Some advanced methods in extreme value analysis
description
Transcript of Some advanced methods in extreme value analysis
![Page 1: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/1.jpg)
Some advanced methods in extreme
value analysis
Peter Guttorp
NR and UW
![Page 2: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/2.jpg)
Outline
Nonstationary models
Extreme dependence (Cooley)
When a POT approach is better than a block max approach (Wehner and Paciorek)
A Bayesian space-time model
An extreme climate event
![Page 3: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/3.jpg)
Trends
![Page 4: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/4.jpg)
What do we mean by trends in extreme
values?
![Page 5: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/5.jpg)
Time-dependent location estimates
Stockholm data(Guttorp and Xu, Environmetrics 2011)
![Page 6: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/6.jpg)
Simple model
Model Estimate -LLR
Fixed μ, all -16.6 706.0
Fixed μ, early -18.1336.8
Fixed μ, late -14.2 350.2
Early + late 687.0
Linear model in μ (-18.9,-14.0) 687.9
A linear change in mean value for annual minima seems a good model.
Modal prediction for 2050: -11.5°C
2100: -10.5°C
![Page 7: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/7.jpg)
Extreme dependence
![Page 8: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/8.jpg)
Measuring dependence
![Page 9: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/9.jpg)
Storm surges and wave heights
Risk region
Most of these data are not extreme!
![Page 10: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/10.jpg)
![Page 11: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/11.jpg)
Describing “tail dependence”
![Page 12: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/12.jpg)
Estimating H
Transform margins to Frechet; keep largest 150 obs (95th %ile)
![Page 13: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/13.jpg)
Density of H
![Page 14: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/14.jpg)
Probability of risk region
![Page 15: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/15.jpg)
Block extremes vs peaks over threshold
![Page 16: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/16.jpg)
Climate model output
Daily precip from 450 year control run of climate model (long stationary series)
Fit GEV to seasonal max
![Page 17: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/17.jpg)
POT analysis
99th percentile
![Page 18: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/18.jpg)
Why are the two analyses so different?
Desert regions: large amount of lack of precipitation. GEV therefore will include many zeros, while GPD only uses data where high values are actually recorded.
![Page 19: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/19.jpg)
Space-time data
![Page 20: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/20.jpg)
Some temperature data
SMHI synoptic stations in south central Sweden, 1961-2008
![Page 21: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/21.jpg)
Annual minimum temperatures and
rough trends
![Page 22: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/22.jpg)
Location slope vs latitude
![Page 23: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/23.jpg)
Dependence between stations
Sveg Malung Karlstad
Sundsvall 19 12 13
Sveg 25 11
Malung 10
Common coldest day in 48 years5 common to all 4 northern stations
![Page 24: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/24.jpg)
Max stable processes
Independent processes Yi(x)
(e.g. space-time processes with weak temporal dependence)
A difficulty is that we cannot compute the joint distribution of more than 2-3 locations. So no likelihood.
![Page 25: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/25.jpg)
Spatial model
where
Allows borrowing estimation strength from other sites
Can include more sites in analysis
![Page 26: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/26.jpg)
Trend estimates
Posterior probability of slope ≤ 0 is very small everywhere
![Page 27: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/27.jpg)
Spatial structure of parameters
![Page 28: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/28.jpg)
Location slope vs latitude
![Page 29: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/29.jpg)
Prediction Borlänge
![Page 30: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/30.jpg)
A different kindof extreme event
![Page 31: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/31.jpg)
What made Gudrun so destructive?
Hurricane Gudrun, Sweden, January 2005. 15 deaths, 340 000 households without power up to 4 weeks.
![Page 32: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/32.jpg)
Consequences
75 million cubic feet of forest fell (normal annual production)
Wind speeds up to 40 m/s
Forest damage due to large amounts of precipitation, temperatures around 0°C, high winds
Much work on multivariate extreme asymptotics needs all components to be extreme–here temperature is not.
Want (limiting) conditional distribution of wind given temperature and previous precipitation
![Page 33: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/33.jpg)
The Heffernan-Tawn approach
Assume we can find normalizing factors a-i(yi), b-i(yi) so that
where G-i has non-degenerate margins.
Pick aji(yi) so that
and
where hji is the conditional hazard function of Yj given Yi=yi.
![Page 34: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/34.jpg)
Asymptotic properties
Let .
Then given Yi>u,
Yi - u and Z-i are asymptotically independent as . Furthermore
Fitting using observed data; forecasting using regional model output
![Page 35: Some advanced methods in extreme value analysis](https://reader031.fdocuments.in/reader031/viewer/2022020721/56814dba550346895dbb0fce/html5/thumbnails/35.jpg)
Some R software
ExtRemes
ismev
evlr
SpatialExtremes