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