XXXV CONVEGNO NAZIONALE DI IDRAULICA

E COSTRUZIONI IDRAULICHE

Bologna, 14-16 Settembre 2016

The Metastatistical Extreme Value Distribution

Metodi Statistici per le Applicazioni Idrologiche

Enrico Zorzetto1, Gianluca Botter2,

Marco Marani1,2,*1Earth and Ocean Science Division, Duke University

2 DICEA, Universita di Padova* marco.marani@unipd.it

Classical Extreme Value Theory (EVT)[Fischer-Tippett-Gnedenko, 1928-1943]

Block Maxima:

Three-Type Theorem:

- As n -After renormalization, 3 possible asymptotic distributions, summarized by GEV (e.g. Von Mises, 1936):

= Maxima n-event blocks

h [

mm

]

1937 19381936 1939 1940 19411942

= max

()

for i.i.d =

= exp 1 +

+

1

Marani and Ignaccolo, AWR, 2015

Weibull-distributed, synthetic, daily rainfall data# events/year & Weibull parameters from Padova (Italy)

GEV fitted on 30-year windows

Considerations on the validity of the classical EVT

- Incomplete convergence to limiting distribution: n

A Metastatistical Extreme Value distribution (MEV)

= ;

for i.i.d. .

F(X; ) = cdf of ordinary events

The Block-maxima distribution

Expected block-maxima distribution compounding stochastic n and :

Marani and Ignaccolo, AWR, 2015; Zorzetto et al., GRL, 2016

G(n,) = joint prob distrib. of the parameters.

Approximating expectations with sample averages.

Parameters ofordinary distributions

A Metastatistical Extreme Value distribution (MEV)

Marani and Ignaccolo, AWR 2015; Zorzetto et al., GRL 2016

1

=1

(; ) T = # years over which n

and are estimated

approximating expectations with sample averages:

MEV:

MEV distribution conceptual interpretation

Zorzetto et al., GRL 2016

A choice for F(x) - the pdf of daily ordinary rainfall

=

= 1

Weibull Parent

distribution

=precipitation efficiency

=specific humidity

m=advection mass

[Wilson e Tuomi, 2005]

-Simple two-layers atmospheric model-Temporal average

MEV-Weibull distribution

Marani and Ignaccolo, AWR, 2015; Zorzetto et al., GRL, 2016

The MEV expression:

1

=1

(; ) T = # sub-periods over which n

and are estimated

In the Weibull case becomes:

1

=1

1

Marani and Ignaccolo, AWR, 2015

Weibull-distributed synthetic dataGEV and MEV fitted on 30-year windows

n random, c and w constant

n, C, and w are constant

n constant, C and w random

How about reality?

36 daily rainfall timeseries, 106 -275 years of daily observations,( =135 yrs) Less than 5% of missing data

OXFORD

SHEFFIELDHOOFDOORP

PUTTEN

ZURICH

HEERDE

S. BERNARD

MELBOURNE

MILANO

PADOVA

BOLOGNA

CAPE TOWN

SAN FRANCISCO

ROOSVELTASHEVILLE

PHILADEPHIAKINGSTON

ALBANY

DUBLIN

ZAGREB

WORCESTER

DUBLIN

SYDNEY

Method of analysis

To eliminate correlation and non-stationarity Preserving the true (unknown) distribution of the

parameters and numbers of wet days.

Fit on a sample of size s Test on remaining data. Non dimensional Root

Mean Square Error:

Which is studied as a function of sample size s.

Bootstrap - Reshuffling of daily data preserving(1) yearly number of events, and(2) observed values (i.e. Pdfs)

ORIGINAL TIME SERIES

=1

(

)2

RANDOMLY RESHUFFLED TIME SERIES

T Years

h [mm]

h [mm]

t [days]

t [days]

Ratio of MEV to GEV estimation errors (using LMOM, but use of ML or POT gives same results)

NOAA-NCDC

Worldwidedataset

Zorzetto et al., GRL, 2016

Estimation error as a function of Tr/(sample size)MEV vs. GEV (LMOM)

Zorzetto et al., GRL, 2016

Return time/sample size

MEV error 50% of GEV error

Conclusions

MEV ouperforms classical EV distributions:

- Reliable assessment of high quantiles and smallsamples (50% improvement over GEV)

- Better use of the available daily data- Removal of the asymptotic hypothesis

Future:

1.MEV is general approach (floods, wind, storm surges ...)

2. MEV is arguably suited to tackle non-stationarity

Grazie per lattenzione

Some thoughts on non stationarity

Bologna (Italy) original 180 years time-seriesSliding and overlapping windows analysis

GEV and POT estimated quantile shows higher variance

MEV shows a positivetrend in est. quantiles

Due to trends in parameters of Weibull distribution

Tr=100 years

i-th temporal window

An interesting observation: GEV performs better if calibration data=testing data

Tr=100 daily rainfall from TRMM observations (17 yrs)

Estimation error as a function of Tr/(sample size)

Distribution of the error computed over 1000 random reshuffling, for all the analyzed datasets.Quantiles (Tr=100 yrs) estimated by GEV, POT, MEVcalibrated over 30-years samples

Error distribution

=

= 1 1 1

from the observational (independent) sample

Distribution of the error computed over 1000 random generations, for all the analyzed datasets.Theoretical quantiles (Tr=100 yrs) estimated by GEV, POT, MEV calibrated over 30-years samples

Error distribution

=

= 1 1 1

from the observational (independent) sample

Global QQ-Plots

Sample size=45 years 100 random reshuffling

Global QQ plots

GEV/ POT are a good fit for the calibration sample but they fail in describing the stochastic process from which the sample has been generated

MEV allows a better description of the underlying process; less variance in high quantile estimation

. 2

The MEV domain

N

1982 1986198519841983t

h [mm]

1 = 97 2 = 105 3 = 89 4 = 94 5 = 114

1, 1 2, 2 3, 3 4, 45, 5

2. Fit Weibull to the singl ,

1. Sampling n from the distribution p(n|C,w)

The MEV distribution

= 1

Assuming Weibull as a pdf for daily rainfall Fit performed using Probability Weighted Moments (Greenwood et al, 1979)

Number of events/ year

=

=1

, , 1

Weibull parameters = , and are random variables themselves

The CDF of annual maximum is the mean on all their possible realizations

The Metastatistical Extreme Value Distribution

n

()

()

()

Density frequencies

Non stationary analysis

GEV and POT estimated quantiles show oscillations with sameamplitude

Due to the variance in the parameter estimates

In the case of MEV the varianceof estimated quantiles is muchsmaller; Stationary behaviour

Tr=100 years

i-th window

Bologna (Italy) randomly reshuffled time seriesSliding and overlapping windows analysis

Daily rainfall observations in Padova 1725-2015

The Padova observatory

(Marani and Zanetti, 2015)

The Padova daily precipitation time series 1725-2006

(Marani andIgnaccolo, 2015)

Padova series:

Wide fluctuations in pdf parameters and in number of events.

Peak Over Threshold Method (POT)[Balkema, De Haan & Pickand, 1975; Davison and Smith, 1990]

Exceedances arrivals Poisson

Distribution of excesses Generalized Pareto

Advantages:

1. Better description of the tail

2. Consistent with GEV

< x =

=1

=

=1

! 1 1 +

1/

For a fixed threshold q Exceedances = . . . . .

=

Performance when testing sample = calibration sample

Ratio of MEV estimation error to GEV-POT error