year, which is a factor of 40 more parameters. Consistent with the greater number of parameters, simple averaging poorly predicts samples that were not included in the estimation of the climatological mean, compared to the spectral method. In addition to being more accurate, the spectral method also accommodates leap years and missing data simply, results in a greater degree of data compression, and automatically produces smooth time series.

The coevolution of the Indian Ocean dipole (IOD) and El Niño-Southern Oscillation (ENSO) is examined using both observational data and coupled global climate model simulations. The covariability of IOD and ENSO is analyzed by applying the extended empirical orthogonal function (EEOF) method to the surface and subsurface ocean temperatures in the tropical Indian Ocean and western Pacific. The first EEOF mode shows the evolution of IOD that lags ENSO, whereas the second mode exhibits the transition from a dipole mode to a basinwide mode in the tropical Indian Ocean that leads ENSO. The lead-lag relationships between IOD and ENSO are consistent with two-way interactions between them. A comparison between two 500-year model simulations with and without ENSO shows that ENSO can enhance the variability of IOD at interannual time scale. The influence of ENSO on the IOD intensity is larger for the eastern pole than for the western pole, and further, is stronger in the negative IOD phase than in the positive phase. The influence of IOD on ENSO is demonstrated by the improvement of ENSO prediction using sea surface temperature (SST) in the tropical Indian Ocean as an ENSO precursor. The improvement of the ENSO forecast skill is found at both a short lead time (0 month) and long leads (10-15 months). The SST in the western pole has more predictive value than in the eastern pole. The eastward propagation of surface and subsurface temperature signals from the western Indian Ocean that precedes the development of heat content anomaly in the tropical western Pacific is the key for extending the lead time for ENSO prediction. Our results are consistent with previously reported findings but highlight the spatialtemporal evolution of the ENSO-IOD system. It is also illustrated that IOD would have been more helpful in predicting the 1997/98 El Niño than the 2015/16 El Niño.