Covariate-dependent modeling of                     extreme events by nonstationary                      Peaks Over Thresh...
Motivation, I     We all love the extreme value theory (EVT) applied to the analysis of     climatic hazard: its elegant, ...
Motivation, I     We all love the extreme value theory (EVT) applied to the analysis of     climatic hazard: its elegant, ...
Motivation, I     We all love the extreme value theory (EVT) applied to the analysis of     climatic hazard: its elegant, ...
Motivation, II     Most studies up to now focused on identifying temporal trends in the     occurrence of extreme events, ...
Motivation, II     Most studies up to now focused on identifying temporal trends in the     occurrence of extreme events, ...
SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection in...
SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection in...
SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection in...
SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection in...
Short review of NSPOT analysis, I                                200                                150         Intensity ...
Short review of NSPOT analysis, II                                200                                150         Intensity...
Short review of NSPOT analysis, IIIStationary POT: assuming independent inter-arrival times, the POT datafollows a General...
Short review of NSPOT analysis, IVApproaches for assessing non-stationarity in POT modeling:     Split-sample approach (Li...
Short review of NSPOT analysis, GPar distribution                    Return period curves,                                ...
Short review of NSPOT analysis, VI  2110                                                                                  ...
Short review of NSPOT analysis, VIIStationary POT:                                                                      −1...
Short review of NSPOT analysis, VIIISome examples of NSPOT analysis of climatic variables:     Time dependence of T and P ...
Teleconnections aecting precipitation in IP, IThe North Atlantic Oscillation (NAO).    (santiago.begueria@csic.es)        ...
Teleconnections aecting precipitation in IP, IIThe North Atlantic Oscillation (NAO).    (santiago.begueria@csic.es)       ...
Teleconnections aecting precipitation in IP, IIIThe Mediterranean Oscillation (MO, Palutikof 2003).    (santiago.begueria@...
Teleconnections aecting precipitation in IP, IVThe Mediterranean Oscillation (MO, Palutikof 2003).    (santiago.begueria@c...
Teleconnections aecting precipitation in IP, VThe Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2...
Teleconnections aecting precipitation in IP, VIThe Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins ...
Teleconnections aecting precipitation in IP, VII                  1.5                                                     ...
Dataset, I                                                                         Stations network                  48000...
Dataset, IITeleconnection indices (Reykjavik, Padova, Lod and Gibraltar). Sources: http://www.cru.uea.ac.uk,http://www.ub....
Dataset, III                       1                       0          NAOi                       −1                       ...
Dataset, IV                       1                       0          NAOi                       −1                       −...
Dataset, V                       1                       0          NAOi                       −1                       −2...
Dataset, VI                       1                       0          NAOi                       −1                       −...
Analysis, I                x |X  x0 ) = 1 − λ 1 + κ x −x0                                                                −...
Analysis, IIR, package ismev (Stuart Coles, ported to R by Alec Stephenson)....m0  gpd.t(xdat=dat, threshold=u, npy=rate)....
Analysis, III                                Histogram of tel$naoi                                                     His...
Example: Valencia, I                                   Location of station: VALENCIA, X8416    4800000    4600000    44000...
Example: Valencia, II                                        Threshold (u = q0.9 = 23.5, n = 229)                         ...
Example: Valencia, III29)                                                                         Stationary model (AIC=18...
Example: Valencia, IV                                      200                                                  q         ...
Example: Valencia, V                                      35                                      30                    Sc...
0.0        0.2             0.4        0.6       0.8         1.0Example: Valencia, VI                                      ...
1.0                                                  −2              −1           0           1        2Example: Valencia,...
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis
Upcoming SlideShare
Loading in …5
×

Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis

445 views
355 views

Published on

Published in: Technology, Travel
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
445
On SlideShare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
2
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Santiágo Beguería: Covariate-dependent modeling of extreme events by non-stationary Peaks Over Threshold analysis

  1. 1. Covariate-dependent modeling of extreme events by nonstationary Peaks Over Threshold analysis A review and a case study in Spain Santiago Beguería santiago.begueria@csic.es Estación Experimental de Aula Dei, Consejo Superior de Investigaciones Cientícas (EEADCSIC), Zaragoza, España Liberec, February 17th 2012(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 1 / 52
  2. 2. Motivation, I We all love the extreme value theory (EVT) applied to the analysis of climatic hazard: its elegant, simple, and provides useful and understandable results in terms of magnitude / frequency curves. The stationarity assumption, though, was an important limitation of the EVT. Relatively recent extensions of the EVT allow for non-stationary analysis (Coles, 2001), and an increasing number of authors are exploring their possibilities for the analysis of climatic hazard. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52
  3. 3. Motivation, I We all love the extreme value theory (EVT) applied to the analysis of climatic hazard: its elegant, simple, and provides useful and understandable results in terms of magnitude / frequency curves. The stationarity assumption, though, was an important limitation of the EVT. Relatively recent extensions of the EVT allow for non-stationary analysis (Coles, 2001), and an increasing number of authors are exploring their possibilities for the analysis of climatic hazard. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52
  4. 4. Motivation, I We all love the extreme value theory (EVT) applied to the analysis of climatic hazard: its elegant, simple, and provides useful and understandable results in terms of magnitude / frequency curves. The stationarity assumption, though, was an important limitation of the EVT. Relatively recent extensions of the EVT allow for non-stationary analysis (Coles, 2001), and an increasing number of authors are exploring their possibilities for the analysis of climatic hazard. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52
  5. 5. Motivation, II Most studies up to now focused on identifying temporal trends in the occurrence of extreme events, i.e. making time a covariate. In the last few years other covariates with an expected inuence on the occurrence of extreme events are being used, too high relevance for the statistical downscaling of reanalysis or model data, which typically cannot be used for local impact studio. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 3 / 52
  6. 6. Motivation, II Most studies up to now focused on identifying temporal trends in the occurrence of extreme events, i.e. making time a covariate. In the last few years other covariates with an expected inuence on the occurrence of extreme events are being used, too high relevance for the statistical downscaling of reanalysis or model data, which typically cannot be used for local impact studio. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 3 / 52
  7. 7. SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-stationary EVT techniques. The talk is organized as follows: A review of non-stationary Peaks Over Threshold analysis Case study: relationship between teleconnection indices and extreme rainfall events in Spain Conclusions and future work (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52
  8. 8. SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-stationary EVT techniques. The talk is organized as follows: A review of non-stationary Peaks Over Threshold analysis Case study: relationship between teleconnection indices and extreme rainfall events in Spain Conclusions and future work (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52
  9. 9. SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-stationary EVT techniques. The talk is organized as follows: A review of non-stationary Peaks Over Threshold analysis Case study: relationship between teleconnection indices and extreme rainfall events in Spain Conclusions and future work (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52
  10. 10. SummaryIn this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-stationary EVT techniques. The talk is organized as follows: A review of non-stationary Peaks Over Threshold analysis Case study: relationship between teleconnection indices and extreme rainfall events in Spain Conclusions and future work (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52
  11. 11. Short review of NSPOT analysis, I 200 150 Intensity (mm day-1) 100 50 0 1950 1960 1970 1980 1990 2000 2010 Time (n = 266)Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0 (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 5 / 52
  12. 12. Short review of NSPOT analysis, II 200 150 Intensity (mm day-1) 100 50 0 1950 1960 1970 1980 1990 2000 2010 Time (n = 266)Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0 (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 6 / 52
  13. 13. Short review of NSPOT analysis, IIIStationary POT: assuming independent inter-arrival times, the POT datafollows a Generalized-Pareto distribution.Probability of exceedance: −1/κ x − x0 P (X x |X x0 ) = 1 − λ 1 + κ (1) αQuantile corresponding to a return period T: κ α 1 XT = x0 + 1− (2) κ λT(beware of alternative conventions: x0 = u , α = σ, κ = ξ ) (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 7 / 52
  14. 14. Short review of NSPOT analysis, IVApproaches for assessing non-stationarity in POT modeling: Split-sample approach (Li et al., 2005) Moving kernel (Hall and Tajvidi, 2000) Non-stationary POT (NSPOT) modeling (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 8 / 52
  15. 15. Short review of NSPOT analysis, GPar distribution Return period curves, V 2500 NAO− 2000 1500 daily EI NAO+ 1000 500 0 1 2 5 10 20 50 100 return period (years)Split-sample approach: independent models for positive and negative phases of NAO (Angulo et al., 2011). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 9 / 52
  16. 16. Short review of NSPOT analysis, VI 2110 ´ S. BEGUERIA et al. X287 X164 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 1930 1950 1970 1990 2010 1930 1950 1970 1990 2010 X9198 X2234 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1930 1950 1970 1990 2010 1930 1950 1970 1990 2010 Figure 6. Model uncertainty: time variation of the 10-year return period event intensity standard error (mm by a nonparametric NSPOT d−1 ) model (dots), a linear NSPOT model (solid line), a high order polynomial NSPOT model (short-dashed line) and a stationary POT modelMoving kernel approach: time variability in the stations. Location of the stations a moving Figure 1. of the previous 20 years of (long-dashed line) in four P10 quantile, based on is shown in windowdata (Beguería et al., 2011). by more than 50% (e.g. series X164). The time evolution and the most parsimonious (i.e. with the lowest polyno- of q10 depended on the evolution (santiago.begueria@csic.es) of both α (scale) and mial order) was chosen. Following this approach, statis- 10 / 52 NSPOT with covariates Liberec, 17/02/2012
  17. 17. Short review of NSPOT analysis, VIIStationary POT: −1/κ x − x0 P (X x |X x0 ) = 1 − λ 1 + κ αNon-stationary POT: −1/κ(c ) x − x0 (c ) P (X x |X x0 , C ) = 1 − λ(c ) 1 + κ(c ) (3) α(c ) (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 11 / 52
  18. 18. Short review of NSPOT analysis, VIIISome examples of NSPOT analysis of climatic variables: Time dependence of T and P (Smith, 1999) Nogaj et al. (2006) time trends of T extremes over the NA region Laurent and Parey (2007), Parey et al. (2007), T extremes in France Méndez et al. (2006), trends and seasonality of POT wave height Yiou et al. (2006) trends of POT discharge in the Czech Republic Abaurrea et al. (2007) trends of POT T in the IP Acero et al. (2011), Beguería et al. (2011), trends in POT P, IP Friederichs (2010), Kallache et al. (2011), downscaling of POT P based on reanalysis / GCM data Tramblay et al. (2012), covariation between POT P extremes and atmospheric covariates, SE France (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 12 / 52
  19. 19. Teleconnections aecting precipitation in IP, IThe North Atlantic Oscillation (NAO). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 13 / 52
  20. 20. Teleconnections aecting precipitation in IP, IIThe North Atlantic Oscillation (NAO). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 14 / 52
  21. 21. Teleconnections aecting precipitation in IP, IIIThe Mediterranean Oscillation (MO, Palutikof 2003). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 15 / 52
  22. 22. Teleconnections aecting precipitation in IP, IVThe Mediterranean Oscillation (MO, Palutikof 2003). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 16 / 52
  23. 23. Teleconnections aecting precipitation in IP, VThe Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 17 / 52
  24. 24. Teleconnections aecting precipitation in IP, VIThe Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 18 / 52
  25. 25. Teleconnections aecting precipitation in IP, VII 1.5 1.5 q q q q q q q q q q q q q q 1.0 1.0 qq qq q qq qqq qq q q q q q q qq q qq q q qq q q q q q q q q q q qq q q q qq q q q q q q qq q q q q q q q qqq q q q q q q q qq q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q qq q qq q q qq q q qq q qq q q q q qq q 0.5 0.5 q q q qqq q q q q q q q q q q qq q qqqq qqq q qq q q q qq q q qq qq qq q q q qqqq q qqq q q q q qqqq q q q q qq q q q qq qq q q q q q qq q q q q qq qq q q qq q qqqqq qq qqq q q qqq q q q q q qq q q qq q qq q q qqqq qq q qqqq q q q qq q q q q q qq q q qqq q q qq q q q q qq q q qqq q q qq q q qqqq q q qq q q q qq q q qqq qq qq q qqq q q q q q q q qq q q q qq q q qq q q q qqq q qqqq q q qqq qq q qqq q q q q q q q q q q q q q q qqq qq q qq qqqq qq q q q q q qq q q qqqq q qqqq q q qq q q q q q q q qq qq qqqqq qq qqqqq q qq q q qq qqqqqq q q qq q qq q q q q qq qq qq qqq q q qq q qq q q q q q q qqqqq q q q q q q qqqq qq qqq q q qqq q qq qq q q q NAOi NAOi q qqqqqqq qq qqqq q q q q q q q q qq qq qq q q q qq qqqqqqqqqqqq q q q q q q q qq qqqq qqqq qqqqqqq qq q qq q qqq qq qq q q q q q q q q q q qqqqqq q qqq q q q q q q qq q 0.0 0.0 q qq q q q q q qq q q q q q q qq qq qq q q q q q q qqq qqqq q qqqq q qqq qq q q q q q q qq qq q qqqqqqq q qq qq q qq q q qq q q q q q q q q q q qq q q q q qqq qq qqqq qq qq qqqq qqq q q q qq q qq q q q q q q qq q qq qq q q q q q qqqqqqqqq q q q qq qqq q q qqqq q q q qqq q qq q q q q qq qq q q q qqqq qq q q qq q qq q q q q qqq qq q q qqq q q qq q q qq q q q q q qq q qq q qq q q q q q q qq q qqqqqq q qq q q qqq q q qq q qq q q q qq qq q q qqqq q q q q q q q qq qqqq qq q qq qq q q q q q qqq q q qq q q q q q q qq q q q q q q qq q qq qq q q q q q q qq qqq q q q q qq q q q qq q q q q q q q q q qq qq qqq q q q q qq q q qq q −0.5 −0.5 q qq q q q q q qqq q qq q q q qqq q q qqq qqqqqqq qq qqq q q q q q q qqqq q q qq q q qqqqq qq q q q q q qq q q q q qq q qq q q q qq qq q q qq q q qq q q q q q q qq qq qqq q qq q q q qq q q qq q qqqqq qq q q q qq q q q q qq qqq q q q q q qq q q q q q q q q q qq q qq q q q q qq q q qq q qq q q q q q q qq q qq q qqqqqq q q q q qqq q qq q q q q q q qq q q q q qqq q qq q q q q q q qq q q q q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q qq q −1.5 −1.5 r2=0.21 r2=0.005 q q −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 MOi WEMOi 1.5 q 1.0 q q q qqq q qq q qq q q q q qq q qq qq q q q q q qqq q q q q qq q q qq qq q q q qq q q q qq q q q q q qq q q q q qq q 0.5 q qq q qq qqqq q qq q qq q q qqqq q q qq q q q qq q q q qq q q q q qqq q q qqq q q q qq qq q q q q q q q qq qq qq qq qqq q q q qq qq q q q q q q qq q q qqq qq q q qq q q qq qq qqq q q qqq q q qq q q qq qqq qq q qq q qqq qq q qq q q q q q q qqq q q qq WEMOi q q q q qqqq qqqqqq qq qqqq q q qqq q q q qq qqqq qq qqq qqq q qqq q q q q q q q q q q qqq q qq q q q qq qq qqqq qq qqqq q q q q q q q qq qq q q qq q q q qq q q qqqq qq qqqqqq qqqqq q qq q qq qq q q qq q qq q qqqq qq qqq q qq 0.0 q q q q q q q q q qq q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q qq qq qq q qq qqq qqqq q q q q qqqq q q q q q q qqqq qqqq q qq q q qq qq q qq q q qq qq qq q qq qq qqqq qqq q q q qqqqq qq qqqqqqqqqq q q q q q qqq q q q q q qq q q q q q qq qq q q q qq q q q q q qq q qqqq qq qq q q q qq q q q q qqq q q q q q q q q qqq q qqqq q q qq qq q qqq q q −0.5 q q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q q qqq q q q q q q q q q q qq q q −1.5 r2=0.047 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 MOiCorrelations between teleconnection indices. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 19 / 52
  26. 26. Dataset, I Stations network 4800000 q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q 4600000 q q q q q q q q q q q q q q qq q q q q 4400000 q q q q q qq qq q q qq q q q q q 4200000 q q q q q q qq q q qq q q q q q q 4000000 q q q q106 stations, 58 daily precipitation 2e+05 reconstructed for6e+05 period 8e+05 0e+00 series 4e+05 the 1950-2009 1e+06 (source: AEMET). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 20 / 52
  27. 27. Dataset, IITeleconnection indices (Reykjavik, Padova, Lod and Gibraltar). Sources: http://www.cru.uea.ac.uk,http://www.ub.es. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 21 / 52
  28. 28. Dataset, III 1 0 NAOi −1 −2 0 20 40 60 80 2 1 WEMOi 0 −1 −2 0 20 40 60 80 15 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001)Declustering: intensity and magnitude series and associated teleconnection indices. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 22 / 52
  29. 29. Dataset, IV 1 0 NAOi −1 −2 0 20 40 60 80 2 1 WEMOi 0 −1 −2 0 20 40 60 80 20 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001)Declustering: intensity and magnitude series and associated teleconnection indices. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 23 / 52
  30. 30. Dataset, V 1 0 NAOi −1 −2 0 20 40 60 80 20 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001) 2 1 WEMOi 0 −1 −2 0 20 40 60 80 20 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001)Declustering: intensity and magnitude series and associated teleconnection indices. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 24 / 52
  31. 31. Dataset, VI 1 0 NAOi −1 −2 0 20 40 60 80 15 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001) 2 1 WEMOi 0 −1 −2 0 20 40 60 80 15 P (mm/day) 10 5 0 0 20 40 60 80 Time (days since 01/11/2001)Declustering: intensity and magnitude series and associated teleconnection indices. (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 25 / 52
  32. 32. Analysis, I x |X x0 ) = 1 − λ 1 + κ x −x0 −1/κM0: P (X α −1/κM1: P (X x |X x0 , C ) = 1 − λ 1 + κ x −x0 (c ) α(c ) x0 = β0 + βi c (4) α = γ0 γic (5) κ = δ (6) λ = ε (7)Likelihood ratio test: D = − 2 ( 1 (M1 ) − 0( M0 )) (8)distributed according to χ2 (with d.f. k k = 4). (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 26 / 52
  33. 33. Analysis, IIR, package ismev (Stuart Coles, ported to R by Alec Stephenson)....m0 gpd.t(xdat=dat, threshold=u, npy=rate)...uu predict(lm(v∼cdat))m1 gpd.t(xdat=dat, threshold=uu, npy=rate, ydat=cdat,sigl=1, siglink=exp) (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 27 / 52
  34. 34. Analysis, III Histogram of tel$naoi Histogram of pnorm(tel$naoi) 4000 1000 1200 3000 800Frequency Frequency 2000 600 400 1000 200 0 0 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 tel$naoi pnorm(tel$naoi)Covariates: NAOi and pnorm(NAOi) (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 28 / 52
  35. 35. Example: Valencia, I Location of station: VALENCIA, X8416 4800000 4600000 4400000 q 4200000 4000000Spatial location 0e+00 5e+05 1e+06 (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 29 / 52
  36. 36. Example: Valencia, II Threshold (u = q0.9 = 23.5, n = 229) 50 100 45 50 Return period (years) 40 25 Mean Excess 35 10 30 5 25 2 20 1 q q q q qqq qq q q qq q q q q q 10 20 30 40 50 60 70 80 q q q q q 50 q q q q q q q qq q q u q q q qStationary model: xed threshold (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 30 / 52
  37. 37. Example: Valencia, III29) Stationary model (AIC=1875) 100 q 50 Return period (years) q 25 q q q 10 q q q q q q q q 5 q q q qq q q q q q q qqq q 2 q q qq q q qq q qq q qq qq q qq 1 q q q q qqq qq q q qq qq q q q 70 80 q q q q q 50 100 150 200 250 qq q q q q q q q q q q q q q Intensity (mm day−1) Intensity (mm) Intensity (days) Stationary model: quantile plot (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 31 / 52
  38. 38. Example: Valencia, IV 200 q 35 q 30 150 q q Intensity (mm day−1) Scale parameter q q 25 q q q q q q q 100 qq qq qq q 20 q qq q q q qq q q q q q q q qq q q q q q q q qq q qq qq q q q qq q q q 50 qq q q 15 q q q q q q q q q q q qq q q q qq qqqq q q qq q q q q q q qq qq q q q q q q q q q q q q qqq qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q qq q q q q q qqqq qq q q q q q q q q qqq q q qq q q qqqq qq q q q q qqq q q q qq q q q q qq q q q q qq q q q q qq q qq q q q q qq q q q qq q qqq q q q q q q qq qqqq qqq q q q qqq q qq qq q qqq qq q q q qqq q qqq qq q q q q qq qq q qq q q qq q q q q qq q qq q qq q q q q q q q q q q qq qq q q q qqq q q q q qq q 10 q qq qq qqqqq qq q qqq q q q q q qq qq qq qqq qqq q qqq q qq qqq qqq q q q q qq qq q qq qq q q q q q q q qq q q qqq qq q qqq q q q q q qq q q qq q q q q q q qq q q q qqq q q q q qqq q q q qqqqqqqqqqqqq qqqqqqqqqq qqqqqq qqqqqqqqqq q qqqqqqqqqqq qqqq q qq qqq q qq qq q qq q q qqq qqqq q qqq q qqqq q q qqqq q q qqqqqqq qqqq qqqqq qqqqq qqqqqq qqq q qqq q q q qqqq qqqq qqqqq qq q q q q q q q q qqq q q q q qqqqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqqqqqq q qqqqqqqqqqqqq qqq q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qq qqq q qq q qqqqqq q q q qq qq qq q qq q q q qqqqq q q qqqqqqqq qq qqq q qq qqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq q qq qqqqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqqqqqqqq qq q q q q q q q q q q q qq q q q q q qq q q q q q q q qq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqq qq q qq q q qq qq q q q q q q qq q qqqqq qq q qq q q qqqqqqqqq qqqq qqqq q qqqq qqqqqqqq qq qqq qqqqqqq q q qqqqqqq qq qq q q qq q qq q qqqq qqq q qqq qq q qqq qqqqq qq q qqq q q qq qq q 0 0.0 0.2 0.4 0.6 0.8 1.0 WEMOi (n = 200) 100Non-stationary model: threshold model 100 2 1 0.5 0 −0.5 −1 −2 50 50 (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 32 / 52 s) s)
  39. 39. Example: Valencia, V 35 30 Scale parameter 25 20 15 q q q qq q q qq q q q q qqq q 10q qqq qqq qqq qq q q q qq q qq qq qq qqqqqqqqqq qqqq q qq qqq q q qq qq q q qq q qqqqqqqqqqqqqqqqqqqqq q qqqqqqqq qqqq qqqqq q q qqqqqqqqqqq q qq q qqq qqqqqqqqq q q q qqqqqqqqqqqqq q qq 1.0 −2 −1 0 1 2 WEMOi (AIC=1569*) 100 Non-stationary model: scale parameter model −2 220 210 200 50 190 180 (santiago.begueria@csic.es) 0 17 160 NSPOT with covariates Liberec, 17/02/2012 33 / 52 )
  40. 40. 0.0 0.2 0.4 0.6 0.8 1.0Example: Valencia, VI WEMOi (n = 200) 100 100 2 1 0.5 0 −0.5 −1 −2 50 50 Return period (years) Return period (years) 25 25 10 10 5 5 2 2 1 1 50 100 150 200 250 300 Intensity (mm day−1)Non-stationary model: quantile plot (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 34 / 52
  41. 41. 1.0 −2 −1 0 1 2Example: Valencia, VII WEMOi (AIC=1569*) 100 −2 220 210 200 50 190 180 170 160 Return period (years) 25 150 140 130 120 10 110 100 90 5 80 70 60 50 2 40 30 1 300 −2 −1 0 1 2 WEMOiNon-stationary model: quantile plot (santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 35 / 52

×