Estimating extremes in climate model databy the POT method with non-stationarythreshold   Jan Kyselý, Jan Picek, Romana Be...
Session 2: Nonstationary peaks-over-threshold (POT) method   11:30-11:50   J. Kyselý (TUL/IAP): Estimating extremes in cl...
Non-stationary extreme value modelsMost studies:non-stationary block maxima (e.g. Kharin and Zwiers, 2005; Laurent and Par...
Block maxima vs. peaks-over-threshold (POT) method                                                           ‘block maxima...
Block maxima vs. peaks-over-threshold (POT) method                                                           ‘block maxima...
Block maxima vs. peaks-over-threshold (POT) method                                                           ‘block maxima...
Peaks-over-threshold (POT) method in non-stationary data                                            POT with stationary th...
Peaks-over-threshold (POT) method in non-stationary data                                            POT with stationary th...
Peaks-over-threshold (POT) method in non-stationary datawhen significant trend is present in the data (e.g. warming on the...
Peaks-over-threshold (POT) method in non-stationary data                                               POT with stationary...
×                                                Peaks-over-threshold (POT) method in non-stationary data                 ...
Non-stationary POT methodnon-stationary POT model:     •     threshold modelled in terms of 95% quadratic regression quant...
Non-stationary POT methodseveral models for the Generalized Pareto distribution (GPD) of     exceedances fitted and compar...
Stationary POT method                        Fig. 1: Projected                        changes in 20-yr                    ...
Non-stationary vs. stationary POT method                                    Fig. 2: Differences                           ...
Non-stationary POT method                            Fig. 3: Differences                            between 20-yr return  ...
}    spatial patterns of changes in high    quantiles related to two sources:    1) changes in the location/    threshold ...
pronounced warming in very high    quantiles of TMAX over western and    central Europe around 45-50°N due to=   shift in ...
SUMMARY 1/2• The proposed non-stationary POT model with time-dependent thresholdand a homogeneous Poisson process is     ...
SUMMARY 2/2Regression quantiles• a useful concept in mathematical statistics, rarely used in environmental andclimatologic...
Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT method with non-stationary th...
Upcoming SlideShare
Loading in …5
×

Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT method with non-stationary threshold

828 views

Published on

Published in: Technology, Economy & Finance
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
828
On SlideShare
0
From Embeds
0
Number of Embeds
75
Actions
Shares
0
Downloads
6
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Jan Kyselý, Jan Picek, Romana Beranová: Estimating extremes in climate model data by the POT method with non-stationary threshold

  1. 1. Estimating extremes in climate model databy the POT method with non-stationarythreshold Jan Kyselý, Jan Picek, Romana Beranová
  2. 2. 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 threshold why? (overview, application) 11:50-12:10 J. Picek (TUL): Statistical aspects of the regression quantiles methodology in the POT analysis how? (details of the methodology) 12:10-12:30 M. Schindler (TUL): How to choose threshold in a POT model? justifying how (specific setting)
  3. 3. Non-stationary extreme value modelsMost 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-dependentThis work:different approach based on a time-dependent threshold estimated usingquantile regression (Koenker and Basset, 1978)
  4. 4. Block maxima vs. peaks-over-threshold (POT) method ‘block maxima’ (block size = usually 1 year/season) 40 35TMAX in July, AR(1) simulation [°C] 30 25 20 15 10 0 31 62 93 124 155 186 217 248 279 what is modelled: magnitude of extremes (GEV)
  5. 5. Block maxima vs. peaks-over-threshold (POT) method ‘block maxima’ (block size = usually 1 year/season) 40 35TMAX in July, AR(1) simulation [°C] 30 25 20 15 10 0 31 62 93 124 155 186 217 248 279 what is modelled: magnitude of extremes (GEV)
  6. 6. Block maxima vs. peaks-over-threshold (POT) method ‘block maxima’ ‘peaks-over-threshold’ (POT) (block size = usually 1 year/season) (threshold = ‘sufficiently high’ quantile) × ‘optimum threshold’: maximum 40 40 information is used & events are ‘extreme’ and independent 35 35TMAX in July, AR(1) simulation [°C] TMAX in July, AR(1) simulation [°C] 30 30 ? 25 25 20 20 15 15 10 10 0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279 what is modelled: magnitude of what is modelled: 1) magnitude of extremes (GEV) excesses (GPD); 2) frequency of excesses (Poisson process)
  7. 7. Peaks-over-threshold (POT) method in non-stationary data POT with stationary threshold & non-homogeneous Poisson process (intensity depends on time) 40 35TMAX in July, AR(1) simulation [°C] 30 25 20 15 10 0 31 62 93 124 155 186 217 248 279 (e.g. Abaurrea et al., 2007; Parey et al., 2007; Yiou et al., 2006)
  8. 8. Peaks-over-threshold (POT) method in non-stationary data POT with stationary threshold & POT with non-stationary threshold non-homogeneous Poisson process & homogeneous Poisson process (intensity depends on time) × (threshold depends on time) 40 40 35 35TMAX in July, AR(1) simulation [°C] TMAX in July, AR(1) simulation [°C] 30 30 25 25 20 20 15 15 10 10 0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279 (e.g. Abaurrea et al., 2007; Parey et al., 2007; Yiou et al., 2006)
  9. 9. Peaks-over-threshold (POT) method in non-stationary datawhen significant trend is present in the data (e.g. warming on the long-termscale 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 andtime-dependent intensity (non-homogeneous Poisson process) 40 TMAX in July, AR(1) simulation [°C] 35 30 25 20 15 10 0 31 62 93 124 155 186 217 248 279
  10. 10. Peaks-over-threshold (POT) method in non-stationary data POT with stationary threshold & non- POT with non-stationary threshold & homogeneous Poisson process homogeneous Poisson process (intensity depends on time) (threshold depends on time) × 40 40 35 35 ?TMAX in July, AR(1) simulation [°C] TMAX in July, AR(1) simulation [°C] 30 30 25 25 20 20 15 15 10 10 0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279 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’
  11. 11. × Peaks-over-threshold (POT) method in non-stationary data POT with stationary threshold & non- POT with non-stationary threshold & homogeneous Poisson process homogeneous Poisson process (intensity depends on time) (threshold depends on time) × 40 40 95% 35 35 regression quantileTMAX in July, AR(1) simulation [°C] TMAX in July, AR(1) simulation [°C] 30 30 25 25 20 20 15 15 10 10 0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279 independence of excesses: declustering (only maxima of clusters taken)
  12. 12. Non-stationary POT methodnon-stationary POT model: • threshold modelled in terms of 95% quadratic regression quantiles • models estimated over 2001-2100data: 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 modelsmodels’ 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)
  13. 13. Non-stationary POT methodseveral 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 (σ(t)) = σ0 ξ = ξ0 --- 2 log (σ(t)) = σ0 + σ1t ξ = ξ0 1 3 log (σ(t)) = σ0 + σ1t ξ(t) = ξ0 + ξ1t 2 4 log (σ(t)) = σ0 + σ1t + σ2t2 ξ = ξ0 2 5 log (σ(t)) = σ0 + σ1t + σ2t2 ξ(t) = ξ0 + ξ1t 4pairs of models 1 to 5 compared in terms of likelihood ratio testsin 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
  14. 14. Stationary POT method 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.
  15. 15. Non-stationary vs. 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.
  16. 16. Non-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.
  17. 17. } 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)
  18. 18. pronounced warming in very high quantiles of TMAX over western and central Europe around 45-50°N 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 50-55°N and a cooling south of 45°N
  19. 19. SUMMARY 1/2• The proposed non-stationary POT model with time-dependent thresholdand 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 theproposed method: changes in the threshold (95% quantile) & changes in the scaleparameter → climatological interpretation• Changes in the scale parameter of the distribution of extremes should not beignored in climate change studies, as they to a large extent influence spatial patternsof extremes• The method may be adjusted to include e.g. circulation indices as othercovariates in addition to time
  20. 20. SUMMARY 2/2Regression quantiles• a useful concept in mathematical statistics, rarely used in environmental andclimatological 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 thedistribution 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 remainunchanged

×