Distribution of wind direction recorded at maximum wind speed: A case study of Malaysian wind data for 2005

Transcription

1 Iteratoal Joural of the Physcal Sceces Vol. 6(7), pp , 4 Aprl, Avalable ole at ISSN Academc Jourals Full Legth Research Paper Dstrbuto of wd drecto recorded at maxmum wd speed: A case study of Malaysa wd data for 5 N. A. B. Kamsa, A. G. Huss*, Y. Z. Zubar ad S. F. Hassa Cetre for Foudato Studes Scece, Uversty of Malaya, 563 Kuala Lumpur, Malaysa. Accepted 5 February, I ths study, four types of crcular probablty dstrbuto, amely crcular uform dstrbuto, vo Mses dstrbuto, wrapped-ormal dstrbuto ad wrapped-cauchy dstrbuto are cosdered search of a crcular probablty dstrbuto provdes the best ft for Malaysa wd drecto data recorded at maxmum speed. The data collected were classfed as aual, ortheast mosoo ad southwest mosoo. Seve stes were detfed for ths study based o ther geographc locato, that s, whether they are stuated o the east coast or west coast of Malaysa. The obectve of the study was to see whether there are ay dffereces the crcular probablty dstrbuto of the wd drecto based o these two coasts. Graphcal measures were used to gauge whch dstrbuto gves the best fttg. To further support the fdgs, two performace dcators or goodess of ft tests are used the aalyss, amely the mea crcular dstace ad the. Geerally, t was foud that the mosoo-based data fts well wth the vo Mses dstrbuto as compared to the aual dataset. Key words: Wd data, crcular dstace,, crcular probablty dstrbuto, performace dcator. INTRODUCTION Malaysa has bee fortuate that extreme atural catastrophes rarely occur the coutry, wth the excepto of the typhoo Utor that caused severe floodg Johor 6 ad recetly, recurret evet of heavy ra dowpour caused massve floodg the orther, souther ad easter part of the pesular. Such cdeces have made us realze that somethg should be doe to uderstad meteorologcal dsasters lke typhoos, for example. To uderstad what costtutes typhoos, we eed to start wth wd. Wd s whe ar flows, bee caused by ueve heatg o a surface. Dfferet types of lads ad water absorb heat at dfferet rates. Thus wd vares stregth accordg to the dfferet parts of the earth ad dfferet surfaces. A typhoo ca be classfed as a example of a extremely strog wd that ca be dsastrous. The obectve of ths paper s to model parametrcally wd drectos *Correspodg author. E-mal: ghapor@um.edu.my. Malaysa recorded at maxmum speed usg crcular probablty dstrbutos. Some of the beefts of ths research are terms of statstcal cotrbutos predctg ad modellg wd patters whch ca be used for may purposes ad for further research edeavour. Wd drecto s oe of the features that should be cosdered buldg wd turbes as well as structural ad evrometal desg aalyss (Wzelus, 7). I may coutres, proect maagers ad archtects have started to ackowledge the mportace of esurg a safe ad comfortable wd evromet the vcty of ew buldgs. The developmet of tall buldgs wdy evromets may lead to problems lke fuellg ad dowwash. Other tha that, wd drecto s mportat predctg the weather. Ths s possble because by predctg wd drecto, oe ca have a dea of how the weather wll chage ad forecast o the weather. Predctos of wd drecto are also mportat for sports lke golf ad skydvg. Data o wd drecto are comprsed of agles measured degrees or rada.

2 Kamsa et al. 84 Fgure. Locatos of the maxmum wd data that has bee recorded daly 5. Source: These types of data are kow as crcular data ad thus wd drecto ca be modelled by usg crcular dstrbutos; for example, Razal et al. (8) ther study o surface wd drecto for Bag, a locato Malaysa, foud that the vo Mses seems to be the best dstrbuto to descrbe the patter of surface wd drecto. I ths study, the wd drectos at maxmum wd speed have bee recorded daly for seve locatos ad ftted wth four crcular dstrbutos. The seve locatos selected are, Lagkaw, Melaka, Sea, Kota Bharu, Kuala Tereggau ad Kuata as show Fgure. These places are selected based o ther locatos, that s, whether they are stuated o the east coast or west coast of Malaysa. Ths s because the two coasts are affected dfferetly by the two mosoos that are prevalet Malaysa, the ortheast mosoo ad the southwest mosoo. I the frst stage of the study, the aual data o the wd drectos were aalysed as a whole. I the secod stage, data were aalysed based o the mosoos whch relate to the locatos. Oly four crcular dstrbutos were selected as the software Axs, 3 could oly support these four dstrbutos. CIRCULAR PROBABILITY DISTRIBUTION A crcular probablty dstrbuto s a probablty dstrbuto whose total probablty s cocetrated o the crcumferece of a ut crcle (Jamalamadaka ad

3 84 It. J. Phys. Sc. SeGupta, ). The rage of crcular radom varables ca be measured rada whch ca be defed as [,π ) or [ π, π ). I ths artcle, four crcular probablty dstrbutos are dscussed, whch are the crcular uform dstrbuto, vo Mses dstrbuto, wrapped-ormal dstrbuto ad wrapped-cauchy dstrbuto. Crcular uform dstrbuto Crcular uform dstrbuto s the basc dstrbuto o the crcle whch s varat uder rotato ad reflecto (Bogda et al., ). If the total probablty s spread out uformly o the crcumferece, the we have crcular uformty (CU) dstrbuto wth the costat desty fucto of g ( θ ), θ π π The vo Mses dstrbuto The most useful dstrbuto o the crcle ad was troduced 98 by vo Mses order to study the devatos of measured atomc weghts from tegral values (Marda, 97). Ths dstrbuto s also kow as crcular ormal dstrbuto. A crcular radom varable θ s sad to have a vo Mses dstrbuto f t has desty fucto of κ cos( θ µ ) g ( θ; µ, κ ) e, θ < π πi ( κ ) where µ < π ad κ are mea drecto ad cocetrato parameters, respectvely. The fucto I deotes the modfed Bessel fucto of the frst kd ad order zero ad s defed as: I r κ. r r! ρ g ( θ; µ, ρ), θ < π π + ρ ρ cos( θ µ ) ρ σ e. The equalty of the two aforemetoed where expressos s verfed by equatg the real parts of the k a geometrc seres detty a wth a k ( θ µ ) a ρe. The dstrbuto s umodal ad symmetrc. Wrapped-ormal dstrbuto A wrapped-ormal dstrbuto s obtaed by wrappg a N ( µ, σ ) dstrbuto aroud the crcle where σ log ρ,.e fucto s gve by g( θ; µ, ρ) σ π ρ / σ e. Its probablty desty m ( θ µ πm) exp σ I partcular, the mea drecto s µ (mod π ) ad the mea resultat s ρ. From the theory of theta fucto, a alterate ad more useful represetato of ths desty ca be show to be + p g ( ; µ, ρ) ρ cos p( θ µ ) π p ρ. θ, It s clear that the desty ca be adequately approxmated by ust the frst few terms of the fte σ seres, depedg o the value of. It s umodal ad symmetrc about the value of θ µ. The wrappedormal dstrbuto possesses the addtve property whch s the covoluto of two wrapped-ormal varables s also wrapped-ormal ulke the vo Mses dstrbuto.. Wrapped-Cauchy dstrbuto The wrapped-cauchy dstrbuto s obtaed by wrappg the Cauchy dstrbuto o the real le wth desty of σ f ( x), < x < π σ + ( x µ ) aroud the crcle. It has the probablty desty fucto PERFORMANCE INDICATOR I measurg the sutablty of a model, oe could use performace dcators to check whether the data follows the specfed model. Performace dcator s a basc measuremet of how close a set of data follows a dstrbuto. However, the case of drectoal or crcular data, oe must ot treat the data lke lear data sce drectoal data has dstctve features (Jammalamadaka ad SeGupta, ). Two performace dcators are

4 Kamsa et al. 843 where θ s the observed, A [,]. θˆ s the predcted value ad If the shows a small value, t meas that the data follows the specfed dstrbuto. It ca be see as the weghted average of the of the. crcular dstace Fgure. Chord. proposed for ths study, amely the mea ad mea crcular dstace. Chord s the of the curve segmet betwee two adacet data pots. As ca be see from Fgure, a (broke le) s the. Jammalamadaka ad SeGupta () proposed θ as the smallest agle betwee θ ad θ whch s gve by θ d ( θ, θ ) π π θ θ where θ [, π ]. Oe ca use ths cocept θ to obta the betwee θ ad θ. The of a betwee two pots θ ad θ s calculated accordg to the formula θ crd( θ ) s r where crd( θ ) [,] ad r s the radus. I ths study. The mea s proposed ad gve by π π θ ˆ s θ A Crcular dstace calculates the betwee ay two pots alog the crcumferece ad t takes the smaller of the two arc s betwee the pots. crcular dstace calculates the mea for all of the crcular dstaces for each pot the data. Let us assume that there are pots θ,..., located o the crcumferece of a ut crcle. Let, θ θ, d be the crcular dstace betwee θ ad θ for,,,, where d s gve by d cos( θ θ ) ad where d [,]. I ths study, a ew measure of mea crcular dstace s proposed ad gve by ( cos( θ ˆ θ ) D ) where θ s the observed data ad data ad D [,]. θˆ s the predcted Values that are approxmately close to zero would dcate that the data follows the specfed dstrbuto. Watso-Wllams test The Watso-Wllams test s a crcular aalogue of two sample t-test. It assesses whether the mea drectos of two or more groups are detcal or ot (Beres, 9). The hypotheses are gve below: H All of s groups share a commo mea drecto, that s, α... α s H Not all s groups have a commo mea drecto A The test statstc (Marda ad Jupp, ) s gve as

5 844 It. J. Phys. Sc. F ( N s)( K ( s )( N s R s R) R ) where R s the mea resultat whe all samples are pooled ad R the mea resultat vector computed o the th group aloe (smlar to total varace ad wth group varace the ANOVA settg). 3 The correcto factor K s computed from K +, 8κ where κ s the maxmum lkelhood estmate of the cocetrato parameter of a vo Mses dstrbuto wth resultat vector r. We compute κ va the approxmato gve by Fsher (993). Here w r w s the mea resultat of the s resultat vector computed for each group dvdually. The obtaed value s the compared to a crtcal value at the δ level obtaed from the F- table ad a small p value meas reecto of the ull hypothess, H. FITTING THE WIND DIRECTION WITH CIRCULAR PROBABILITY DISTRIBUTION Parameter estmates For all crcular dstrbutos, the three mportat parameters are mea agle, cocetrato parameter ad mea resultat respectvely. The mea agle, µ s gve by S ta S >, C > C S µ ta + π C < C S ta + π S <, C > C where C cosθ ad S sθ. The mea resultat of crcular dstrbuto, ρ s gve by ρ ( C + S ), r where C cos θ ad S s θ. The cocetrato parameter of crcular dstrbuto κ (Fsher, 993) s gve by: ρ + ρ + ρ 3.43 κ ρ + ρ 3 ( ρ 4ρ + 3ρ ) ρ < ρ <.85 ρ.85 The parameters estmates for all seve locatos ths study are gve subsequetly. Graphcal ad umercal measures for aual wd drecto Table shows the parameter estmated for each data set. After the parameters have bee obtaed from the observed data, a ew set of data s geerated from these parameters. The the geerated data s compared wth the observed data graphcally by usg cumulatve dstrbuto fucto (cdf) plot ad umercally by usg three goodess-of-ft tests as prevously dscussed. Oe example of a cdf plot s show subsequetly for the locato of whch the observed values ad ftted values for all the dstrbuto cosdered are plotted o the same graph. It ca be see that oe of the cdf plots fts well wth the observed cdf plot. I fact, smlar plots doe for the remag sx locatos also show a smlar result to the oe Fgure 3. Ths dcates that oe of these four crcular dstrbutos s a good ft for all of the specfed locatos for the aual dataset. To verfy ths outcome, three goodess-of-ft tests were used to calculate the ftess. The results are show Tables to 4. As ca be see from the Tables to 4, the smaller the value of the mea ad mea crcular dstace, the better the specfed dstrbuto fts wth the wd drecto. The same apples for the Watso- Wllams test, the smaller the F value compared to the p- value dcates the better the ft wth the wd drecto. But as ca be observed, both cdf plot ad the three goodess-of-ft tests do ot show a good ft. Ths could very well to be due to the fact that the aual wd drecto data comprse the wd drecto for both mosoos, amely the southwest ad ortheast mosoos. As a result, the aual wd drecto data have a mxture of both etrely two dfferet two mosoos, thus hderg the search for the best crcular dstrbuto to descrbe the aual wd drecto. The ext step of the aalyss to exame the data wth respect to the two mosoos.

6 Kamsa et al. 845 Table. Parameter estmates for all seve locatos. Locato Parameter agle ( rada) Cocetrato parameter resultat Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata Wrapped - Cauchy.7.6 Crcular Uform vo Mses Wrapped - ormal Fgure 3. The cdf plot of. Table. The mea for aual wd drecto. Locato Crcular dstrbuto Crcular uform Wrapped Cauchy Wrapped ormal vo Mses Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata

7 846 It. J. Phys. Sc. Table 3. The mea crcular dstace for aual wd drecto. Locato Crcular dstrbuto Crcular uform Wrapped Cauchy Wrapped ormal vo Mses Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata Table 4. The Watso-Wllams test for aual wd drecto. Locato Crcular dstrbuto Crcular uform Wrapped ormal Wrapped Cauchy vo Mses Statstcal test p F p F p F p F Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata Table 5. Parameter estmates for ortheast (N.e) ad southwest (S.w) mosoos. Locato Parameter agle (rada) Cocetrato parameter resultat Mosoo N.e S.w N.e S.w N.e S.w N.e S.w Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata FITTING MONSOON WIND DIRECTION WITH CIRCULAR PROBABILITY DISTRIBUTION Parameter estmates As metoed earler, sce the results obtaed from the cdf plot ad the goodess of ft do ot provde cosstet results, the aual data were the categorsed accordg to the two maor mosoos Malaysa amely, the southwest mosoo ad the ortheast mosoo (the southwest mosoo hts the west coast of Malaysa from November tll March ad the ortheast mosoo the east coast of Malaysa from May tll September every year). The same procedure appled for studyg the aual data was repeated for the mosoo-based data. Table 5 shows the parameters estmate for both mosoos

8 Kamsa et al crcular uform vo Mses.5.4 Wrapped - ormal Wrapped - Cauchy Fgure 4. The cdf plot of ortheast mosoo for. Kota Bharu.9 vo Mses.8 Kota Bharu.7.6 wrapped-ormal.5 wrapped-cauchy.4 crcular uform Fgure 5. The cdf plot of ortheast mosoo for Kota Bharu. accordg to the locatos. As ca be see from Table 5, locatos that faced a partcular mosoo have larger cocetrato parameters compared to the oes that dd ot face the partcular mosoo. For stace, Lagkaw ad Melaka have a larger cocetrato parameter durg the southwest mosoo compared to Kota Bharu, Kuala Tereggau ad Kuata whch have larger cocetrato durg the ortheast mosoo. Larger cocetrato parameter suggests that the data s closer to each other. Graphcal ad umercal measures for mosoo wd drecto After the parameters had bee obtaed, ew data were geerated ad compared wth the prevous data. Followg are the results for both the cdf plot (for two locatos) ad the goodess-of-ft tests (for all the locatos). As a llustrato, the cdf plots for two locatos amely ad Kota Bharu s as gve o Fgures 4 to 7. We oted that a umber of fdgs ca be draw. Ftted wth probablty crcular dstrbutos by

9 848 It. J. Phys. Sc..9.8 wrapped - ormal.7 crcular uform.6 wrapped - Cauchy.5.4 vo Mses Fgure 6. The cdf plot of southwest mosoo for. Kota Bharu.9 Kota Bharu.8 wrapped-ormal.7 vo Mses.6.5 wrapped-cauchy.4 crcular uform Fgure 7. The cdf plot of southwest mosoo for Kota Bharu. mosoos; locatos o the east coast lke Kota Bharu seems to ft wth vo Mses durg the ortheast mosoo but t s ot the case durg the southwest mosoo (Fgures 5 ad 7). A smlar patter ca be detected durg the southwest mosoo where vo Mses seems to ft wth (Fgure 6) but ot for Kota Bharu. To support these fdgs, the three goodess-offt tests results are show Tables 6 to 9.

10 Kamsa et al. 849 Table 6. ad crcular dstace of ortheast (N.e) mosoo. Locato Crcular dstrbuto Crcular uform Wrapped Cauchy Wrapped Normal vo Mses crcular dstace Performace dcator crcular dstace crcular dstace crcular dstace Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata Table 7. Watso-Wllams test for ortheast (N.e) mosoo. Locato Crcular uform Wrappedormal Wrapped- Cauchy Crcular dstrbuto Vo Mses Crcular dstrbuto Statstcal test Crcular uform Wrappedormal Wrapped- Cauchy p F p F p F p F Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata Table 8. ad mea crcular dstace for southwest (S.w) mosoo Locato Crcular dstrbuto Crcular uform Wrapped Cauchy Wrapped Normal vo Mses Performace dcator crcular dstace crcular dstace crcular dstace crcular dstace Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata The goodess-of-ft tests results are cosstet wth the cdf plot. Vo Mses fts well wth the locatos alog the east coast of Malaysa durg the ortheast mosoo ad locatos alog the west coast durg the southwest

11 85 It. J. Phys. Sc. Table 9. Watso-Wllams test for southwest (S.w) mosoo Locato Crcular dstrbuto Crcular uform Wrapped-ormal Wrapped-Cauchy Vo Mses Statstcal test p F p F p F p F Lagkaw Melaka Sea Kota Bharu Kuala Tereggau Kuata mosoo. Although for Kuala Tereggau, the Watso- Wllams test does ot show a acceptable p-value, ths s uderstadable sce the data for Kuala Tereggau cotas a large amout of mssg values durg the ortheast mosoo compared to the other locatos. Where Sea s cocered, whe t was tested usg mea ad mea crcular dstace, for both mosoos, t shows a good ft though the values for both performace dcators are larger tha for other locatos whe ftted wth vo Mses. However, whe the Watso- Wllams test was appled, Sea do ot ft well wth ay of the dstrbuto cosdered for both mosoos. Ths s because of Sea s partcular locato. It s stuated at the souther part of Malaysa where t s exposed to both mosoos. Two approaches of performace dcators, the mea ad the mea crcular dstace are cosstet wth the graphcal ad Watso-Wllams test whch mea that the two performace dcators ca be used to test the goodess-of-ft for a comparso betwee two sets of crcular data. REFERENCES AXIS (3). A program for the statstcal aalyss of crcular data, Verso.. Bogda M, Bogda K, Futschk A (). A data drve smooth test for crcular uformty. A. Ist. Stat. Math., 54(): Beres P (9). CrcStat: A MATLAB toolbox for crcular statstcs. J. Stat. Softw., 3(): -. Fsher NI (993). Statstcal Aalyss of Crcular Data. Cambrdge Uversty Press, Cambrdge, pp Jammalamadaka SR, SeGupta A (). Topcs I Crcular Statstcs. World Scetfc, Sgapore. pp. 33: Marda KV, Jupp PE (). Drectoal Statstcs. Joh Wley, Chchester, pp Marda KV (97). Statstcs of Drectoal Data. Academc Press, New York, pp Razal AM, Ahmad A, Zaharm A, Sopa K (8). Fttg the probablty dstrbuto to wd drecto data Bag. Proc. Sem. Eg. Math., pp Wzelus T (7). Developg wd power proects: Theory ad practce. Earthsca Publcato Ltd., Lodo. Cocluso The aual wd drecto recorded at maxmum wd speed 5 dd ot ft wth ay of the four crcular dstrbutos cosdered for all the seve locatos. Ths s because the data gathered were spread betwee the two maor mosoos Malaysa. Whe the data were dvded accordg to the two mosoos whch are the ortheast ad the southwest mosoos, vo Mses dstrbuto seems to be the best crcular dstrbuto that fts well wth the data ad depedg o the locato ad the mosoo. I cocluso, t ca be sad that vo Mses dstrbuto s the best crcular dstrbuto to descrbe the wd drecto recorded at maxmum wd speed ad t depeds o the locato ad the mosoo.