Fig. 4: Supporting the emergent relationship in the statistical framework.

Probability ratios of extreme precipitation changes as a function of gamma distribution parameters (x-axis: the scale parameter \(\theta\); y-axis: the stretch parameter \(\nu\)). Background shadings are statistical calculations of probability ratios based on idealized transformations (using Eqs. 2–4; see Methods) based on multi-model medians of present-day mean precipitation (in mm/day) and its fractional change under warming (\(\alpha\); noted in top-right). The dots are individual models, and their filled colors indicate model empirically estimated probability ratios (based on the frequency of occurrence of extreme events in model data). The location of model results in the parameter space is determined by fitting model data into gamma distributions (see Methods). Comparing how the background shadings and filled dots vary in the parameter space shows that the statistical argument supports the model results qualitatively. Regional results are shown for 5-day precipitation events (pr5d) for Northern hemisphere mid-latitudes (45°−70°N) in June-to-August (JJA; a), Southern hemisphere mid-latitudes (45°−70°S) in JJA (b), Northern Asia in JJA (c), West North Pacific in JJA (d), Europe in December-to-February (DJF; e) and Eastern North America in DJF (f).