A theoretical relationship between probabilistic relative operating characteristic skill and deterministic correlation skill in dynamical seasonal climate prediction

Understanding the relationship between probabilistic and deterministic prediction skills is of important significance for the study of seasonal forecasting and verification. Based on the Brier skill score methodology, we have previously found a theoretical relationship between the probabilistic resolution skill and the deterministic correlation (i.e., anomaly correlation; AC) skill and a lack of necessary or consistent relationship between the probabilistic reliability skill and the deterministic skill in dynamical seasonal prediction. Here, we further theoretically investigate the relationship between the probabilistic relative operating characteristic (ROC) skill and the deterministic skill. The ROC measures the discrimination attribute of probabilistic forecast quality, another important attribute besides the resolution and reliability. With some simplified assumptions, we first derive theoretical expressions for the hit and false-alarm rates that are basic ingredients for the ROC curve, then demonstrate a sole dependence of the ROC curve on the AC, and finally analytically derive a relationship between the related ROC score and the AC. Such a theoretically derived ROC-AC relationship is further examined using dynamical models' ensemble seasonal hindcasts, which is well verified. The finding here along with our previous findings implies that the discrimination and resolution attributes of probabilistic seasonal forecast skill are intrinsically equivalent to the corresponding deterministic skill, while the reliability appears to be the fundamental attribute of the probabilistic skill that differs from the deterministic skill, which constitutes an understanding of the fundamental similarities and difference between the two types of seasonal forecasting skills and predictability and can offer important implications for the study of seasonal forecasting and verification.