Supplementary Material for On the evaluation of climate model simulated precipitation extremes

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1 Supplementary Material for On the evaluation of climate model simulated precipitation extremes Andrea Toreti 1 and Philippe Naveau 2 1 European Commission, Joint Research Centre, Ispra (VA), Italy 2 Laboratoire des Sciences du Climat et l Environnement (LSCE) CNRS, Orme des Merisiers, Gif-sur-Yvette, France andrea.toreti@jrc.ec.europa.eu

2 precipitation extremes 2 1. Methods As stated in the Appendix, the statistic A can be derived as follows, A = nm N N 1 i=1 (Ḡm F n ) 2 1 i N 1 N = 1 N 1 (M i N ni) 2 nmn (N i) i=1 where M i = nf n H 1 N ( i ), i.e. the number of X s less equal the i-th smallest value N in the pooled sample. Under the null hypothesis of equally distributed samples and by noticing that M i is hypergeometric distributed (with population size equal to N, sample size given by i and number of successes equal to n), the expected value of A is given by (1) E(A) = 1 N 1 nmn i=1 N 2 V(M i) (N i) = 1 N 1 N i=1 The variance of A can derived as follows V(A) = N 2 n 2 m E 2 ( N 1 i M 2 i N i ) 2 i N 1 = 1 2 (2) (3) where M i = M i ni. Now, the expected value in (3) can be subdivided into two N parts, namely A 1 and A 2. A 1 is given by A 1 = E = { N 1 } M i 4 (N i) 2 i nm N 2 (N 1)(N 2)(N 3) ( mnn 3 4 3mnN mnN 4 while A 2 can be derived (M i and M j being independent) as follows 3mn ) (4) 2 Thus, N 1 A 2 = E i = 2n 2 m 2 N 4 (N 1) 2 N 1 j=1 j i 2 M i 2 M 2 j N i N j ( N 4 4 5N N 2 4 N 6 V(A) = N 2 n 2 m (A A 2 ) (6) = 1 ( N 3 (N 1)(N 2)(N 3) 4 3N N 3 ) 4 2 ( 2 N 3 + N(N 1) 2 4 5N N 4 1 ) (7) 6 ) (5)

3 precipitation extremes 3 The approximation of the upper tail area of the survival distribution function of the limiting distribution of A suggest by Sinclair et al (1990) is: F (x) = x e x (8) The Kullback Leibler directed divergence can be estimated as follows (Naveau et al 2013): Î(f e ; g e ) = N e (Ḡ(X ) i log e /ˆµ Xe 0 ), (9) N e Ḡ(0) i=1 where N e is the number of X-excesses and Ḡ(t) = 1 (m + 1) 1 m i=1 I {Ye i /ˆµ Ye 0 t} with m equal to the number of Y -excesses. The divergence method of Naveau et al (2013) is based on: D(f, g) = I(f e ; g e ) + I(g e ; fe), (10) where I can be estimated as in eq. (8). The multiple testing approach of Genovese and Wasserman (2004) is based on the False Discovery Rate - FDR (Benjamini and Hochberg 1995). Given m tests, let M 0 (M 1 ) be the number of true (false) null hypothesis. The FDR is defined as: { E { V F DR = R} if R > 0 (11) 0 if R = 0 where R represents the number of rejected null hypotheses and V the true null hypotheses wrongly rejected. Associated to the m null hypotheses, there are m p- values P i (i = 1,..., m) having a marginal distribution G = (1 a)u + af, where U represents the uniform distribution on (0, 1), a is a coefficient varying in [0, 1] and F denotes an unknown distribution. Thus, the FDR can be controlled at a chosen level α (here, 0.1) by rejecting all null hypotheses associated with a p-value less equal a specific threshold T (a, G) that can be estimated by: { } T (â, Ĝ) = sup (1 â)t t : Ĝ(t) = (12) α where Ĝ is the empirical cumulative distribution function of the P i and the estimation of a is given by: ( ) Ĝ(t) t ɛ m 1 2 â = max, ɛ m = t 1 t 2m log (13) α

4 precipitation extremes 4 Table 1: Number of wrong decisions (based on 95% critical values) derived by testing (10 3 times) samples coming from three different tails (see the bold red values) against samples from tails having a different ξ (see bold numbers on rows) and using three sample sizes (10 2,10 3,10 4 ). Values in percentage. ξ = ξ = ξ = ξ = Table 2: CMIP5 Global Climate Models used in this study. We acknowledge the World Climate Research Programmes Working Group on Coupled Modelling, which is re- sponsible for CMIP, and we thank the climate modeling groups (listed in the table) for producing and making available their model output. For CMIP the U.S. Department of Energys Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. Model CMCC-CM (CMCC) CNRM-CM5 (CNRM) HadGEM2-CC (HadCC) HadGEM2-ES (HadES) INM-CM4 (INM) IPSL-CM5A-MR (IPSL) MIROC5 (MIROC) MRI-CGCM3 (MRI) Institution Centro Euro-Mediterraneo sui Cambiamenti Climatici Centre National de Recherches Meteorologiques - Centre Europeen de Recherche et Formation Avancees en Calcul Scientifique Met Office Hadley Centre Met Office Hadley Centre - Instituto Nacional de Pesquisas Espaciais Institute for Numerical Mathematics Institut Pierre-Simon Laplace Atmosphere and Ocean Research Institute (The University of Tokyo), National Institute for Environmental Studies, and Japan Agency for Marine-Earth Science and Technology Meteorological Research Institute References Benjamini Y and Y Hochberg 1995 Controlling the False Discovery Rate: a practical and powerful approach to multiple testing J. Roy. Stat. Soc. B

5 precipitation extremes 5 Genovese C and L Wasserman 2004 A stochastic process approach to false discovery control Ann. Stat Naveau P, Guillou A and Rietsch T 2013 A non-parametric entropy-based approach to detect changes in climate extremes J. Roy. Stat. Soc. B in press Sinclair C D, Spurr B D and M I Ahmad 1990 Modified Anderson-Darling test Comm. Stat. A

precipitation extremes 6 Figure 1: Spearman-based spatial correlation matrix of the tail scaling factors, estimated for the eight GCMs, ˆµ model 0, and the gridded observations E-OBS, ˆµ obs 0, in

6 precipitation extremes 6 Figure 1: Spearman-based spatial correlation matrix of the tail scaling factors, estimated for the eight GCMs, ˆµ model 0, and the gridded observations E-OBS, ˆµ obs 0, in the autumn period The colors and the shape of the ellipses are associated with the correlation values. The last column refers to the same analysis without the southern part of the domain (south of degrees North)..

7 precipitation extremes 7 Figure 2: Rescaled-tail comparison of model simulations during the historical autumn period and E-OBS. Colors are associated with the values of the 2-sample modified Anderson-Darling statistic with the sign given by the estimated KLD-divergence. Blank areas are associated with non-significant values..

8 precipitation extremes 8 Figure 3: Boxplots of the ratio between the estimated conditional means of the excesses for the future winter time periods ( : blue; : green) and the historical simulations, ˆµ scenario 0 /ˆµ hist 0, derived for each grid point in the domain..

9 precipitation extremes 9 Figure 4: Results of the rescaled-tail comparison of the autumn period w.r.t. the historical simulation ( ). Colors are associated with the values of the 2- sample modified Anderson-Darling statistic with the sign given by the estimated KLDdivergence. Blank areas are associated with non-significant values..

10 precipitation extremes 10 Figure 5: As for Figure 4, but for

Supplement of Insignificant effect of climate change on winter haze pollution in Beijing

Supplement of Insignificant effect of climate change on winter haze pollution in Beijing Supplement of Atmos. Chem. Phys., 18, 17489 17496, 2018 https://doi.org/10.5194/acp-18-17489-2018-supplement Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License.