Download Rethink on the inference of annual maximum wind speed distribution

The 4th Conference on Extreme Value Analysis: Probabilistic and statistical models and their applications, Gothenburg, 2005
Rethink on the inference of annual
maximum wind speed distribution
Hang CHOI
GS Engineering & Construction Corp., Seoul, KOREA
[email protected]
Jun KANDA
Inst. Environmental Studies, The University of Tokyo, JAPAN
[email protected]
01 Theoretical Frameworks of EVA
1) Stationary Random Process (Sequence)
- Distribution Ergodicity
- (In)dependent and Identically Distributed
random variables (I.I.D. assumption)
→ strict stationarity
* Conventional approach
2) Non-stationary Random Process (Sequence)
- (In)dependent but non-identically Distributed
random variables (non-I.I.D. assumption)
→ weak stationarity, non-stationarity
3) Ultimate (Asymptotic) and penultimate forms
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1) I.I.D. random variable Approach
Ultimate form : Fisher-Tippett theorem (1928)
X1 ,
, X n  F ( x), Zn  max  X1 ,..., X n 
 Z n  an

lim P 
 x   lim F n (an  bn x)  G( x)  F
n
 bn
 n
F  { Gumbel, Fréchet, Weibull }
GEVD, GPD, POT-GPD + MLM, PWM, MOM etc.
R.A. Fisher & L.H.C. Tippett (1928), Limiting forms of the frequency distribution of the largest or smallest
member of a sample, Proc. Cambridge Philosophical Society, Vol 24, p180~190
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2) non-I.I.D. random variable Approach
X1  F1 ( x),
, X n  Fn ( x), Zn  max  X1,
, X n
n
 Z n  cn

lim P 
 x   lim  Fj (d j  c j x)  Q( x)  S
n
 dn
 n j 1
FS
S  { Gumbel, Fréchet, Weibull,…..}
The class of EVD for non-i.i.d. case is much larger.
* Falk et al., Laws of Small Numbers: Extremes and Rare Events, Birkhäuser, 1994
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3) Ultimate / penultimate form and
finite epoch T in engineering practice
In engineering practice, the epoch of interest, T is finite.
e.g. annual maximum value, monthly maximum value and
maximum/minimum pressure coefficients in 10min etc.
As such, the number of independent random variables m
in the epoch T becomes a finite integer, i.e. m<∞, and
consequently, the theoretical framework for ultimate form
is no longer available regardless i.i.d. or non-i.i.d.case.
Following discussions are restricted on the penultimate d.f.
for the extremes of non-stationary random process
in a finite epoch T.
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02 Practical example:
What kinds of problems do exist in practice?
non-stationarity and the law of large numbers
(case study for max/min pressure coefficients)*
Tap #25
wind
Side face
* Choi & Kanda (2004), Stability of extreme quantile function estimation from relatively short records
having different parent distributions, Proc. 18th Natl. Symp. Wind Engr., p455~460 (in Japanese)
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Non-stationarity: mean & standard deviation
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The law of large numbers (3030 extremes)
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Example: Peak pressure coefficients
C p (t ) 
p(t )  ps
1/ 2 U H2
p(t ) : instant total pressure, ps : static pressure
 : air density, U H : wind velocity at model height
* 50 blocks contain about 480,000 discrete data
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Example: 10min mean wind speed (Makurazaki)
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mean
stdv
skewness
kurtosis
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Limitation of the i.i.d. rvs approach
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03 Assumptions and Formulation
According to the conventional approach in wind engineering, let
assume the non-stationary process X(t) can be partitioned with
a finite epoch T, in which the partitioned process Xi(t),(i-1)T≦
t < iT, can be assumed as an independent sample of stationary
random process, and define the d.f. FZ(x) as follows.
FZ ( x)  P( Z  x; Z : sup X (t ), t [0, T  ))
Then, by the Glivenko-Cantelli theorem and the block maxima
approach
1 n
1 n
FZ ( x)  lim  I (  , x ]  Z i : sup X i (t )   lim  P( Z i  x)
n  n
n  n
i 1
i 1
where I c ( x)  1 if x  c else 0
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04 Equivalent random sequence (EQRS)
approach to non-stationary process
By partitioning the interval [(i-1)T,iT) into finite subpartitions
[( j  1)h, jh), 1  j  [T / h]  mi in the manner of that
mi 
P( Z i  x)   P  Z *j  x; Z *j : sup  X i (t ), t  [( j  1)h, jh    FZmi i ( x)
j 1
the required d.f. FZ(x) can be defined as follows.
1 n mi
FZ ( x) : lim  FZi ( x)
n  n
i 1
If it is possible to assume that all mi≈m, then
1 n m
FZ ( x) : lim  FZi ( x)
n  n
i 1
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05 practical application:
Annual maximum wind speed in Japan
1) Observation records and Historical annual maximum wind
speeds at 155 sites in Japan
Observation Records: JMA records (CSV format)
1961~2002 : 10 min average wind speeds
2) Historical annual maximum wind speeds record:
1929~1999 : A historical annual max. wind speed data set
compiled by Ishihara et al.(2002)*
2000~2002 : extracted from JMA records (CSV format)
*T. Ishihara et al. (2002), A database of annual maximum wind speed and corrections for anemometers in Japan,
Wind Engineers, JAWE, No.92, p5~54 (in Japanese)
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05 practical application:
Annual maximum wind speed in Japan
3) Approximation of the annual wind speed distribution
Based on the probability integral transformation,
( z )  FX i ( x )  FX i  g ( z )   
x  g ( z )  a  bz  cz2  dz3

FX i ( x )    g 1 ( x ) 
The coefficients a,b,c and d can be estimated from the given
basic statistics of annual wind speed, i.e. mean, standard
deviation, skewness and kurtosis (Choi & Kanda 2003).
H. Choi and J. Kanda (2003), Translation Method: a historical review and its application to simulation on non-Gaussian
stationary processes, Wind and Struct., 6(5), p357~386
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4) Number of independent rvs required in EQRS
From Rice’s formula and Poisson approximation, normal
quantile function is given as follows:
z  2 log 0T  yT 
in whcih 0 : mean zero crossing rate, yT   log( log  )
From  m ( z ) ,
z
2 log  m / 4   yT  *
By comparison,
m 40T
* Choi & Kanda (2004), A new method of the extreme value distributions based on the translation method, Summaries
of Tech. Papers of Ann. Meet. of AIJ, Vol. B1, p23~24 (in Japanese)
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5) non-Normal process
To Rice formula for the expected number of crossings, i.e.


0
0
 ( x)   x  f ( x, X  x)dx  f ( x)   x  f ( x)dx
applying translation function g(z)

dx

 x  exp   x /  z  g ( z )  / 2
1
 ( x) 
  g 1 ( x)   
0
g ( z )
2
2   z  g ( z )
 {g 1 ( x)}2 
 {g 1 ( x)}2 
z

 exp  
   0  exp  

2
2
2




From Poisson approximation
x  g ( z )  g

2log( 0T )  yT 

With the same manner
m  4 0T  30000
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6) Simulation of the non-identical parent distribution
parent distribution function for each year
→Generalized bootstrap method＋
Translation method (Probability Integral Transform)
For the year i,
Fi ( x  gi ( z ))  ( z ), gi ( z )  ai  bi z  ci z 2  d i z 3
a, b, c, d i  ( i ,  i ,  1i ,  2i )
max( x1 ,
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, xm )i 1,
,n
 gi (max( z1 ,
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, zm )i 1,
,n
)
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Example : Tokyo (1961~2002)


1
2
Estimated from historical records
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Monte Carlo Simulation
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No. of Simulation：n=100 year x 1000 times
○：annual mean and standard dev. from historical records
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No. of Simulation：n=100 years x 1000 times
○：skewness and kurtosis from historical records
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Examples for
the Monte Carlo Simulation
results
Aomori
Sendai
Kobe
Tokyo
Shionomisaki
Makurazaki
Naha
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7) Comparison of the quantile functions from MCS and
historical records in normalized form
( Z n  am ,n ) / bm ,n
Aomori
95% confidence intervals
Mean of 1000 samples
(Z-am,n)/bm,n
○：before 1961 ●：after 1962
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Sendai
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Tokyo
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Kobe
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Shionomisaki
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Makurazaki
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Naha
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am , n
from Monte Carlo Simulation
○：n>50 (136 sites), ■：n≦50 (19 sites)
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bm , n
from historical data
from historical data
8) Comparison of the attraction coefficients from MCS and
the historical records
from Monte Carlo Simulation
lim am,n  am , lim bm ,n  bm
n 
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n 
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am , n
from historical records
from historical records
9) Comparison of the attraction coefficients from the
i.i.d. rvs approach and the historical records
from i.i.d. rvs approach
From i.i.d. rvs approach
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bm , n
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06 Conclusions
• EVA for natural phenomena such as annual max wind
speeds and peak pressure due to the fluctuating wind
speeds never be free from the law of large numbers.
• It is confirmed that the class of EVD for non-stationary
processes/sequences is much larger than that for
stationary processes/sequences as indicated by Falk et
al.(1994)
• The i.i.d. rvs (mean distribution) approach for nonstationary random process, which is conventionally
adopted in practice, significantly underestimates the
variance of extremes but not for the mean.
• Generalized bootstrap method incorporated with Monte
Carlo Simulation for the EVA of non-stationary
processes/sequences is a quite effective tool.
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