Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT...

Estimating extremes in climate model data by the POT method with non-stationary threshold

Jan Kyselý, Jan Picek, Romana Beranová

Session 2: Nonstationary peaks-over-threshold (POT) method

11:30-11:50 J. Kyselý (TUL/IAP): Estimating extremes in climate model data by the POT method with nonstationary thresholdwhy? (overview, application)

11:50-12:10 J. Picek (TUL): Statistical aspects of the regression quantiles methodology in the POT analysishow? (details of the methodology)

12:10-12:30 M. Schindler (TUL): How to choose threshold in a POT model?justifying how (specific setting)

Most studies:

non-stationary block maxima (e.g. Kharin and Zwiers, 2005; Laurent and Parey, 2007)

& non-stationary POT models (e.g. Abaurrea et al., 2007; Parey et al., 2007; Yiou et al., 2006) with invariable (fixed) threshold to delimit extremes

the intensity of the Poisson process (i.e. the frequency of upcrossings) is time-dependent

Non-stationary extreme value models

This work:

different approach based on a time-dependent threshold estimated using quantile regression (Koenker and Basset, 1978)

Block maxima vs. peaks-over-threshold (POT) method

‘block maxima’(block size = usually 1

year/season)

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what is modelled: magnitude of extremes (GEV)

Block maxima vs. peaks-over-threshold (POT) method

‘block maxima’(block size = usually 1

year/season)

?

‘peaks-over-threshold’ (POT)

(threshold = ‘sufficiently high’ quantile)

what is modelled: 1) magnitude of excesses (GPD);

2) frequency of excesses (Poisson process)

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‘optimum threshold’: maximum information is used & events are

‘extreme’ and independent

what is modelled: magnitude of extremes (GEV)

Peaks-over-threshold (POT) method in non-stationary data

POT with stationary threshold & non-homogeneous Poisson

process(intensity depends on time)

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(e.g. Abaurrea et al., 2007; Parey et al., 2007; Yiou et al.,

2006)


POT with stationary threshold & non-homogeneous Poisson

process(intensity depends on time)

POT with non-stationary threshold & homogeneous

Poisson process(threshold depends on time)

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(e.g. Abaurrea et al., 2007; Parey et al., 2007; Yiou et al.,

2006)

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when significant trend is present in the data (e.g. warming on the long-term scale as in climate change simulations) & effective sample size is small

model with a time-dependent threshold and constant intensity (homogeneous Poisson process) superior to a model with a fixed threshold and time-dependent intensity (non-homogeneous Poisson process)

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POT with stationary threshold & non-homogeneous Poisson process

(intensity depends on time)

?

POT with non-stationary threshold & homogeneous Poisson process(threshold depends on time)

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a constant threshold in a POT model cannot be suitable over longer periods of time: there are either too few exceedances above the threshold in an earlier part of record (which enhances the variance of the estimated model), or too many exceedances towards the end of the examined period (which violates asymptotic properties of the model and leads to bias), or both the deficiencies are present in the examined samples of ‘extremes’


POT with stationary threshold & non-homogeneous Poisson process

(intensity depends on time)

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95% regression quantile

POT with non-stationary threshold & homogeneous Poisson process(threshold depends on time)

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independence of excesses: declustering (only maxima of

clusters taken)

non-stationary POT model:

• threshold modelled in terms of 95% quadratic regression quantiles

• models estimated over 2001-2100

data: coupled GCMs CM2.0, CM2.1, ECHAM5 over Europe; several SRES emission scenario simulations over 2001-2100 (A2, A1B, B1, A1FI)

comparison of stationary POT models over selected 30-yr time slices (2021-2050, 2071-2100) with non-stationary POT models

models’ performance evaluated in terms of 20-yr return values of TMAX

(20-yr return value in a non-stationary model defined analogously to the conventional meaning as a value occurring with a probability 1/20 in a given year)

Non-stationary POT method

several models for the Generalized Pareto distribution (GPD) of exceedances fitted and compared:

Model

Scale parameter modeled as

Shape parameter modeled as

Tested against

1 log ((t)) = 0 = 0 ---

2 log ((t)) = 0 + 1t = 0 1

3 log ((t)) = 0 + 1t (t) = 0 + 1t 2

4 log ((t)) = 0 + 1t + 2t2 = 0 2

5 log ((t)) = 0 + 1t + 2t2 (t) = 0 + 1t 4

pairs of models 1 to 5 compared in terms of likelihood ratio tests

in all examined GCM scenarios, the non-stationary extreme value model selected is model 2, i.e. model with a linear trend in logarithm of the scale parameter and constant shape parameter


Fig. 1: Projected changes in 20-yr return values of TMAX estimated for 30-yr time slices using the stationary POT model in 2071-2100 relative to the control period 1961-1990.

Stationary POT method

Fig. 2: Differences between 20-yr return values of TMAX estimated using the non-stationary POT model for year 2100 and the stationary POT model over 2071-2100.

Non-stationary vs. stationary POT method

Fig. 3: Differences between 20-yr return values of TMAX estimated using the non-stationary POT model for years 2100 and 2071.


spatial patterns of changes in high quantiles related to two sources:

1) changes in the location/ threshold (which capture shifts in the location of the GPD),

2) changes in the scale parameter of the GPD (related e.g. to interannual variability of extremes)

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pronounced warming in very high quantiles of TMAX over western and central Europe around 45-50N due to

shift in the location of the distribution of extremes (threshold)

BUT maxima in the spatial patterns of the changes in the 20-yr return values and the location/threshold do not correspond exactly to each other, the former being shifted northward in the A2, A1B and A1FI scenarios; this is because of additional changes in the scale parameter of the GPD, with a maximum warming around 50-55N and a cooling south of 45N

+

=

• The proposed non-stationary POT model with time-dependent threshold and a homogeneous Poisson process is

computationally straightforward does not violate assumptions of the extreme value analysis (unlike models with an invariable threshold and a non-homogeneous Poisson process used in some previous climate change studies, and/or stationary POT models)

• Two sources of increases in high quantiles are disaggregated using the proposed method: changes in the threshold (95% quantile) & changes in the scale parameter climatological interpretation

• Changes in the scale parameter of the distribution of extremes should not be ignored in climate change studies, as they to a large extent influence spatial patterns of extremes

• The method may be adjusted to include e.g. circulation indices as other covariates in addition to time

SUMMARY 1/2

Regression quantiles

• a useful concept in mathematical statistics, rarely used in environmental and climatological studies (mainly for the detection of trends)

• the most natural and intuitive solution to the problem of setting a (time-dependent) threshold in the POT analysis, corresponding to a high quantile of the distribution of the examined variable

• the results are not dependent on the particular choice of the threshold: if the 96% or 97% quantiles are used instead of the 95% quantile, the main findings remain unchanged

SUMMARY 2/2

Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT...

Technology

Transcript of Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT...