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Some advanced methods in extreme value analysis. Peter Guttorp NR and UW. 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. Trends. - PowerPoint PPT Presentation
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Some advanced methods in extreme
value analysis
Peter Guttorp
NR and UW
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
Trends
What do we mean by trends in extreme
values?
Time-dependent location estimates
Stockholm data(Guttorp and Xu, Environmetrics 2011)
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
Extreme dependence
Measuring dependence
Storm surges and wave heights
Risk region
Most of these data are not extreme!
Describing “tail dependence”
Estimating H
Transform margins to Frechet; keep largest 150 obs (95th %ile)
Density of H
Probability of risk region
Block extremes vs peaks over threshold
Climate model output
Daily precip from 450 year control run of climate model (long stationary series)
Fit GEV to seasonal max
POT analysis
99th percentile
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.
Space-time data
Some temperature data
SMHI synoptic stations in south central Sweden, 1961-2008
Annual minimum temperatures and
rough trends
Location slope vs latitude
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
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.
Spatial model
where
Allows borrowing estimation strength from other sites
Can include more sites in analysis
Trend estimates
Posterior probability of slope ≤ 0 is very small everywhere
Spatial structure of parameters
Location slope vs latitude
Prediction Borlänge
A different kindof extreme event
What made Gudrun so destructive?
Hurricane Gudrun, Sweden, January 2005. 15 deaths, 340 000 households without power up to 4 weeks.
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
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.
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
Some R software
ExtRemes
ismev
evlr
SpatialExtremes