1. Verification of clustering properties of extreme daily temperatures in winter and summer using the extremal index in five downscaled climate models José.A. LÓPEZ Climatological Techniques Unit AEMET Spain EMMS & ECAM 2011, Berlin

2. Outline Methodology –Extremal Index θ: definition, example –Estimation of θ –Declustering procedure –Bootstrapping technique for C.I. –Data, deviation index Verification for Dec-Jan lowest daily temperatures –Observed θ values –Statistics of verification of θ for AR4 models –Some results including AR3 models Verification for Jul-Aug highest daily temperatures –.... –... EMMS & ECAM 2011, Berlin

3. 3 The Extremal Index: Definition The Extremal Index θ is a statistical measure of the clustering in a stationary series. It varies between 0 and 1, with 1 corresponding to absence of clustering (Poisson process) Formal definition: Let X(i), i=1,..,n be a stationary series of r.v. with cdf F (with F*= 1-F); define M(n)= max(X(i): 1 ≤ i ≤ n). We say that the process X(i) has extremal index θ ε [0, 1] if for each τ > 0 there is a succession u(n) such that for n -> ∞, a) n F* (u(n)) -> τ (mean nº of exceedances = τ) b) P ( M(n) ≤ u(n) ) -> exp (- θ τ) If θ = 1 the exceedances of progressively higher thresholds u(n) occur independently, i.e. They for a Poisson process (this is the case of independent r.v X(i)

4. EMMS & ECAM 2011, Berlin4 The Extremal Index : interexceedance times The extremal index is the proportion of interexceedance times that may be regarded as intercluster times. This fact is used for declustering.

5. EMMS & ECAM 2011, Berlin5 The Extremal Index : simulation with an ARMAXprocess The ARMAX process is defined by: where de Z’s are standard independent Fechet variables, i.e. prob(Z < x) = Exp (-1/x) This process has an extremal index: Θ = 1 - α

6. EMMS & ECAM 2011, Berlin6 The Extremal Index : simulation for θ = 1 (Poisson)

7. EMMS & ECAM 2011, Berlin7 The Extremal Index : simulation for θ = 0.8

8. EMMS & ECAM 2011, Berlin8 The Extremal Index : simulation for θ = 0.5

9. EMMS & ECAM 2011, Berlin9 The Extremal Index : simulation for θ = 0.2

10. EMMS & ECAM 2011, Berlin10 If the T i are the successive times between exceedances of the high threshold u the Extremal Index is estimates by: (Ferro, C.A.T. “Inference for cluster of extreme values”, J.R.Statist.Soc. B(2003), 65, Part 2, 545-556). Estimation of the Extremal Index

11. EMMS & ECAM 2011, Berlin11 Objective: Define the clusters in a series of exceedances The times between exceedances are classified as inter-cluster times or intra-cluster (belonging to the same cluster) ones according to their length. The criterion used is “objective” and simple, it depends only the Extremal Index θ. More specifically the longest θ N inter-exceedance times are assigned an inter-cluster character, the rest are assigned an intra-cluster character. Between two successive inter-cluster times there is a set (which may be void) of intra-cluster times Declustering procedure

12. EMMS & ECAM 2011, Berlin12 In order to build confidence intervals for the θ of a series, a “bootstrapping” technique was used: a)Sample with replacement successively from the set of inter-cluster times, and then from the set of sets of intra-cluster times to build a fictitious process b)Compute the θ of this fictitious process c)Repeat the above steps the desired nº of times to build the confidence interval Bootstrapping technique

13. EMMS & ECAM 2011, Berlin13 Data and models used Period: 1961-1990 Data used: observed and dowscaled daily temperature at 16 observatories of Spain Models AR4: cccma-cgcm3 (CA), gfdl-cm2 (US), inmcm3 (RU), mpi-echam5 (AL), mri-cgcm2 (JA) Models AR3: ECHAM4, HadAM3, CGCM2 The statistical downscaling technique was analog-based

14. EMMS & ECAM 2011, Berlin14 The thresholds used to build the exceedances (on 15-day moving windows) –90th percentile for Jul-Aug –10th percentile for Dic-Jan (in this case the values below the threshold are found) In order to assess the differences in θ between observations and downscaled data the following deviation index was used where 1000 bootstrap samples where used to compute the medians and the IQR Verification of the Extremal Index in extreme temperature for downscaled climate models

15. EMMS & ECAM 2011, Berlin15 Dec-Jan (occurrances below the 10th percentile of daily temperature)

16. EMMS & ECAM 2011, Berlin16 Observed values of θ Dec-Jan (in percent) Median= 37 Max = 57 Min = 23

17. EMMS & ECAM 2011, Berlin17 Observed values of θ Dec-Jan : values above (1) and below (-1) the median

18. EMMS & ECAM 2011, Berlin18 Observed values of θ Dec-Jan : spatial distribution Lowest values of θ (more clustering) in the NE and interior Highest values of θ (less clustering) in the western half

19. EMMS & ECAM 2011, Berlin19 Verification of θ for AR4 downscaled models Dec-Jan Histogram of absolute deviation index of θ (on the y-axis nº of observatories, on the x-axis accumulated frequencies) Aver. absol. dev. Index: CA (1.3), US(2.3), RU(1.3) AL(0.8) JA (1.2) Aver. dev. Index: CA (0.9), US(2.3), RU(0.2) AL(-0.3) JA (-0.2)

20. EMMS & ECAM 2011, Berlin20 Verification of θ for AR4 downscaled models Dec-Jan: leading models At each observatory the downscaled model that leads the others in terms of absolute deviation index (in no case by more than 1.0)

21. EMMS & ECAM 2011, Berlin21 Verification of θ for downscaled models in Dec- Jan: leading models AR4+ 3 AR3 models Aver. dev. Index AR3 : EC (-2.3) HA ( -1.5) CG (-0.5) Aver. dev. Index AR4: CA (0.9), US(2.3), RU(0.2) AL(-0.3) JA (-0.2) Four models of AR4 show little or moderate global bias in θ, whereas with AR3 only one shows little bias (the rest show more clustering)

22. EMMS & ECAM 2011, Berlin22 Jul-Aug (occurrances above the 90th percentile of daily temperature)

23. EMMS & ECAM 2011, Berlin23 Observed values of θ Jul-Aug (in percent) Median = 48 Max = 81 Min = 36

24. EMMS & ECAM 2011, Berlin24 Observed values of θ Jul-Aug : values above (1) and below (-1) the median

25. EMMS & ECAM 2011, Berlin25 Observed values of θ Jul-Aug : spatial distribution It is more difficult than in the Dic-Jan case to discern spatial patterns of the θ index The northern coast and obsevatories on the Iberian mountain range show above average θ values (less clustering) The contrary (more clustering) happens at the NE extreme (Catalonia)

26. EMMS & ECAM 2011, Berlin26 Verification of θ for AR4 downcaled models Jul-Aug Histogram of absolute deviation index of θ (on the y-axis nº of observatories, on the x-axis accumulated frequencies) Aver. absol. dev. Index: CA (1.6), US(1.8), RU(3.0) AL(1.6) JA (1.3) Aver. dev. Index: CA (-1.4), US(-1.6), RU(-2.9) AL(-1.2) JA (-0.5)

27. EMMS & ECAM 2011, Berlin27 Verification of θ for AR4 downcaled models Jul-Aug All the downscaled AR4 models show a bias towards excessive clustering (the Japanese little) in Jul-Aug (though less than in the three AR3 models)

28. EMMS & ECAM 2011, Berlin28 Verification of θ for AR4 downscaled models Jul-Aug: leading models At each observatory the downscaled model that leads the others in terms of absolute deviation index (with an asterisk when the difference to the others is >1.0)

29. EMMS & ECAM 2011, Berlin29 Verification of θ for downscaled models in Jul-Aug: leading models AR4+ 3 AR3 models Aver. dev. Index AR3: EC (-2.9) HA ( -4.1) CG (-2.4) Aver. dev. Index AR4: CA (-1.4), US(-1.6), RU(-2.9) AL(-1.2) JA (-0.5) There is a clear decrease in the amount of bias (excess clustering) in AR4 models with respect to AR3·

30. EMMS & ECAM 2011, Berlin30 END