1. Rethink on the inference of annual maximum wind speed distribution Hang CHOI GS Engineering & Construction Corp., Seoul, KOREA number3choi@unitel.co.kr Jun KANDA Inst. Environmental Studies, The University of Tokyo, JAPAN kandaj@k.u-tokyo.ac.jp The 4 th Conference on Extreme Value Analysis: Probabilistic and statistical models and their applications, Gothenburg, 2005

2. Choi & KandaEVA2005, Gothenburg2 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

3. Choi & KandaEVA2005, Gothenburg3 1) I.I.D. random variable Approach Ultimate form : Fisher-Tippett theorem (1928) { 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

4. Choi & KandaEVA2005, Gothenburg4 2) non-I.I.D. random variable Approach { Gumbel, Fréchet, Weibull,…..} * Falk et al., Laws of Small Numbers: Extremes and Rare Events, Birkhäuser, 1994 The class of EVD for non-i.i.d. case is much larger.

5. Choi & KandaEVA2005, Gothenburg5 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.

6. Choi & KandaEVA2005, Gothenburg6 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)* * Choi & Kanda (2004), Stability of extreme quantile function estimation from relatively short records having different parent distributions, Proc. 18 th Natl. Symp. Wind Engr., p455~460 (in Japanese) Side face wind Tap #25

7. Choi & KandaEVA2005, Gothenburg7 Non-stationarity: mean & standard deviation

8. Choi & KandaEVA2005, Gothenburg8 The law of large numbers (3030 extremes)

9. Choi & KandaEVA2005, Gothenburg9 Example: Peak pressure coefficients * 50 blocks contain about 480,000 discrete data

10. Choi & KandaEVA2005, Gothenburg10 Example: 10min mean wind speed (Makurazaki) mean stdv skewness kurtosis

11. Choi & KandaEVA2005, Gothenburg11 Limitation of the i.i.d. rvs approach

12. Choi & KandaEVA2005, Gothenburg12 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 X i (t),(i-1)T ≦ t < iT, can be assumed as an independent sample of stationary random process, and define the d.f. F Z (x) as follows. Then, by the Glivenko-Cantelli theorem and the block maxima approach

13. Choi & KandaEVA2005, Gothenburg13 04 Equivalent random sequence (EQRS) approach to non-stationary process By partitioning the interval [(i-1)T,iT) into finite subpartitions in the manner of that the required d.f. F Z (x) can be defined as follows. If it is possible to assume that all m i ≈m, then

14. Choi & KandaEVA2005, Gothenburg14 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)

15. Choi & KandaEVA2005, Gothenburg15 05 practical application: Annual maximum wind speed in Japan 3) Approximation of the annual wind speed distribution Based on the probability integral transformation, 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

16. Choi & KandaEVA2005, Gothenburg16 4) Number of independent rvs required in EQRS From Rice’s formula and Poisson approximation, normal quantile function is given as follows: in whcih From, By comparison, * * 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)

17. Choi & KandaEVA2005, Gothenburg17 5) non-Normal process To Rice formula for the expected number of crossings, i.e. applying translation function g(z) From Poisson approximation With the same manner

18. Choi & KandaEVA2005, Gothenburg18 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,

19. Choi & KandaEVA2005, Gothenburg19 Example : Tokyo (1961~2002) Estimated from historical records Monte Carlo Simulation

20. Choi & KandaEVA2005, Gothenburg20 ○ ： annual mean and standard dev. from historical records No. of Simulation ： n=100 year x 1000 times

21. Choi & KandaEVA2005, Gothenburg21 ○ ： skewness and kurtosis from historical records No. of Simulation ： n=100 years x 1000 times

22. Choi & KandaEVA2005, Gothenburg22 Aomori Sendai Tokyo Shionomisaki Makurazaki Naha Examples for the Monte Carlo Simulation results Kobe

23. Choi & KandaEVA2005, Gothenburg23 7) Comparison of the quantile functions from MCS and historical records in normalized form Aomori 95% confidence intervals Mean of 1000 samples ○ ： before 1961 ● ： after 1962 (Z-a m,n )/b m,n

24. Choi & KandaEVA2005, Gothenburg24 Sendai

25. Choi & KandaEVA2005, Gothenburg25 Tokyo

26. Choi & KandaEVA2005, Gothenburg26 Kobe

27. Choi & KandaEVA2005, Gothenburg27 Shionomisaki

28. Choi & KandaEVA2005, Gothenburg28 Makurazaki

29. Choi & KandaEVA2005, Gothenburg29 Naha

30. Choi & KandaEVA2005, Gothenburg30 8) Comparison of the attraction coefficients from MCS and the historical records ○ ： n>50 (136 sites), ■ ： n ≦ 50 (19 sites) from historical data from Monte Carlo Simulation

31. Choi & KandaEVA2005, Gothenburg31 9) Comparison of the attraction coefficients from the i.i.d. rvs approach and the historical records from historical records From i.i.d. rvs approach from i.i.d. rvs approach

32. Choi & KandaEVA2005, Gothenburg32 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 non- stationary 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.