ENVIRONMENT CANADA ENVIRONNEMENT CANADA. Verification of extreme events. Barbara Casati (Environment Canada) D.B. Stephenson (University of Reading). Extreme events can be defined by: Maxima/minima Magnitude Rarity Impact/losses. Gare Montparnasse, 22 October 1895.

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Verification of extreme events

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As the threshold increases TS,ETS,HSS,KSS converge to zero (no skill) for all the cases

As the threshold increasesodds’ ratio, ROC, Yule’s Q, separates the cases

Why do the scores behave differently ?

Base rate versus threshold

Base Rate = Probability of the event

Threshold increases, base rate decreases

Intense/extreme/rare events when 0

Categorical scores versus base rate

Plots in logarithmic scale: the rate of convergence of the statistics plays a key role in discriminating the NIMROD cases

Behavior of the hits

Asymptotic model:

= slope parameter

b > 1 for ROC curve regularity

b = 2 for random forecast:

b > 2 no skill

b < 2 skill

Asymptotic behaviour of the joint distribution (un-biased forecast)

2 degrees of freedom: (, ) fully describe the joint distribution

Express joint probabilities and verification statistics as functions of ,

Analyze statistics asymptotic behavior when the base rate 0

Scores asymptotic behaviour (no bias)

TS, ETS,HSS, KSS magnitude converges to zero as 0 either for skilful, random or worse than random forecasts ( acts as adiminishing factor)

Odds Ratio exhibits different asymptotic behaviours depending on whether a/n converge to zero faster, at the same rate or slower than a/n for a random forecast

Asymptotic behaviour of the joint distribution (biased forecast)

3 degrees of freedom: (B, , ) fully describe the joint distribution

Express joint probabilities and verification statistics as functions of B, ,

Analyze statistics asymptotic behavior when the base rate 0

Analyze statistics sensitivity to the bias B in the limit 0

Scores asymptotic behaviour (bias)

magnitude of TS, ETS, HSS, KSS monotonically increase as B increases: encourage over-forecasting !!

The odds’ ratio is not affected by the bias

Extreme dependency score

forecast and obs values are transformed into empirical cumulative probabilities

uniform marginal distributions, no bias

cumulative probability:

p = 1 – base rate

as 0, thenp 1 intense/extreme/rare events

b’

a’

d’

c’

Extreme Dependency Score

does not depend on the base rate

is not affected by the BIAS

it depends only on the parameter (rate of convergence of the joint probability a/n to zero, as the events get rarer)

separate the case studies

EDS measures forecast and obs extreme dependency

TS, ETS,HSS, KSS magnitude converges to zero either for skilful, random or worse than random forecasts: not suitable to verify extremes

Odds ratio, Yule’s Q, ROC are more suitable for detecting the skill in forecasting extreme/rare events

TS, ETS,HSS, KSS are overly sensitive to the bias in extreme event situations and encourage over-forecasting

the odds ratio, Yule’s Q, ROC are not affected by the bias when verifying extreme/rare events

The Extreme Dependency Score provides a bias and base rate independent measure of extreme dependency: very suitable to verify extremes

Conclusions

References

S. Coles, J. Heffernan, J. Tawn (1999) “Dependence Measures for Extreme Value Analyses”, Extremes 2:4, pp. 339-365

C. Ferro (2007) “A probability model for verifying deterministic forecasts of extreme events”, Weather & Forecasting, in press

B. Casati (2004) “New approaches for the verification of spatial precipitation forecasts”, PhD Thesis (Chapter 6), available at http://www.met.rdg.ac.uk/~swr00bc/PhD.html

D.B. Stephenson, B. Casati, C. Wilson (2004) “Skill measures for forecasts of rare extreme events”, presentation given at the 2nd international workshop on verification methods, Montreal, Sept 2004.