# Jos© .A. L“PEZ Climatological Techniques Unit AEMET Spain

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Verification of clustering properties of extreme daily temperatures in winter and summer using the extremal index in five downscaled climate modelsJos .A. LPEZClimatological Techniques UnitAEMETSpain

EMMS & ECAM 2011, Berlin

EMMS & ECAM 2011, Berlin

Outline Methodology Extremal Index : definition, exampleEstimation of Declustering procedureBootstrapping technique for C.I.Data, deviation index

Verification for Dec-Jan lowest daily temperatures Observed valuesStatistics of verification of for AR4 modelsSome results including AR3 models

Verification for Jul-Aug highest daily temperatures ....

...

EMMS & ECAM 2011, Berlin

EMMS & ECAM 2011, Berlin

The Extremal Index: Definition The Extremal Index 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 [0, 1] if for each > 0 there is a succession u(n) such that for n -> ,

a) n F* (u(n)) -> (mean n of exceedances = )

b) P ( M(n) u(n) ) -> exp (- )

If = 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

The Extremal Index : interexceedance timesThe 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

The Extremal Index : simulation with an ARMAXprocessThe ARMAX process is defined by:where de Zs are standard independent Fechet variables, i.e. prob(Z < x) = Exp (-1/x) This process has an extremal index: = 1 -

EMMS & ECAM 2011, Berlin

The Extremal Index : simulation for = 1 (Poisson)

EMMS & ECAM 2011, Berlin

The Extremal Index : simulation for = 0.8

EMMS & ECAM 2011, Berlin

The Extremal Index : simulation for = 0.5

EMMS & ECAM 2011, Berlin

The Extremal Index : simulation for = 0.2

EMMS & ECAM 2011, Berlin

Estimation of the Extremal IndexIf 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

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 .

More specifically the longest 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

Bootstrapping techniqueIn order to build confidence intervals for the 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 of this fictitious process

Repeat the above steps the desired n of times to build the confidence interval

EMMS & ECAM 2011, Berlin

Data and models usedPeriod: 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

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 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

Dec-Jan (occurrances below the 10th percentile of daily temperature)

EMMS & ECAM 2011, Berlin

Observed values of Dec-Jan (in percent)Median= 37Max = 57Min = 23

EMMS & ECAM 2011, Berlin

Observed values of Dec-Jan: values above (1) and below (-1) the median

EMMS & ECAM 2011, Berlin

Observed values of Dec-Jan : spatial distributionLowest values of (more clustering) in the NE and interior

Highest values of (less clustering) in the western half

EMMS & ECAM 2011, Berlin

Verification of for AR4 downscaled models Dec-JanHistogram of absolute deviation index of (on the y-axis n 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

Verification of for AR4 downscaled models Dec-Jan: leading modelsAt 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

Verification of for downscaled models in Dec-Jan: leading models AR4+ 3 AR3 modelsAver. 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 , whereas with AR3 only one shows little bias (the rest show more clustering)

EMMS & ECAM 2011, Berlin

Jul-Aug (occurrances above the 90th percentile of daily temperature)

EMMS & ECAM 2011, Berlin

Observed values of Jul-Aug (in percent)Median = 48Max = 81Min = 36

EMMS & ECAM 2011, Berlin

Observed values of Jul-Aug: values above (1) and below (-1) the median

EMMS & ECAM 2011, Berlin

Observed values of Jul-Aug : spatial distributionIt is more difficult than in the Dic-Jan case to discern spatial patterns of the index The northern coast and obsevatories on the Iberian mountain range show above average values (less clustering)The contrary (more clustering) happens at the NE extreme (Catalonia)

EMMS & ECAM 2011, Berlin

Verification of for AR4 downcaled models Jul-AugHistogram of absolute deviation index of (on the y-axis n 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

Verification of 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

Verification of for AR4 downscaled models Jul-Aug: leading modelsAt 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

Verification of for downscaled models in Jul-Aug: leading models AR4+ 3 AR3 modelsAver. 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

EMMS & ECAM 2011, Berlin

END

EMMS & ECAM 2011, Berlin

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