Supplemental Material for. Statistical Emulation of Climate Model Projections based on Precomputed GCM Runs

Transcription

1 Supplemental Material for Statistical Emulation of Climate Model Projections based on Precomputed GCM Runs Stefano Castruccio Department of Statistics, University of Chicago, Chicago, Illinois David J. McInerney + Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois Michael L. Stein and Feifei Liu Department of Statistics, University of Chicago, Chicago, Illinois Robert L. Jacob Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois Elisabeth J. Moyer Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois

2 Regions used for emulation See Figure S. ARL ARO ALA HBO CGI NNA NEU WNA CNA ENA SEU NNE MED NAT CAMCAR SAH NPE WAF EPW EPE EAT EAF AMZ SAF SPW SPE SAT SSA NAS CAS TIB NNW EAS SAS SEAWPN NPW WPE IND WPS NAU SAU SIO AOP AOI ANL Figure S: Regions used for statistical emulations described in the text. Regions are a variant of those described in [], with some ocean areas further subdivided to capture qualitative differences in transient precipitation response Details about the statistical fit We give a more detailed description of how the profile likelihood method has been used to fit temperature. If we have only a single scenario with n observations, and we call σt σ :=, then equation () can be reformulated φ as T = X(ρ)β +ε ε N (,σ TK(φ) ) () where K(φ) is the correlation matrix of an AR() process with parameter φ and X(ρ) is a n matrix whose entries are X t, = X t, = log[co ](t)+ log[co ](t ) X t, = X t, (ρ) = ( ρ) + i= ρi log[co ](t i)

3 Dropping the dependence of the design matrix X on ρ to simplify the notation, the loglikelihood for model () is l(ρ,β,φ,σ T,[CO ]) = n log(π) log σ T K(φ) (T Xβ) t K(φ) (T Xβ). σt () 4 For given values of ρ and φ, the Maximum Likelihood Estimator (MLE) of β is ˆβ = (X t K(φ) X) X t K(φ) T, () and the MLE for σt is ˆσ T = n (T Xˆβ) t K(φ) (T Xˆβ), (4) 5 6 We can compute K(φ) = ( φ ) n, and plugging () and (4) into () we have, if we call W = X t K(φ) X l(ρ,φ T,[CO ]) = n log(π) n + log( φ ) n log(tt (K(φ) K(φ) XW X t K(φ) )T) (5) which is the profile likelihood (see e.g. [] ch.4) and is a function only of the parameters ρ and φ and can be easily and quickly maximized. If more than one scenario/realization is present, the approach can be easily generalized in those cases where we can assume that all the processes are independent. Parameter estimates with associated variability Table S gives the parameter estimates for the temperature emulator () along with their standard errors, based on large sample approximations when the emulator is trained with one realization each of the fast and jump scenario. Specifically, the variability of (ˆβ, ˆβ, ˆβ ) is computed conditionally on (ˆρ, ˆφ,ˆσ ) via generalized least squares, whereas the variability of (ˆρ, ˆφ) is computed by first estimating the Fisher information matrix.

4 Region ˆρ ˆβ ˆβ ˆβ ˆφ ˆσ ARL.999(.7) 55.8(.4) 4.9(.8).54(.7).7(.9).4474 ARO.(.) 55.97(.57) 4.98(.) 587.(4.).7(.7).557 ANL.9(.) 9.7(.487).8(.8).77(.57).7(.).577 AOP.8(.78) 7.5(.444).5(.).(.8).54(.79).58 AOI.9(.7) 68.7(.54).(.6).9(.7).4(.).97 ALA.9(.59) 64.4(.5).(.5).58(.54).(.4).4 CGI.9(.6) 65.9(.44).69(.).8(.49).4(.9).5777 WNA.4(.8) 76.4(.5).(.64).5(.568).5(.).665 CNA.(.) 8.7(.48).5(.84) 59.77(.68).(.).656 ENA.997(.5) 78.79(.4).(.).7(.587).5(.).67 CAM.99(.).4(.).7(.6).54(.659).(.).84 AMZ.99(.5).8(.58).7(.45).7(.49).9(.).6 SSA.9(.7) 88.8(.77).(.45).86(.545).6(.).56 NEU.(.) 74.4(.68).6(.7) 98.6(9.746).(.).84 SEU.999(.) 86.5(.).66(.6).59(.5).(.).45 SAH.985(.).75(.8).49(.97).7(.99).6(.).977 WAF.7(.9) 97.7(.6).54(.8).89(.8).7(.).4 EAF.987(.).(.).4(.4).68(.44).6(.).7 SAF.984(.) 9.64(.58).(.56).9(.57).5(.).55 NAS.(.) 65.55(.458).58(.88) 69.7(84.688).(.).655 CAS.(.6) 84.9(.7).9(.4).4(.54).(.).58 TIB.98(.8) 7.5(.64).65(.68).7(.6).5(.).59 EAS.997(.) 79.9(.97).78(.64).89(.86).9(.).46 SAS.99(.44).49(.4).7(.58).58(.67).5(.). SEA.989(.5) 98.(.9).6(.7).46(.9).(.5).69 NAU.98(.46).47(.7).79(.97).(.98) -.(.).47 SAU.988(.5) 88.4(.99).6(.6).7(.644).8(.).748 MED.999(.5) 87.5(.).4(.6).57(.58).7(.).487 CAR.99(.6).4(.4).6(.).5(.48).9(.6).88 IND.99(.5).77(.85).58(.4).58(.66).(.5).6 SPW.988(.) 9.8(.74).4(.).68(.4).8(.8).69 SPE.989(.6) 9.4(.9).4(.5).78(.79).7(.8).4 EPW.989(.4) 98.5(.78).5(.5).7(.57) -.6(.7).6 EPE.986(.6).4(.).49(.7).(.754) -.8(.8).486 NPW.99(.) 97.4(.5).54(.5).5(.4).45(.).8 NPE.(.).84(.9).8(.65).8(.64).4(.).4 SAT.988(.5) 89.46(.7).54(.6).9(.8).8(.).86 EAT.98(.9) 98.48(.98).6(.6).7(.6).8(.7). NAT.997(.) 9.68(.9).8(.46).7(.75).47(.9).68 NNA.(.) 74.7(.5).6(.) 4.6(.987).64(.55). WPS.989(.9) 97.45(.9).5(.7).55(.9).7(.7).5 WPE.99(.6) 99.5(.9).4(.6).4(.9).4(.6).85 WPN.99(.7) 97.6(.9).7(.).4(.68).7(.7).49 NNE.99(.8) 8.79(.7).(.).5(.6).48(.9).9 NNW.997(.8) 78.65(.8).89(.8).5(.4).4(.). SIO.99(.9) 8.9(.66).(.44).(.58).5(.8). HBO.99(.5) 6.4(.7) 4.(.).44(.).5(.).87 Table S: Parameter estimates for () in all regions, with their standard deviations in parentheses. In several regions (ARO, CNA, NEU, NAS and NNA), ˆρ is numerically equal to, which implies that + i= w i log[co r ](t i) so ˆβ is highly unstable.

5 Emulator prediction for different realizations Figure S shows the small variations in predictions one gets by using different realizations of the same scenarios in the training set. The only feature that is noticeably affected is the emulation near the sudden drop in CO. Note that because all five realizations of the fast and jump are used in this exercise, the five estimated curves are not quite independent because some pairs of curves will have been trained with identical climates during 87-, although for any one curve, the two training runs were selected so that they never shared a restart year. T anomaly ( C) (a) T anomaly ( C) (b) P anomaly (mm yr ) (c) Year P anomaly (mm yr ) (d) Year Figure S: An example of temperature emulation for different realizations in the North Pacific West (NPW) region (a-b), and of precipitation emulation in the Equatorial Pacific West (EPW) region (c-d). Panels a and c show the emulated slow scenario and b and d the drop scenario. Each emulator was trained by one realization each of the fast and jump scenarios. The solid lines in different colors represent the emulated value for each of the realization, and the gray lines represent the five realizations for the scenarios. Emulation appears indistinguishable for the slow, while for drop different realizations result in somewhat different estimated values near the drops. 4

6 4 4 5 Empirical coverages See Figure S Long term predictions See Figure S References [] A.E. Gelfand et al. Handbook of Spatial Statistics. Chapman& Hall/CRC, Boca Raton, 8. [] K. Ruosteenoja, T.R. Carter, and K. Jylha. Future climate in world regions: an intercomparison of model-based projections for the new IPCC emissions scenarios. Technical report, Finnish Environment Institute, Helsinki,. 5

7 T, "slow" 9 9 T, "drop" Figure S: Empirical coverage (multiplied by ) of the nominal % prediction bands for temperature. The training set is one realization of the fast and jump scenario. Top panel shows the slow and bottom panel the drop scenarios. The coverage for slow is very close to the nominal value, for drop there is a sign of misspecification for some regions in the southern hemisphere, as noticed for the mean emulation. 6

8 5.5 4 T anomaly ( C).5.5 T anomaly ( C) Year Year Figure S4: Training set a fast (until year 449) and a jump realization (until year 99). Example of emulation for a very long time scale (until year 4599) for the fast scenario for the SAT region (on the left) and NAU region (on the right). The red line represents the estimated mean and the dashed red lines represent the % prediction intervals. 7