Fig. 5: Graphical representation of Granger causality using logistic regression.

a The results of Granger causality analysis of severe CAOs in terms of the POV model according to the reanalysis data. b As shown in (a), but for moderate CAOs. c, d As shown in (a) and (b), respectively, but for the Granger causality analysis in terms of the combined model. Note that we add the significant links between the polar vortex and the cold air mass transport in (c) and (d), respectively. Circles represent the predictands, namely, the current states of CAOs at high latitudes and mid-latitudes. Squares denote predictors (e.g., POV or PCAM), namely, past states of the polar vortex and the cold air mass transport. The arrows denote the direction of the influence, with dashed arrows denoting that the predictor 6–10 days before provides predictive information to the predictand and solid arrows for the predictor 1–5 days before. The arrow colors denote the logistic regression coefficients with only statistically significant links (p < 0.05) being presented (the color bar on the bottom; the contour interval is 0.1). The node colors in (a) and (c) indicate the ratio between the ΔG² of severe CAOs and that of moderate CAOs (the color bar to the left; the contour interval is 1.0). A larger value of ΔG² means a better model performance. See “Methods” for the definition of ΔG². We do not color the node over mid-latitude Europe, since there is not a significant link for both severe and moderate CAOs.