[[conferencetype]]國際[[conferencedate]]20110710~20110715[[booktype]]電子版[[iscallforpapers]]Y[[conferencelocation]]Amsterdam, Netherlan

[[corresponding]]Yuan-Lung Lo

Kanda, Jun

國際20110710~20110715電子版YAmsterdam, Netherlan

Yuan-Lung Lo

Tamkang University Institutional Repository

Application of Polynomial Translation Method to Prediction of Annual Maximum Wind Speeds Jun Kandaa, Yuan-Lung Lob, Takayuki Wadac  a The University of Tokyo, Kashiwa, Chiba, Japan, kandaj@k.u-tokyo.ac.jp b The University of Tokyo, Kashiwa, Chiba, Japan, ohlalaloren@hotmail.com c The University of Tokyo, Kashiwa, Chiba, Japan, takayuki0120w@hotmail.com 1 INTRODUCTION In order to provide an appropriate design wind speed for the wind resistant design of a structure, it is essential and important to estimate the probability distribution of annual maximum wind speeds. The annual maximum value of 10 minute mean wind speeds is only one value among 52,560 samples in one year. It seems reasonable to assume that the statistical nature of 10 minute mean wind speeds, such as four moments, provide suffi-cient information for the characteristics of annual maximum wind speeds. In this research, samples of 10 minute mean wind speeds at 155 meteorological sites are utilized to investigate the general relationships between four moments and the probability distributions of annual maxima. Then, the application of polynomial transla-tion method is verified by utilizing these four moments and a simulation procedure based on polynomial translation method is developed. Simulation results of annual maximum wind speeds of several meteorological sites show that such examinations of probabilistic characteristics of 10 minute mean wind speeds may provide good information for wind hazard models for individual sites. 2 CHARACTERISTICS OF FOUR MOMENTS OF PARENT DISTRIBUTION 2.1 Yearly variation of four moments 10 minute mean wind speeds from 1961 to 2002 at 155 meteorological sites in Japan are utilized to calculate four moments (mean, standard deviation, skewness, and unbiased kurtosis) to represent the yearly statistical nature. Figure 1 shows the parent distribution and its probability distribution at Tokyo in 2002. Four moments calculated from these samples are 3.192 for mean, 1.528 for the standard deviation, 0.916 for the skewness and 1.255 for the unbiased kurtosis (“kurtosis” hereafter), which can be indicated as the sta-tistical nature of the probability distribution. The annual maximum sample is pointed as the tail point of the probability distribution. Once the statistical nature is properly esti-mated, the simulation of annual maxima can be expected by estimating the four moments.   Figure 1 Parent distribution and probability distribution of 10 minute mean wind speeds at Tokyo in 2002 0.02.04.06.08.010.012.00 1000 2000 3000 4000 5000 6000 7000 8000 9000U(m/sec)Hour0.000.050.100.150 2 4 6 8 10ProbabilityU (m/sec)Yearly variations of four moments for one specific site, such as Tokyo in Figure 2, can then be obtained to observe the long term characteristics. It is found that in Figure 2 the variation of skewness and kurtosis are highly correlated. Among 155 meteorological sites, correlation coefficients between four moments are calculated and taken mean and standard deviation values as Table 1. A high mean value and a relatively low standard deviation value of 155 correlation coefficients indicate that the high correlation exists be-tween yearly skewness and kurtosis in most sites.    Figure 2 Yearly variations of four moments and relationship between skewness and kurtosis at Tokyo from 1961 to 2002  Table 1 Mean and standard deviation value of correlation coefficients between four moments  μ – σ μ – β3 μ – β4 σ – β3 σ – β4 β3 – β4 mean 0.59 -0.09 -0.06 0.17 0.09 0.91 standard deviation 0.33 0.30 0.26 0.32 0.27 0.05 ※μ: mean, σ: standard deviation, β3: skewness, β4: kurtosis Histograms of four moments at Tokyo from 1961 to 2002 can also be drawn as Figure 3. It is observed that the histograms of skewness and kurtosis show a tail which makes the distribution similar to a lognormal distribution rather than a normal distribu-tion. Among 155 sites, the common features can also be observed in most sites. For the simulation of annual maxima, a proper assumption of the probability distribution model for four moments may contribute to a better prediction.   (a) μ (in m/sec)        (b) σ (in m/sec)            (c) β3               (4) β4 Figure 3 Histograms of four moments at Tokyo from 1961 to 2002 2.2 Regional variation of four moments After examining statistical characteristics of yearly variations of four moments in 155 sites, differences from site to site are further examined as the effect of regional variation of four moments. Mean and standard deviation of yearly four moments are calculated and denominated as moment parameters, such as E(μ), E(σ), E(β3), E(β4), σ(μ), σ(σ), σ(β3), and σ(β4). From the histograms of these 155 sets of moment parameters, it indicates that y = 4.4142x - 2.9314R² = 0.89430.01.02.03.04.05.06.00.8 1.0 1.2 1.4 1.6 1.8KurtosisSkewness0246810122.6 2.9 3.2 3.5 3.8Frequency02468101.5 1.7 1.9 2.1 2.3Frequency024681012140.8 1.0 1.2 1.4 1.6 1.8Frequency024681012140.5 1.5 2.5 3.5 4.5 5.5Frequency0123456781961 1966 1971 1976 1981 1986 1991 1996 2001yearmean (m/sec)standard deviation (m/sec)skewnesskurtosissome sites show relatively large scattering in yearly skewness and kurtosis. Once the es-timation of skewness and kurtosis is large, the simulation of annual maxima will results in longer tail in the probability distribution model so that the simulation will be overesti-mated (Lo and Kanda, 2010).  (a) E(μ)                (b) E(σ)                (c) E(β3)               (d) E(β4) (e) σ(μ)                (f) σ(σ)                (g) σ(β3)               (h) σ(β4) Figure 4 Histograms of moment parameters calculated from four moments at 155 sites in Japan 3 APPLICATION OF POLYNOMIAL TRANSLATION METHOD 3.1 Brief introduction From the aforementioned statistical nature of parent distributions in all sites, it is essential to simulate annual maximum wind speeds with the proper estimation of four moments, especially those with extremely large skewness or kurtosis. A set of non-Gaussian ran-dom variables, Y, whose four moments are given, is written in a polynomial form with respect to a set of Gaussian random variables, X, as 32 dXcXbXaY   (1) where the coefficients of the polynomial form can be approximated by the following equations,   0caYE   (2)   1d15c2bd6bYVar 222   (3)    2d105bd24bc2Y 223   (4)       222224 d225c141bd4812dbd28b1cbd24Y   (5) An available approximation based on least square method for nonlinear parameters was introduced by Edgeworth (1898) to solve Equation (2) ~ (5) simultaneously. Table 2 shows the application sheet provided by Edgeworth to estimate the coefficients of the polynomial form. By substituting the given skewness and kurtosis, coefficients, a, b, c, and d, can be obtained by Equation (6) and Table 2. A set of non-Gaussian random varia-0510152025303540Frequency0510152025303540Frequency05101520253035Frequency0510152025303540Frequency010203040506070Frequency01020304050Frequency0102030405060Frequency010203040506070Frequencybles can then be obtained by assuming X a set of Gaussian random variables in Equation (1). After adding the given mean and multiplying the given standard deviation, a set of non-Gaussian random variables, which possess the statistical nature of given four mo-ments, is translated from a set of Gaussian random variables.   161jjjbTb  161jjjcTc  161jjjdTd  ca   (6)  Table 2 Application sheet for approximation of coefficients of the polynomial form  j Tj bj cj dj β4<1.5 β4>=1.5 β4<1.5 β4>=1.5 β4<1.5 β4>=1.5 1 1 1.0000 0.9698 0.0000 0.0012 0.0000 0.0112 2 β3 -0.0014 -0.0305 0.1668 0.1566 0.0007 0.0129 3 β4 -0.1238 -0.0765 0.0000 -0.0009 0.0412 0.0236 4 β32 0.1224 0.0558 0.0019 -0.0024 -0.0469 -0.0177 5 β42 0.0353 0.0054 0.0000 0.0002 -0.0131 -0.0018 6 β33 -0.0491 -0.0348 0.0653 0.0466 0.0258 0.0216 7 β43 -0.0085 -0.0002 0.0001 0.0000 0.0033 0.0001 8 β3β4 0.0027 0.0181 -0.0397 -0.0155 -0.0009 -0.0061 9 β32β4 -0.0768 -0.0130 0.0178 0.0236 0.0314 0.0087 10 β3β42 -0.0075 -0.0041 0.0183 0.0026 0.0021 0.0016 11 β33β4 0.0134 0.0029 -0.0068 0.0023 -0.0108 -0.0009 12 β3β43 0.0007 0.0003 -0.0018 -0.0002 0.0010 -0.0001 13 β32β42 -0.0101 -0.0002 -0.0071 -0.0029 0.0018 -0.0005 14 β32β43 0.0103 0.0001 0.0136 0.0001 0.0002 0.0000 15 β33β42 -0.0322 -0.0002 -0.0167 -0.0011 0.0165 -0.0001 16 β33β43 0.0127 0.0000 0.0207 0.0000 -0.0033 0.0000 3.2 Testing of application sheet for polynomial translation method To verify the applicability of the application sheet (Table 2) and the polynomial transla-tion method in simulating non-Gaussian random variables, a testing flow is attempted as Figure 5. A specific set of four moments is first given to generate a new set of non-Gaussian random variables. Comparing the given set and the resultant set of four mo-ments, the applicability can be verified.  Figure 5 Testing flow for verification of polynomial translation method  All the four moments calculated from 155 sites are then utilized to verify the appli-cation range. The maximum and minimum values of all four moments are defined as the testing ranges. To simplify the testing procedure, whenever one given moment is tested, other three given moments are fixed at their mean values. As shown in Figure 6, the test-ing flow is carried out for the verification of each given moment. In the topmost four fig-ures, the first moment value, mean, is varying from 1.003 to 9.887 and other three mo-ment values are fixed at 2.041, 1.175, and 2.087 respectively. For the varying given mean and given standard deviation, the simulation of non-Gaussian random variables can fit the original statistical nature well. However, when the given skewness is larger than 1.4, the polynomial translation method based on Table 2 will provide a biased resultant statistical nature. The same tendency occurs in the given kurtosis when the given value is larger than about 12.  Figure 6 Verification results of each varying given moment (−: given moment values; +: resultant moment values) 3.3 Modification of application sheet based on observed statistical nature From the testing results in 3.2, it is indicated that extremely large given skewness or kur-tosis is not suitable for the approximation sheet proposed by Edgeworth. To avoid the complexity of solving the nonlinear equations (2) ~ (5) and to improve the approximation method proposed by Edgeworth, the modification is conducted by utilizing the observed four moments from 155 sites. As shown in Figure 7, all sets of four moments from 155 sites are substituted into the testing flow in Figure 5 again. Given and resultant moment values are drawn respectively to show the biases. It can be presumed that the bias in standard deviation figure is caused by large given skewness or kurtosis. In the figures of skewness and kurtosis, there seems a bilinear relationship between given and resultant values. 02468100 2 4 6 8 10μμ012340 2 4 6 8 10σμ01230 2 4 6 8 10β 3μ012340 2 4 6 8 10β 4μ02468100 1 2 3 4 5 6μσ01234560 1 2 3 4 5 6σσ01230 1 2 3 4 5 6β 3σ012340 1 2 3 4 5 6β 4σ02468100 1 2 3μβ301234560 1 2 3σβ3012340 1 2 3β 3β3051015202530350 1 2 3β 4β302468100 4 8 12 16 20 24μβ401234560 4 8 12 16 20 24σβ4-4-3-2-1012340 4 8 12 16 20 24β 3β40102030405060700 4 8 12 16 20 24β 4β4Tpresumpvalues c20% anrepresenwhich nsmaller attemptgiven v 33  44 F4 SIM4.1 SimBased otion me02468100μ*-1012345670β 3*02468100μo eliminatetion is attean be firstd those withts the appeeds to be given valued as Equaalues for fo249.1307.0751.1 5.250.4468.14  igure 8 BilinFigure 10 VeULATION ulation pron the statisthod aforem2 4 6 8μ1 2β3Error <= 20%Error > 20%2 4 6 8μ the biases mpted. As calculated  error larglicable rangmodified. Ie is inversetion (7) andur moment751.1  468.143235   Figure 7 Verear relationshrification resuOF ANNUcedure bastical natureentioned, a1001234560σ*30102030405060700β 4*1001234560 1σcaused by e illustrated and categoer than 20%e of givenn order to ly estimated (8). Figurs. ification resulips in β3 and lts of 155 setAL MAXIMed on polyn of four mo simulation1 2 3 4 5σ4 8 12 16β4Error <= 20%Error > 20%2 3 4 5σxtremely lin Figure 8rized as tw. The form values; thmake all th as shown e 10 showsts of 155 sets β4     Figur s of given fouUM WINomial tranments and  procedure6012345670β 3*20 24601230β 3*arge skewn, errors beo groups, ter group foe latter groe resultant in Figure 9 the modifi of given foure 9 Inverse esr moments afD SPEEDSslation metthe applicat is introduc1 2β31 2β3ess or kurtotween givehose with ellows the slup followsvalues foll. The inversed resultan moments  timation diagter inverse ca hod ion of polyed as Figur30102030405060700β 4*3048121620240β 4*sis, an invn and resulrrors less tope, L1, wh the slope, ow L1 slope estimatiot values veram for β3 anlculation  nomial trane 11 (Choi 4 8 12 16β44 8 12 16 2β4erse tant han ich L2, e, a n is rsus (7)(8)d β4 sla-and 20 240 24  Kanda, 2003, 2005). To estimate the annual maximum wind speed distribution for a spe-cific site, 8 moment parameters, E(μ), E(σ), E(β3), E(β4), σ(μ), σ(σ), σ(β3), and σ(β4), are calculated in advance. By utilizing these moment parameters and the observation of probability distribution models, the generation of skewness and kurtosis can be conducted as normally distributed or log-normally distributed. According to the linear relationship between yearly skewness and kurtosis, a fully correlated assumption is also included. Then the polynomial translation method is applied to generate non-Gaussian samples. From these samples, the maximum value is picked as the annual maximum wind speed. In this research, 100 sets of four moments are generated as samples of 100 years. Every year has its annual maximum value and can be drawn as an extreme value distribution on the Gumbel model paper. To keep the randomness of simulation procedure, the extreme value distribution drawing is repeated for 11 times and taken the median values as the best estimation of annual maximum wind speed distribution for the specific site.          Figure 11 Simulation procedure for 100 annual maxima based on polynomial translation method  4.2 Simulation results of several sites in Japan The aforementioned simulation procedure and the general statistical nature of 155 mete-orological sites are then applied to estimate the annual maximum wind speeds. Four sites, Aikawa, Sapporo, Ishigakijima, and Kumijima, are chosen to demonstrate the agreement of simulation results and observed records. Table 3 lists moment parameters of these four sites. Since the standard deviation of skewness and kurtosis is quite large in Ishigakijima and Kumijima, it is predictable to generate a longer tail in the probability distribution of skewness and kurtosis and further results in large annual maximum wind speeds. For Aikawa and Sapporo relatively, the values of moment parameters are considered moderate among 155 sites.  Table 3 Moment parameters of four meteorological sites  E(μ) E(σ) E(β3) E(β4) σ(μ) σ(σ) σ(β3) σ(β4) Aikawa 4.32 3.33 1.31 1.40 0.41 0.30 0.12 0.48 Sapporo 2.51 1.71 0.94 0.88 0.28 0.22 0.14 0.49 Ishigakijima 4.51 2.15 1.28 6.32 0.38 0.23 0.65 6.47 Kumijima 3.70 1.96 0.92 5.05 0.18 0.15 0.60 5.30  Figure 12 shows the simulation results based on two probability distribution models for generated skewness and kurtosis. For Aikawa and Sapporo, skewness and kurtosis are generated based on moderate moment parameters so that the assumption of probability distribution can be normal or lognormal. However, for Ishigakijima and Kumijima, a lognormal distribution may be more appropriate to generate skewness and kurtosis. The errors between the observed annual maxima and the simulation results are 2.97%, 9.67%, 16.93%, and 16.21% by Equation (9). 100EEEn1(%)Ern1i2*i*ii    (9) Generate 100 sets of four momentsGenerate 100 sets of non-Gaussian samples for 100 yearsGenerate 100 sets of Gaussian samples for 100 yearsPolynomial translation methodPickup 100 annual maxima and plot on Gumbel paperwhere Emaximato be 1Amongjima.                        5 CONA simunual mmethodrange oof momthe simunature, nomial 6 REF Choi, H.,GaussChoi, H.,variabEdgeworRoyaLo, Y.L.,butioni represen; and n rep00 and obs 155 sites, tFigure 1CLUSIONlation proceaximum w proposed f skewnessent paramelation resufour mometranslation ERENCES Kanda, J., 20ian stationary Kanda, J., 20les. Extreme th, F.Y., 1898l Statistical So Kanda, J., 20. ASRANet 25101520-2Annual Max. Wind Speed (m/s)020406080-2Annual Max. Wind Speed (m/s)ts i-th estimresents theerved annuhe worst tw2 Simulation S dure basedind speedsby Edgewo or kurtosisters, statistlts. The simnts, are prmethod can 03. Translati process. Win05. Extreme Value Theory. On the reprciety, 61(4), 10. Annual m010 Confere-1 0 1 2Reduced VAikawaONL-1 0 1 2Reduced VIshigakijimaated annu number ofal maximao fitting caresults for Ai on the poly was introdrth was fu for non-Gical nature ulation resoperly estim provide a on method: a d and Structuvalue distribu and Applicaesentation of 670-700. aximum winnce, Edinburg3 4 5 6ariatesbservedormalognormal3 4 5 6ariatesal maxima simulation are linearlses have bekawa, Sappornomial trauced. Therther conduaussian ranof parent dults of 155ated, the fairly good historical revre, 6(5), 357tion of annutions, 59-77.statistics by md simulation h. 152025303540Annual Max. Wind Speed (m/s)01020304050Annual Max. Wind Speed (m/s); Ei* repres years. In ty interpolaen shown o, Ishigakijimnslation me modificatcted to prdom variabistributions sites indicsimulation prediction.iew and its ap-386. al maximum wathematical fbased on mom-2 -1 0 1ReducedSapporo-2 -1 0 1ReducedKumijimaents i-th obhis researchted to calcas Ishigakija, and Kumijthod for esion to the ovide a wiles. With tcould be reate that oncprocedure bplication to sind speeds aormulae (Parent paramet2 3 4 5 Variates2 3 4 5 Variatesserved ann, n is assumulate the erima and Kuima timation ofapproximader applicahe observaflected wee the statistased on pimulation of s I.N.I.D. rant I). Journal oer of parent d66ual ed ror. mi- an-tion tion tion ll to ical oly-non-dom f the istri-

Application of Polynomial Translation Method to Prediction of Annual Maximum Wind Speeds

http://tkuir.lib.tku.edu.tw:8080/dspace/bitstream/987654321/93165/2/C201107%28Application+of+Polynomial+Translation+Method+to+Prediction+of+Annual+Maximum+Wind+Speeds%29.pdf

Application of Polynomial Translation Method to Prediction of Annual Maximum Wind Speeds

Abstract

Similar works

Full text

Available Versions

Tamkang University Institutional Repository