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José .A. LÓPEZ Climatological Techniques Unit AEMET Spain

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. Outline. Methodology

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José .A. LÓPEZ Climatological Techniques Unit AEMET Spain

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  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. 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 indexθε [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 (- θτ) • 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) EMMS & ECAM 2011, Berlin

  4. 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. EMMS & ECAM 2011, Berlin

  5. The Extremal Index : simulation with an ARMAX process 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 EMMS & ECAM 2011, Berlin

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

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

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

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

  10. Estimation of the Extremal Index If the Ti 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) . EMMS & ECAM 2011, Berlin

  11. Declustering procedure 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 EMMS & ECAM 2011, Berlin

  12. Bootstrapping technique In order to build confidence intervals for the θ of a series, a “bootstrapping” technique was used: • 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 • Compute the θ of this fictitious process • Repeat the above steps the desired nº of times to build the confidence interval EMMS & ECAM 2011, Berlin

  13. 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 EMMS & ECAM 2011, Berlin

  14. Verification of the Extremal Index in extreme temperature for downscaled climate models • 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 EMMS & ECAM 2011, Berlin

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

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

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

  18. 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 EMMS & ECAM 2011, Berlin

  19. 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) EMMS & ECAM 2011, Berlin

  20. 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) EMMS & ECAM 2011, Berlin

  21. Verification of θ for downscaled models in Dec-Jan: leading models AR4+ 3 AR3 models 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) Aver. dev. IndexAR3 : 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) EMMS & ECAM 2011, Berlin

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

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

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

  25. 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) EMMS & ECAM 2011, Berlin

  26. 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) EMMS & ECAM 2011, Berlin

  27. 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) EMMS & ECAM 2011, Berlin

  28. 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) EMMS & ECAM 2011, Berlin

  29. Verification of θ for downscaled models in Jul-Aug: leading models AR4+ 3 AR3 models There is a clear decrease in the amount of bias (excess clustering) in AR4 models with respect to AR3· 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) EMMS & ECAM 2011, Berlin

  30. END EMMS & ECAM 2011, Berlin

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