Scale-Selective Ridge Regression for Multimodel Forecasting

This paper proposes a new approach to linearly combining multimodel forecasts, called scale-selective ridge regression, which ensures that the weighting coefficients satisfy certain smoothness constraints. The smoothness constraint reflects the “prior assumption” that seasonally predictable patterns tend to be large scale. In the absence of a smoothness constraint, regression methods typically produce noisy weights and hence noisy predictions. Constraining the weights to be smooth ensures that the multimodel combination is no less smooth than the individual model forecasts. The proposed method is equivalent to minimizing a cost function comprising the familiar mean square error plus a “penalty function” that penalizes weights with large spatial gradients. The method reduces to pointwise ridge regression for a suitable choice of constraint. The method is tested using the Ensemble-Based Predictions of Climate Changes and Their Impacts (ENSEMBLES) hindcast dataset during 1960–2005. The cross-validated skill of the proposed forecast method is shown to be larger than the skill of either ordinary least squares or pointwise ridge regression, although the significance of this difference is difficult to test owing to the small sample size. The model weights derived from the method are much smoother than those obtained from ordinary least squares or pointwise ridge regression. Interestingly, regressions in which the weights are completely independent of space give comparable overall skill. The scale-selective ridge is numerically more intensive than pointwise methods since the solution requires solving equations that couple all grid points together.