Verification of clustering properties of extreme daily temperatures in winter and summer using the...
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Transcript of Verification of clustering properties of extreme daily temperatures in winter and summer using the...
Verification of clustering properties of extreme daily temperatures in winter and summer using the extremal
index in five downscaled climate models
José .A. LÓPEZClimatological Techniques Unit
AEMETSpain
EMMS & ECAM 2011, Berlin
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
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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)
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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.
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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
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The Extremal Index : simulation for θ = 1 (Poisson)
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The Extremal Index : simulation for θ = 0.8
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The Extremal Index : simulation for θ = 0.5
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The Extremal Index : simulation for θ = 0.2
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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)
.
Estimation of the Extremal Index
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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
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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
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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
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• 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
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Dec-Jan (occurrances below the 10th percentile of
daily temperature)
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Observed values of θ Dec-Jan (in percent)
2938
26304138
4223
41
37
41
34
27 344842
3742
52
3740
35 57
4938
35
3733
45
3428
3031
34
36
Median= 37
Max = 57
Min = 23
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Observed values of θ Dec-Jan: values above (1) and below (-1) the median
11
1111
11 1
0
1
1
1 111
01
1
01
1 1
11
1
01
1
11
11
1
1
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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
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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)
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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)
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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)
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Jul-Aug (occurrances above the 90th percentile of
daily temperature)
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Observed values of θ Jul-Aug (in percent)
4237
65415561
3640 56
60
48
49
44 515142
4244
47
4446
57 81
5448
52
5657
57
5650
4039
42
47
Median = 48
Max = 81
Min = 36
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Observed values of θ Jul-Aug: values above (1) and below (-1) the median
11
1111
11 1
1
0
1
1 111
11
1
11
1 1
10
1
11
1
11
11
1
1
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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)
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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)
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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)
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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)
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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·
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END