Songklanakarin Journal of Science and Technology SJST R2 Khamkong

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1 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog A Modfed Box ad Cox Power Trasformato to Determe the Stadardzed Precptato Idex For Revew Oly Joural: Sogklaakar Joural of Scece ad Techology Mauscrpt ID SJST-0-0.R Mauscrpt Type: Orgal Artcle Date Submtted by the Author: -Apr-0 Complete Lst of Authors: Chato, Taachot ; Chag Ma Uversty, Multdscplary Scece Research Cetre, Faculty of Scece Khamkog, Maad; Chag Ma Uversty, Departmet of Statstcs, Faculty of Scece Keyword: drought, o-ormal, postvely skewed dstrbuto, rafall, skewess For Proof Read oly

2 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 Abstract A Modfed Box ad Cox Power Trasformato to Determe the Stadardzed Precptato Idex Taachot Chato,, Maad Khamkog,* Multdscplary Scece Research Cetre, Faculty of Scece, Chag Ma Uversty, Chag Ma, 00, Thalad Departmet of Statstcs, Faculty of Scece, Chag Ma Uversty, Chag Ma, 00, Thalad For Revew Oly * Correspodg author, Emal address: maad.k@cmu.ac.th The objectve of ths research was to create a ew trasformato method based o a modfcato of the Box ad Cox power trasformato (SMBC) by addg the rato of skewess to two sample szes to evaluate drought codtos by determg the stadardzed precptato dex (SPI). Alog wth varous classcal data trasformato methods: the Box ad Cox power trasformato (BC), the expoetal trasformato, the Yeo ad Johso trasformato, ad a modfcato to BC by addg rage, the results of a smulato study showed that the BC ad SMBC methods had smlar effcecy whe trasformg gamma data, Webull data, ad Pearso type III data to a ormal dstrbuto, ad otably, SMBC performed partcularly well wth the latter. Drought codtos were evaluated usg the SMBC trasformato o real-lfe data from ra gaugg statos at Muag (Lamphu), Mae Prk (Lampag) ad Chom Thog (Chag Ma) Thalad, wth whch t proved to be partcularly useful determg the SPI. Keywords: drought, o-ormal, postvely skewed dstrbuto, rafall, skewess. Itroducto For Proof Read oly

3 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 A drought s a atural dsaster caused by a water shortage a area for a exteded tme, severely affectg the local ecoomy ad socety. I Thalad, the ma cause of drought s suffcet ra the ray seaso (Agro-meteorologcal Academc Group Meteorologcal Developmet, 0). There are may ways to evaluate drought codtos, ad the stadardzed precptato dex (SPI) s a popular ad wdely used method to detect ad motor drought. For Revew Oly The SPI was developed by McKee et al. () for evaluatg drought at dfferet tmes. Calculato of the SPI uses cumulatve rafall data durg a specfc tme-perod of terest, for stace,,,,, or -moth tme scales, usg the cumulatve rafall amout trasformed to stadard ormal. The SPI ca be used to classfy codtos as dry ad wet each area (Agro-meteorologcal Academc Group Meteorologcal Developmet, 0). Whle examg the dstrbuto of rafall, McKee et al. () ftted a gamma dstrbuto to the rafall data the SPI calculato, whereas other researchers have tred to produce a approprate model for studyg rafall whe applyg SPI to a partcular area uder study. For stace, Zhag et al. (00) appled a logormal dstrbuto to ft the rafall data from Pearl Rver, Cha, whereas Yusof ad Hu- Mea (0) ftted the rafall data from the state of Johor, Malaysa to be from a Webull dstrbuto because of ts heavy tal. Gabrel (0) used a Pearso Type III dstrbuto wth hs calculato of the SPI for Sao Paulo, Brazl, ad Khamkog ad Bookkamaa (0) ftted a geeralzed extreme value dstrbuto for the aual maxma of daly ad two-day rafall data upper Norther Thalad. Evrometal varables most frequetly dsplay asymmetrc dstrbuto patters wth varous levels of skewess ad kurtoss, ad precptato data are o For Proof Read oly

4 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 excepto sce they are skewed postvely (Twardosz ad Walaus, 0). I partcular, data trasformato methods are used to trasform data that s o-ormally dstrbuted data to a ormal dstrbuto. Maly () stated that a expoetal trasformato s qute effectve at turg a skewed umodal dstrbuto to a early symmetrc ormal dstrbuto, whereas the Yeo ad Johso trasformato (000) s a effectve seres of power trasformatos of a skewed dstrbuto. More recetly, For Revew Oly Watthaacheewakul (0) proposed a modfed Box ad Cox trasformato as a approprate method to trasformed rght-skewed data to a ormal dstrbuto. Our focus ths research s to evaluate the avalable data trasformato methods by applyg them to the stadardzed precptato dex to evaluate drought.. Materals ad Methods. Types of Dstrbuto A gamma dstrbuto s a two parameter famly wth a probablty desty fucto (pdf) wrtte as ( α ) x β x e f ( x; α, β ) = α β Γ( α), for x> 0 ad α, β > 0, where α x Γ ( α ) = x e dx s the gamma fucto, α s the shape parameter, ad β s the scale parameter. 0 A Webull dstrbuto s a two parameter famly wth a pdf expressed as For Proof Read oly ()

5 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 α x f ( x; α, β ) = β β α α x β where α s the shape parameter ad β s the scale parameter. For Revew Oly e, for x> 0 ad α, β > 0, A Pearso Type III dstrbuto (P) s a three-parameter gamma dstrbuto wth the pdf wrtte as α ( x ξ ) β ( x ξ ) e f ( x; α, β, ξ ) = α β Γ( α), for where ξ s the locato parameter, α s the shape parameter, β s the scale parameter, µ s the mea, σ s the stadard devato, ad γ s the skewess. A ormal dstrbuto s a two-parameter famly wth the pdf expressed as x µ σ f ( x; µ, σ ) = e, for x, µ R ad σ > 0, () πσ where µ s the locato parameter ad σ s the scale parameter.. Trasformato Methods outled here. The methods of trasformg o-ormal data to ormal data our study are The Box ad Cox Power Trasformato (BC) Box ad Cox () proposed a data trasformato method for trasformg o-ormal data to ormal data wth homogeety of varaces: x Y = l( x) 0 = 0 () γ > 0, ξ = ( µ σ ) / γ, (), for x > 0, () For Proof Read oly

6 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 where s a trasformato parameter. The Expoetal Trasformato (EP) Maly () proposed a data trasformato method whch s qute effectve at trasformg data from a skewed umodal dstrbuto to a early symmetrc ormal dstrbuto: exp( x) 0 Y = x, = 0 () where s a trasformato parameter. The Yeo ad Johso Trasformato (YJ) Yeo ad Johso (000) proposed a data trasformato method whch was developed from the Box ad Cox trasformato seres ad s effectve o skewed dstrbutos: For Revew Oly ( x+ ) Y = l( x + ) 0, = 0 () where s a trasformato parameter. Watthaacheewakul s (0) Modfed Box ad Cox Power Trasformato (MBC) Ths s a coveet method for trasformg rght-skewed data to a ormal or early ormal dstrbuto. Watthaacheewakul (0) proposed a modfed Box ad Cox trasformato the form For Proof Read oly

7 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 ( x+ c) Y = 0 l( x, () + c) = 0 where s a trasformato parameter ad c s the rage of the data. The proposed trasformatos of the Modfed the Box ad Cox Power trasformato The Proposed Modfed Box ad Cox Power Trasformato (SMBC) s a For Revew Oly mprovemet o the orgal Box ad Cox Power trasformato whch makes t approprate for use trasformg rght-skewed data to a ormal or early ormal dstrbuto. Hece, f c of ths method has cotas zero values, the t becomes a Box ad Cox Power trasformato. BC has better effcacy for trasformg data but s ot approprate whe the data cotas zero values. Rafall data from Thalad have data cotag zero values summer ad wter, thus we proposed a alteratve trasformato method by addg c as the rato of skewess betwee two sample szes. ( x+ c) Y = 0 l( x, () + c) = 0 (x Q ) where s a trasformato parameter ad c =, whch Q s the d (sd) quartle, x s the mea, sd s the stadard devato ad s the sample sze. The estmato of the trasformato parameter of SMBC method s llustrated the Appedx.. Estmato of Trasformato Parameter The value of trasformato parameter (), (), (), ad () ca be estmated by the probablty desty fucto of a ormal dstrbuto as For Proof Read oly

8 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 f ( y µ,σ ) = exp ( y µ ), (0) ( ) πσ σ where y s the trasformed value, µ s the mea ad The lkelhood fucto of a ormal dstrbuto s gve by If y where ( ) σ = πσ For Revew Oly σ s the varace. L( µ, σ y ) = exp ( y µ ). () x = s the lkelhood fucto, the x L( µ, σ, x ) exp µ J ( y;x) ( πσ ) = σ, () = y J ( y;x) = = s the Jacoba of the trasformato. x For a fxed, the maxmum lkelhood estmato for µˆ ad x x ˆ. = = ad σ = Substtute µˆ ad x ˆ µ = = ˆσ are ˆσ to equato (). Thus, the log lkelhood s ( ) ( ) l L( x ) = l π l x x + ( ) l( x ) () = = = The maxmum lkelhood estmate of trasformato parameter () s l l x x x x x d l L( x ) = = = = + + l( x ) = 0, d = x x = = For Proof Read oly ()

9 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 the maxmum lkelhood estmate of trasformato parameter () s x x x e x e e x d l L( x ) = = = = + + x d x = x e e = = For Revew Oly = 0, the maxmum lkelhood estmate of the trasformato parameter for () s ( ) l( ) x + x + = ( ) ( ) l( ) l ( ) x + x + x + d L x = = = + + l( x d = ( + ) ( ) x x + = = + ) = 0, ad the maxmum lkelhood estmate of the trasformato parameter () (detal are gve appedx) s ( ) l( ) x + c x + c = ( ) ( ) l( ) l ( ) x + c x + c x + c d L x = = = + + l( x d = ( + ) ( ) x c x + c = = + c) = 0. Watthaacheewakul (0) recommeded a umercal method such as bsecto to fd a sutable value for the trasformato parameters where the slope of the curvature of the maxmzed log lkelhood fucto s early zero. However, to estmate the parameters, the Bsecto method eeds to be repeated a umber of tmes more tha ether the Secat method or the others. Thus, the Secat method ca be used as a alteratve to the Bsecto method.. Stadardzed Precptato Idex (SPI) () () () The SPI s where the cumulatve amout of rafall has bee trasformed to stadard ormal. A postve SPI dcates that the observed precptato s greater tha For Proof Read oly

10 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 the mea precptato whereas a egatve SPI dcates the cotrary. Iterpretato of the SPI crtero s show Table.. Model Selecto Crtera Two model selecto crtera for cosderg the problems selectg the bestft dstrbuto are as follows: Aderso-Darlg (AD) Test. Aderso ad Darlg () proposed a For Revew Oly goodess-of-ft test whch compares the observed cumulatve dstrbuto fucto (CDF) wth the expected CDF, ad s defed as A = = ( ) where s the sample sze, F() [ l F( x ) + l( F( )] x +, () s the expected CDF, ad x are the ordered data. Akake formato crtero (AIC). Akake () proposed a crtero for the selected model by comparg the actual model wth the proposed model, ad s defed as AIC= k l L, () where k s the umber of parameters model ad L s the maxmzed value of the lkelhood fucto for the estmated model.. Numercal Studes Methodology.. Smulato Study. Geerate the data from a gamma dstrbuto, a Webull dstrbuto ad Pearso type III dstrbuto usg varous parameters ad sample szes wth the R program. For Proof Read oly

11 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page 0 of 0 0. Ivestgate whether data follow a gamma dstrbuto, Webull dstrbuto ad Pearso type III dstrbuto usg the AD test at the 0.0 sgfcace level.. Check whether the data are ormally dstrbuted usg the AD test at the 0.0 sgfcace level.. Estmate the trasformato parameters by applyg the secat method.. Smulate data trasformato by each of the data trasformato methods. For Revew Oly. Re-check to see f the trasformed data have a ormal dstrbuto usg the AD test at the 0.0 sgfcace level.. Repeat steps -,000 tmes.. Cosder the percetage of ormal dstrbutos. Full detals are gve Fgure... Applcato to the SPI Idex of Rafall Data. The rafall data were separated to the three seasos: summer (February to May), ray (Jue to September), ad wter (October to Jauary of the ext year), each of four moths durato. The real-lfe data were gathered from ra gaugg statos at Muag (Lamphu), Mae Prk (Lampag), ad Chom Thog (Chag Ma), Thalad.. For each data set, the ukow parameters of each dstrbuto were estmated by the maxmum lkelhood method ad by selectg the best-ft dstrbuto usg the AD test at the 0.0 sgfcace level ad crtero AIC.. Rafall data were trasformed usg SMBC.. The trasformed rafall data were re-checked to see f they were ormally dstrbuted usg the AD test at the 0.0 sgfcace level. For Proof Read oly

12 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0. If the trasformed rafall data had a ormal dstrbuto, they were evaluated as drought usg the SPI.. Results ad Dscusso. Smulato Study For a gamma dstrbuto, the BC trasformato obtaed the hghest percetage For Revew Oly followed by the SMBC trasformato. Whe the sample sze was small, the EP trasformato gaed the hghest percetage, ad whe skewess ad sample sze creased, the EP the YJ ad the MBC trasformatos showed percetages of reducto (Table ). For a Webull dstrbuto, the BC ad SMBC trasformatos were smlarly effcet. Moreover, whe skewess ad the sample sze creased, SMBC showed the best performace (Table ). For a Pearso Type III dstrbuto, the BC ad SMBC trasformatos were smlarly effcet. Whe the sample sze was small, the MBC trasformato gaed the hghest percetage (Table ).. Rafall Data Study Applcato of the SMBC trasformato to rafall data are show Table. The ra gaugg stato at Muag (Lamphu) had maxmum/mmum cumulatve seasoal rafall of,0 mm/. mm wth a cetral tedecy of. mm. The Pearso type III dstrbuto s the best-ft dstrbuto to these data compared to the others. The ra gaugg stato at Mae Phrk (Lampag) had a maxmum/mmum cumulatve seasoal rafall of,0 mm/. mm wth a cetral tedecy of. mm. For Proof Read oly

13 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 The Webull dstrbuto provded a very good ft to these data alog wth the Pearso Type III ad gamma dstrbutos. The ra gaugg stato at Chom Thog (Chag Ma) had a maxmum/mmum cumulatve seasoal rafall of mm/. mm wth a cetral tedecy of.mm. The gamma dstrbuto was the most approprate compared to the others. For Revew Oly The evaluato of drought by applyg the SPI yelded the followg results. I Fgure, we ca see fdgs from the Muag (Lamphu) stato showed extremely dry codtos, ad, whereas, as show Fgure, those from the Mae Phrk (Lampag) stato dcated extremely dry codtos, 0, 00, ad 00. Lastly, results from the Chom Thog (Chag Ma) stato showed extremely dry codtos,, 00, ad 00 (see Fgure ).. Coclusos The data trasformato method to covert a postvely skewed dstrbuto to a ormal dstrbuto s most essetally for drought aalyss usg the SPI. Mostly, the BC trasformato was the most powerful but may ot be approprate whe the data cotas zero values. We proposed a alteratve trasformato method, SMBC, by addg the rato of skewess to two sample szes. The smulato results showed that the BC ad SMBC trasformatos were smlarly effcet for both gamma ad Webull dstrbutos, ad SMBC performed partcularly well wth a Webull dstrbuto. Therefore, our recommedatos are that the SMBC trasformato s useful whe data follow a Webull dstrbuto or whe data cota zero values for evaluatg drought codtos by applyg the SPI. Moreover, f oe uses the SMBC trasformato for the For Proof Read oly

14 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 Webull dstrbuto to the ormal dstrbuto, the we could be costruct a populato mea cofdet terval estmator as usefuless for further data aalytcal. The exted dea for the future research s to effort the SMBC trasformato to other kd of postvely skewed dstrbutos such as Gumbel dstrbuto, geeralzed expoetal dstrbuto ad so o. Ackowledgmets We are grateful to referees for ther thoughtful commets. The authors are grateful for the facal ad research facltes supported by the Multdscplary Scece Research Cetre, Faculty of Scece, Chag Ma Uversty ad the CMU Md-Career Research Fellowshp program. Refereces Agro-meteorologcal Academc Group Meteorologcal Developmet. 0. Study o For Revew Oly drought dex Thalad. /fo.php?fleid=. [March, 0]. Akake, H.. Iformato theory ad a exteso of the maxmum lkelhood prcple. d Iteratoal Symposum o Iformato Theory, Tsahkadsor, Armea, USSR, -. Aderso, T.W ad Darlg, D.A.. Asymptotc theory of certa "goodess-of-ft" crtera based o stochastc processes. Aals of Mathematcal Statstcs.,. For Proof Read oly

15 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 Box, G.E.P. ad Cox, D.R.. A aalyss of trasformatos. Joural of the Royal Statstcal Socety. Seres B (Methodologcal). (), -. Gabrel, C.B. 0. Stadardzed precptato dex based o Pearso type III dstrbuto. Revsta Braslera de Meteorologa. (), -0. Khamkog, M. ad Bookamaa, P. 0. Developmet of statstcal models for maxmum daly rafall upper orther rego of Thalad. Chag Ma Joural For Revew Oly of Scece. (), 0-0. Maly, B.F. J.. Expoetal data trasformatos. The Statstca. (), -. McKee, T.B., Doeske, N.J. ad Klest, J.. The relatoshp of drought frequecy ad durato o tme scale. Proceedgs of the Eghth Coferece o Appled Clmatology, Calfora, USA, -. Twardosz, R. ad Walaus, A. 0. A practcal method for testg of the sgfcace of dfferece betwee average values precptato seres. Przeglad Geofzyczy. (),. Watthaacheewakul, L. 0. Trasformatos wth Rght Skewed Data. Proceedgs of the World cogress o Egeerg 0, Lodo, UK. -. Yeo, I. ad Johso, N. R A ew famly of power trasformatos to mprove ormalty or symmetry. Bometrka. (), -. Yusof, F. ad Hu-Mea, F. 0. Use of statstcal dstrbuto for drought aalyss. Appled Mathematcal Sceces. (), 0 0. Zhag, Q., Xu, C.Y. ad Zhag, Z. 00. Observed chages of drought wetess epsodes the Pearl Rver Bas, Cha, usg the stadardzed precptato dex ad ardty dex. Theory of Appled Clmatology. (),. For Proof Read oly

16 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 Appedx Estmato of Trasformato Parameter for SMBC The value of trasformato parameter ( ) ca be estmated by the probablty desty fucto of a ormal dstrbuto as For Revew Oly f ( y µ,σ ) = exp ( y µ ), (A) ( ) πσ σ where y s the observed value, µ s the mea, ad σ s the varace. The lkelhood fucto of a ormal dstrbuto s gve by For y L( µ, σ y ) = exp ( y µ ) (A) ( x + c) ( ) σ = πσ =, the lkelhood fucto ca be wrtte as ( ) L( µ, σ x + c,, c x ) exp µ = J(y;x) = Where ( πσ ) σ y J ( y;x) = = s the Jacoba of the trasformato. x For a fxed, the maxmum lkelhood estmators µˆ ad ( x + c) µ ˆ = = ad ( x + ) c ( x + c) Substtute µˆ ad σˆ =. = = ˆσ are derved as ˆσ to equato (A). Thus, the log lkelhood s For Proof Read oly (A)

17 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 l L(µ, σ,, x ) = l π l σ σ = ( x + c) For Revew Oly µ + ( ) l( x + c) = (A) The log lkelhood fuctos of becomes ( ) ( ) x + + c x c = l L( x ) l π l + + ( ) l( x c) = = = (A) The maxmum lkelhood estmate of the trasformato parameter s d l L( x ( ) ( ) + + ) d x c x = c l π l + + ( ) l( x c) d d = = = (A) ( ) l( ) x + c x + c = ( ) ( ) l( ) l ( ) x + c x + c x + c d L x = = = + + l( x d = ( + ) ( ) x c x + c = = For Proof Read oly + c) = 0. (A)

18 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 SPI Category.00 ad above Extremely wet. to. Severely wet.00 to. Moderately wet -0. to0. Near ormal -.00 to-. Moderately dry -. to-. Severely dry.00 ad less Extremely dry For Revew Oly Table The stadardzed precptato dex (SPI) categores based o the tal classfcato of SPI values. α β SK BC EP YJ MBC SMBC Note: The bold-face values were cosdered to dcate the performace trasformato. Table Percetage ormal dstrbuto after trasformato of a gamma dstrbuto. For Proof Read oly

19 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 α β SK BC EP YJ MBC SMBC For Revew Oly Note: The bold-face values were cosdered to dcate the performace trasformato Table Percetage ormal dstrbuto after trasformato of a Webull dstrbuto. For Proof Read oly

20 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 α β ξ SK BC EP YJ MBC SMBC For Revew Oly Note: The bold-face values were cosdered to dcate the performace trasformato Table Percetage ormal dstrbuto after trasformato of a Pearso Type III dstrbuto. For Proof Read oly

21 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page 0 of 0 0 Statos Muag Lamphu Maepk Chom Thog Rafall Gamma Webull P Meda QD Skewess M Max AIC AD AIC AD AIC AD., ,..,..,.0 0.., ,. 0.,. 0., ,. 0.,. 0.,.0 0. Note: The bold-face values were cosdered to dcate the best dstrbuto ftted to the real data ad QD s quartle devato. Table Descrptve statstcs ad model selecto crtera of the four moth rafall perod. For Revew Oly For Proof Read oly

22 Page of Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog 0 0 Start Geerate the data from a gamma dstrbuto, a Webull dstrbuto ad Pearso type III usg varous parameters ad sample szes wth the R program. Ivestgate whether data follow a gamma dstrbuto, Webull dstrbuto ad Pearso type III dstrbuto usg the AD test at the 0.0 sgfcace level. For Revew Oly Check whether the data are ormally dstrbuted usg the AD test at the 0.0 sgfcace level. Estmate the trasformato parameters by applyg the secat method. Smulate data trasformato by each of the data trasformato methods No Re-check to see f the trasformed data have a ormal dstrbuto usg the AD test at the 0.0 sgfcace level. Repeat,000 tmes Yes Cosder the percetage of ormal dstrbutos. Ed Yes No Fgure Steps a smulato study. For Proof Read oly Yes No

23 Sogklaakar Joural of Scece ad Techology SJST-0-0.R Khamkog Page of 0 0 Fgure Four moth SPI for Muag, (Lamphu). Fgure Four moth SPI for Mae Phrk, (Lampag). Fgure Four moth SPI for Chom Thog, (Chag Ma). For Revew Oly For Proof Read oly