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 ModelEstimate-LLR Fixed  all Fixed , early Fixed , late Early + late687.0 Linear model in  (-18.9,-14.0) 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 (95 th %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 99 th 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,

Annual minimum temperatures and rough trends

Location slope vs latitude

Dependence between stations SvegMalungKarlstad Sundsvall Sveg2511 Malung10 Common coldest day in 48 years 5 common to all 4 northern stations

Max stable processes Independent processes Y i (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 kind of extreme event

What made Gudrun so destructive? Hurricane Gudrun, Sweden, January deaths, 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 (y i ), b -i (y i ) so that where G -i has non-degenerate margins. Pick a ji (y i ) so that and where h ji is the conditional hazard function of Y j given Y i =y i.

Asymptotic properties Let. Then given Y i >u, Y i - 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