DERIVATION OF THE PROBABILITY PLOT CORRELATION COEFFICIENT TEST STATISTICS FOR THE GENERALIZED LOGISTIC DISTRIBUTION

Transcription

1 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy DERIVATION OF THE PROBABILITY PLOT CORRELATION COEFFICIENT TEST STATISTICS FOR THE GENERALIZED LOGISTIC DISTRIBUTION by Sooyoung Km (), Hongjoon Shn (), Taesoon Km (3) and Jun-Haeng Heo (4) () Department of Cvl and Envronmental Engneerng, Yonse Unversty, Korea () Department of Cvl and Envronmental Engneerng, Yonse Unversty, Korea (3) Korea Hydro and Nuclear Power Co., Ltd. () Department of Cvl and Envronmental Engneerng, Yonse Unversty, Korea ABSTRACT The selecton of an approprate probablty dstrbuton s very mportant n hydrology to estmate the accurate desgn ranfall. An approprate probablty dstrbuton s generally chosen by usng the goodness of ft tests. The PPCC test has been nown as a powerful test among the goodness of ft tests. Generally, the PPCC test statstcs are calculated by consderng sgnfcance levels, sample szes, plottng poston formulas, and shape parameters of a gven probablty dstrbuton. It s mportant to select an exact plottng poston formula for a gven probablty dstrbuton because the PPCC test statstcs are defned from the correlaton coeffcent values based on the selected plottng poston formula. After Cunnane(978) defned the plottng poston that related wth the mean of data and proposed the general formula that can be appled to varous probablty dstrbutons, varous plottng poston formulas have been developed for consderng the effect of coeffcents of sewness related wth the shape parameter for a gven dstrbuton. In ths study, the PPCC test statstcs are derved by usng a plottng poston formula contaned a term of a coeffcent of sewness that can be consdered as the effect of shape parameters for the generalzed logstc dstrbuton(glo). In addton, the PPCC test statstcs for the GLO s derved based on varous sample szes, sgnfcance levels, and shape parameters of the generalzed logstc dstrbuton. And then, the power test to estmate the rejecton ratos of the derved PPCC test statstcs s performed and the comparsons between derved plottng poston formula and other plottng poston are accomplshed. Keywords: Generalzed logstc dstrbuton, probablty plot correlaton coeffcent tet, test statstcs, plottng poston formula INTRODUCTION An approprate probablty dstrbuton n frequency analyss usng annual maxmum ranfall or flood data provdes an accurate desgn quantle for the hydrologc structures such as ban, retenton, and dam. Therefore, the selecton of an approprate probablty dstrbuton s very mportant for the reaseonable desgn. Generally, the selecton of an approprate probablty dstrbuton s based on the goodness of ft test whch s the decson-mang method to evaluate the ftness between sample data and ts populaton for a gven probablty dstrbuton. Many goodness of ft tests have been developed n lteratures and the Kolmogorov-Smrnov test, the Cramer von Mses test, and the ch-square test are popular especally. The Probablty Plot Correlaton Coeffcent(PPCC) test developed by Fllben(975) has been nown as a powerful test among many goodness of ft tests. Snce then, the PPCC test has been appled to varous probablty dstrbutons. Looney and Gulledge(985) appled varous plottng poston formulas to normal dstrbuton and chose the Blom(958) s plottng poston formula for the dervaton of normal PPCC test statstcs. Vogel(986) proposed the PPCC test statstcs for the Gumbel dstrbuton, and Vogel and Kroll(989) derved the PPCC test statstcs for the -parameter Webull and unform dstrbutons n frequency analyss for low flow data. In addton, the PPCC test statstcs of 5% sgnfcance level for gamma dstrbuton are studed by Vogel and McMartn(99), and the PPCC test statstcs for the GEV dstrbuton are provded by Chowdhury et al.(99). Recently, Heo et al.(7) proposed the regresson equatons to estmate the test statstcs for normal, gamma, Gumbel, GEV and Webull dstrbutons. In ths study, the PPCC test statstcs are derved by usng Monte Carlo smulaton for the generalzed logstc dstrbuton. To estmate the test statstcs of the generalzed logstc dstrbuton, the plottng poston formula contaned a term of a coeffcent of sewness s developed by usng the genetc algotthm. Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton

2 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy In addton, the PPCC test statstcs are derved by usng a plottng poston formula contaned a term of a coeffcent of sewness that can be consdered as the effect of shape parameters for the generalzed logstc dstrbuton(glo). And then, the power test to estmate the rejecton ratos of the derved PPCC test statstcs s performed and the comparsons between derved plottng poston formula and other plottng poston formulas such as Blom(958), Grngorten(963), Fllben(975), and Cunnane(978) are accomplshed. DERIVATION OF THE PPCC TEST STATISTICS. The generalzed logstc dstrbuton The generalzed logstc dstrbuton s defned by Eq. ()(Hosng, 986). x x F( x) = + α where x s a locaton parameter, α s a scale parameter, and s a shape parameter. Then, x α α + x< for < and < x x+ for >. The generalzed logstc dstrbuton was recommended for the regonal flood frequency analyss n England by Flood Estmaton Handboo(Insttute of Hydrology, 999).. The dervaton of plottng poston formula usng genetc algorthm Snce Cunnane(978) dscussed the unbased plottng poston that was related wth the mean of sample data and proposed the general formula for varous probablty dstrbutons, many researchers have developed the plottng poston formulas consderng the nfluence of a coeffcents of sewness related wth shape parameters of a gven probablty dstrbuton. However, those lteratures were restrcted for several probablty dstrbutons such as the GEV, Gumbel, and log-pearson Ⅲ dstrbutons. In case of the generalzed logstc dstrbuton, Grngorten(963) s formula was recommended as an approprate plottng poston formula by FEH(Insttute of Hydrology, 999). Therefore, ths research proposes the plottng poston formula wth the concept of unbased plottng poston realted wth the mean of reduced varates for the generalzed logstc dstrbuton. The mean of densty functon of the rth smallest value n random sample n s defned as follows, n! r n r E[ xr ] = xr F( xr ) [ F( xr )] f ( xr ) dxr ( r )!( n r)! () The reduced varates of the generalzed logstc dstrbuton are assumed as follws, F y = F y / F = F where, Eqs.(3) and (4) are the reduced varates n case of negatve( < ) and postve( > ) shape parameters, respectvely. And then, each reduced varate has the ranges of < y < and < y <, respectvely. The reduced varates are substtuded nto Eq.() and the theoretcal reduced varates are expressed by Eqs. (5)~(6). n! F r n r [ ] = [ ] E y F F df ( r )!( n r)! (5) F! F r n r (6) n E[ y] = F [ F] df ( r )!( n r)! F () (3) (4) Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton

3 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy In addton, ths study aopted the real-coded genetc algorthm(rga) that s one of genetc algorthm to estmate the parameters of plottng poston formula wth the theoretcal reduced varates of the generalzed logstc dstrbuton. The objectve functon of RGA s the root mean square error(rmse) between theoretcal reduced varates and calculated those from plottng poston formula contaned parameters. Then, populaton sze s,, generaton number s,, crossover probablty s.8, and mutaton probablty s.. The RGA runs tmes n total because of the nfluence of seed number(.3). Derved plottng poston s examned by comparng the RMSEs between theoretcal reduced varates and calculated those from other plottng poston formulas such as Blom(958), Grngorten(963), and Cunnane(978). The results wth several coeffcents of sewness are shown n Table I. Accordng to Table I, the RMSEs of derved plottng formula n ths study over all sample szes and coeffcents of sewness related wth shape parameters are the smallest. As the results, the calculated reduced varates by the derved plottng formula are more accurate than those by other plottng formulas for the generalzed logstc dstrbuton. Table I - The comparson of RMSE from varous plottng poston formulas Shape parameters Coeffcents of sewness Plottng Poston Formulas Sample sze Derved Blom Grngorten Cunnane The derved plottng formula n ths study s proposed by Eq.(7). Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 3

4 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy +.76γ P = n+.9 where, n s sample sze, s an order, and γ s a coeffcent of sewness from sample data..3 The dervaton of the PPCC test statstcs The PPCC test by usng the correlaton coeffcent between the ordered observaton and the correspondng ftted quantles was provded by Fllben(975) for normalty test. The correspondng ftted quantles of ths test are determned by plottng poston for each observaton. The correlaton coeffcent CC s expressed as follows, CC= n = ( X X )( M M ) n n ( X X ) ( M M ) = = where X and M represent the mean values of the observaton respectvely, and n s the sample sze. X and the ftted quantles (7) (8) M, If correlaton coeffcent CC s close to., the observatons can be defned by the ftted probablty dstrbuton. The order statstc medan for M by Fllben(975) s explaned as follow. M = ϕ ( m ) (9) where φ ( ) means the nverse of cumulatve dstrbuton functon for the standard normal dstrbuton and the medan value m s gven n Eqs.()~(). / (.5) n m = for = () m = (.375) / ( n+.365) for =,3, L, n () / m (.5) n = for n = () If the followng condton s satsfed, the null hypothess cannot be rejected at the q sgnfcant level. r> rq ( n ) (3) where r q (n) s the test statstc of the PPCC test for a gven sample sze and sgnfcance level. As stated above, the PPCC test statstcs are nfluenced by the characterstcs of varous sgnfcance levels, sample szes, plottng poston formulas, and shape parameters for the ftted probablty dstrbuton. Therefore, the applcaton of a proper plottng poston formula for the ftted probablty dstrbuton s mportant to estmate the PPCC test statstcs. The recommended plottng poston formulas to derve the PPCC test statstcs are dfferent for each probablty dstrbuton n many researches(stednger et al., 993) and the recommended plottng poston formulas are shown n Table II. Table II - The recommended plottng poston formulas Type Plottng poston formula Recommended probablty dstrbutons Blom(958) p = ( 3 / 8) / ( n+ / 4) Normal, gamma, lognormal, log-pearson type Ⅲ Grngorten(963) p = (.44) / ( n+.) Gumbel, Webull Cunnane(978) p = (.4) / ( n+.) GEV, log-gumbel The procedure to estmate the PPCC test statstcs are as follows(vogel and McMartn, 99); Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 4

5 (a) Generate parameters, (b) Calculate X of sample sze n ( Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy = L,, n ) for an assumed parent dstrbuton wth gven shape M usng the nverse of the cumulatve dstrbuton functon and plottng poston, (c) Estmate the correlaton coeffcent between generated sample M, X and calculated plottng poston value (d) Repeat the procedure (from step (a) to step (c)), tmes to obtan, correlaton coeffcents, (e) Select, q th smallest r as r q, Ths study apples the followng condtons to derve of the PPCC test statstcs for the generalzed logstc dstrbuton, - Sample szes( n ) :,, 3, 4, 5, 6, 7, 8, 9,,, 5,, 3, and 5, - Sgnfcance levels :.5,.,.5,.,.5,, 5, 9, and 95, - The range of shape parameters : -.3, -., -., -.5,.5,.,., and.3. In ths study, the derved plottng formula for the generalzed logstc dstrbuton shown n Eq.(7) s used for the estmaton of test statstcs. In addton, the test statstcs of the derved plottng formua are compared wth those of exstng plottng formulas such as Blom(958), Grngorten(963), Fllben(975), and Cunnane(978) for other probablty dstrbutons..4 The test statstcs of the generlazed logstc dstrbuton The values of the PPCC test statstcs usng the derved plottng formula are shown n Fgure. Fgure (a) shows the PPCC test statstcs wth 5% sgnfcance leve ans Fgure (b) shows those wth % sgnfcance level. The PPCC test statstcs of the generalzed logstc dstrbuton are smlar to the same absolute values of negatve shape parameters and postve shape parameters, respectvely. For example, the test statstcs n case that shape parameter s -.3 are smlar to those n case that shape parameter s +.3. Ths tendancy s caused by the symmetry of coeffcents of sewness related wth shape parameters of the generalzed logstc dstrbuton. The PPCC test statstcs ncrease as sample szes and sgnfcance levels ncrease, and the absolute values of shape parameters decrease. The results of test statstcs whch are derved by varous plottng formulas such as the derved plottng poston formula, Blom(958), Grngorten(963), Fllben(975), and Cunnane(978) are plotted n Fgure. The PPCC test statstcs by the derved plottng formula are greater than other values over all sample szes. In addton, the dfferences between the test statstcs by the derved plottng formulas and those by other formulas ncrease as the absolute values of shape parameter ncrease. Test Statstcs.8 q=.5 shape=-.3 shape=-. shape=-. shape=-.5 shape=.5 shape=. shape=. shape=.3 Test Statstcs.8 q=. shape=-.3 shape=-. shape=-. shape=-.5 shape=.5 shape=. shape=. shape= Sample sze(n) Sample sze n (a) Sgnfcance level =.5 (b) Sgnfcance level =. Fgure The PPCC test statstcs wth several sgnfcance levels by derved plottng poston formula Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 5

6 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy Test Statstcs 5.85 Shape =-.3 Derved Blom Grngorten Fllben Cunnane Test Statstcs 5.85 Shape = -. Derved Blom Grngorten Fllben Cunnane Sample sze n Sample sze n (a) Shape parameter = -.3 (b) Shape parameter = -. Test Statstcs 5.85 Shape =+. Derved Blom Grngorten Fllben Cunnane Test Statstcs 5.85 Shape =+. Derved Blom Grngorten Fllben Cunnane Sample sze n Sample sze n (c) Shape parameter = +. (d) Shape parameter = +. Fgure The comparson of the PPCC test statstcs wth varous plottng poston formulas. 3 POWER TEST 3. The procedure of power test The performance of the test statstcs by varous plottng poston formulas are examned by the power test whch used the Monte Carlo smulaton. The power tests wth varous shape parameters, sample szes, sgnfcance levels, and plottng poston formulas for the generalzed logstc dstrbuton are as follows; (a) Assume that the parent probablty dstrbuton s the generalzed logstc dstrbuton, (b) Generate data set wth gven shape parameters consderng varous sample szes and plottng formulas, (c) General frequency analyss contaned the goodness of ft test whch means the PPCC test s appled to the generated data set wth varous sgnfcance levels, (d) Repeat the procedure(from step (a) to step (c)) at, tmes, (e) Estmate the performance of varous plottng poston formulas by calculatng the rejecton ratos that are expressed as percentage. Ths study sets up the followng condtons to accomplsh the power test. - Sample szes :, 5, 5,, and, - Assumed shape parameters : -.3, -., -.,.,., and.3, - Appled plottng formulas : Blom(958), Grngorten(963), Fllben(975), Cunnane(978), and the derved plottng formulas n ths study, - Sgnfcance levels :.5(5%) and.(%), - Appled method of parameter estmaton : PWM(Probablty Wegthed Moments). Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 6

7 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy 3. The results of power test The results of power test are shown n Fgure 3. The rejecton ratos are computed as percentages to dvde counted rejecton numbers by,. The rejecton ratos ncrease as the absolute values of shape parameters decrease because the PPCC test statstcs n the same cases are hgher than others. In addton, the rejecton ratos by derved plottng poston formula are generally hgher than those by other plottng poston formulas over all occasons. Therefore, the test statstcs by the derved plottng poston formula are effectve to estmate the ftness between sample data and the parent dstrbuton the generalzed logstc dstrbuton. The rato of rejecton(%) n=5 Shape=-.3 Shape=-. Shape=-. Shape=. Shape=. Shape=.3 The rato of rejecton(%) n= Shape=-.3 Shape=-. Shape=-. Shape=. Shape=. Shape=.3 Derved Blom Grngorten Fllben Cunnane Plottng poston formulas Derved Blom Grngorten Fllben Cunnane Plottng poston formulas (a) Sgnfcance level=.5 and sample sze=5 (b) Sgnfcance level=.5 and sample sze= The rato of rejecton(%) n=5 Shape=-.3 Shape=-. Shape=-. Shape=. Shape=. Shape=.3 The rato of rejecton(%) n= Shape=-.3 Shape=-. Shape=-. Shape=. Shape=. Shape=.3 Derved Blom Grngorten Fllben Cunnane Plottng poston formulas Derved Blom Grngorten Fllben Cunnane Plottng poston formulas (c) Sgnfcance level=. and sample sze=5 (d) Sgnfcance level=. and sample sze= Fgure 3 The comparson of the rejecton rato(%) by varous plottng poston formulas 4 CONCLUSIONS The exact plottng poston formula for the generalzed logstc dstrbuton was derved by usng genetc algorthm and the theoretcal reduced varates n ths study. In addton, the PPCC test statstcs for the generalzed logstc dstrbuton were developed based on varous sample szes, sgnfcance levels, shape parameters of the generalzed logstc dstrbuton, and plottng poston formulas ncludng the derved plottng poston formula. The PPCC test statstcs by the derved plottng poston formula were generally hgher than those by other plottng poston formulas. The power tests by usng used Monte Carlo smulaton were performed to fgure out the ablty of goodness of ft test by varous plottng poston formulas. As a result, the test statstcs by the derved plottng poston formula shows more powerful rejecton to sample data for the generalzed logstc dstrbuton. 5 ACKNOWLEDGEMENT Ths study was fnancally supported by the Constructon Technology Innovaton Program(8-Tech- Inovaton-F) through the Research Center of Flood Defence Technology for Next Generaton n Korea Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 7

8 Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy Insttute of Constructon & Transportaton Technology Evaluaton and Plannng(KICTEP) of Mnstry of Land, Transport and Martme Affars(MLTM). 6 REFERENCES Blom, G. (958). Statstcal estmates and transformed beta varables. John Wley and Sons, New Yor. Chowdhury, J.D., Stednger, J.R., and Lu, L.H. (99). Goodness-of-ft tests for regonal generalzed extreme value flood dstrbutons. Water Resources Research, 7(7): Cunnane, C. (978). Unbased plottng postons - A revew. Journal of Hydrology, 37(3/4): 5-. Fllben, J.J. (975). The Probablty Plot Correlaton Coeffcent Test for Normalty. Technometrcs, 7(): -7. Grngorten, I.I. (963). A plottng rule for extreme probablty paper. Journal of Geophyscal Research, 68(3): Heo, J., Kho, Y., Shn, H., Km, S., and Km, T. (7). Regresson Equatons of Probablty Plot Correlaton Coeffcent Test Statstcs from Several Probablty Dstrbutons. Journal of Hydrology, 355(-4): -5. Hosng, J.R.M. and Walls, J.R. (986a). Paleoflood hydrology and flood frequency analyss. Water Resources Research, (4): Hosng, J.R.M. and Walls, J.R. (986b). The value of hstorcal data n flood frequency analyss. Water Resources Research, (): Insttute of Hydrology (999). Flood Estmaton Handboo. Wallngford, UK. Looney, S.W. and Gulledge, T.R. (985). Use the correlaton coeffcent wth normal probablty plots. The Amercan Statstcan, 39(): Stednger, J.R., Vogel, R.M., and Foufoula-Georgous, E. (993). Frequency analyss of extreme events - chapter 8 of Handboo of Hydrology(ED. Madment, D. R.), McGraw-Hll, New Yor. Vogel, R.M. (986). The probablty plot correlaton coeffcent test for the normal, lognormal, and Gumbel dstrbutonal hypothess. Water Resources Research, (4): Vogel, R.M. and Kroll, C.N. (989). Low-flow frequency analyss usng probablty plot correlaton coeffcents. Journal of Water Resources Plannng and Management, 5(3): Vogel, R.M. and McMartn. D.E. (99). Probablty plot goodness-of-ft and sewness estmaton procedures for the Pearson type Ⅲ dstrbuton. Water Resources Research, 7(): Km et al., Dervaton of the Probablty Plot Correlaton Coeffcent Test Statstcs for the Generalzed Logstc Dstrbuton 8