World Appled Scences Journal 15 (8): 1181-1185, 011 ISSN 1818-495 IDOSI Publcatons, 011 Determnaton of Plottng Poston Formula for the Normal, Log-Normal, Pearson(III), Log-Pearson(III) and Gumble Dstrbutonal Hypotheses Usng The Probablty Plot Correlaton Coeffcent Test 1 Fuladpanah Mehd and Jorabloo Mehd 1 Department of Cvl Engneerng, Ramhormoz Branch, Islamc Azad Unversty, Ramhormoz, Iran Department of Engneerng, Garmsar Branch, Islamc Azad Unversty, Garmsar, Iran Abstract: The selecton of an approprate plottng plot formula for each statstcal dstrbuton s the most mportant step that s generally chosen by usng the goodness of ft tests. Probablty Plot Correlaton Coeffcent (PPCC) test whch was developed by Fllben for normalty s a powerful tool among the goodness of ft tests. PPCC test s based on probablty plot and correlaton coeffcent. In ths paper, PPCC test was used for data wth Normal, Log-Normal, Pearson, Log-Pearson and Gumbel dstrbutons. Ths statstcal test was used for sample of length n=10, 0, 0, 40, 50 and 100. Usng Chow s formula, frequency factor was determned Calforna, Hazen, Webul, Blom, Chegodayov, Tukey and Grngorton plottng poston formulas. Crtcal ponts of the test statcs were provded for samples of mentoned length. Results showed that Blom, Grngorton and Webul plottng poston formula are approprate for the Normal(Log-normal), the Gumbel and PearsonIII (Log-pearsonIII) dstrbutons, respectvely. Key words: Probablty Plot Correlaton Coeffcent Test Statstcal Dstrbutons Plottng Poston Formula Crtcal Ponts INTRODUCTION comparson of the results n both a graphcal (probablty plot) and a numercal (correlaton coeffcent ) form [, ]. Accurate analyss of recorded data plays an Probablty plots have been used wdely n many mportant role n the decson-makng process n resources nvestgatons. Whle analytc approaches for hydrologcal and hydraulc projects. It have been fttng probablty dstrbutons to observed data are, n developed many goodness of ft tests n lteratures such theory, more effcent statstcal procedures than graphcal as the Kolmogorov-Smrnov test, the Cramer von Mses curve fttng procedures, many hydrologsts wouldn t test and the Ch-square test whch are popular especally make engneerng decsons wthout the use of a graphcal [1]. Fllben developed powerful probablty plot dsplay (probablty plot). Probablty plots were correlaton coeffcent (PPCC) tests for normalty whch recommended by the Natonal Research Councl as a bass have followng attractve features: 1) the test statstcs s for extrapolaton of flood frequeccy curve n dam safety conceptually easy to understand because t combnes two evaluatons [1]. Smlarly, The Federal Emergency fundamentally smple concepts: the probablty plot and Management Agency recommends the use of probablty correlaton coeffcent, ) the test s computatonally plots n determnaton of the probablty dstrbuton of smple snce t s only requres computaton of a smple annual maxmum flood elevatons whch arses from the correlaton coeffcent, ) the test statstcs s readly combned effects of ce jam and storm-nduced floddng extendble for testng some non-normal dstrbuton [1]. Looney and Gulledge appled varous plottng hypotheses, 4) the test compares favorably wth seven poston formulas to normal dstrbuton and chose the other tests of normalty on the bass of emprcal power Blom plottng poston formula for the dervaton of normal studes performed by Fllben, 5) the test s nvarant to the PPCC test statstcs [4]. Vogel proposed the PPCC test parameter estmaton procedure employed to ft the statstcs for the Gumbel dstrbuton [] and Vogel and probablty dstrbuton and 6) the test allows a Kroll derved the PPCC test statstcs for the -parameter Correspondng Author: Fuladpanah Mehd, Department of Cvl Engneerng, Ramhormoz Branch, Islamc Azad Unversty, Ramhormoz, Iran. 1181
World Appl. Sc. J., 15 (8): 1181-1185, 011 Webull and unform dstrbutons n frequency analyss I Start. for low flow data [5]. In addton, the PPCC test statstcs II Generate random data. of 5% sgnfcance level for gamma dstrbuton are III Select statstcal dstrbuton. studed by Vogel and McMartn [6] and the PPCC test IV Ft statstcal dstrbuton (step III) on the generated statstcs for the GEV dstrbuton are provded by data (step II). Chowdhury et al. [7]. Recently, Heo et al., proposed the V Classfy ftted data (step IV) to samples of length n. regresson equatons to estmate the test statstcs for VI Ft statstcal dstrbuton (step III) on classfed data normal, gamma, Gumbel, GEV and Webull dstrbutons (step IV) usng equaton (1). [8]. Sooyoung et al., used PPCC test for generalzed VII Calculate r between data and frequency factor. logstc dstrbuton. They used PPCC test extreme value VIII Determne approprate plottng poston formula for dstrbuton. They found that ths test s very strong tool statstcal dstrbuton (step III) usng maxmum r to fnd approprate dstrbuton [9]. Goda had studed values n PPCC test. dfferent plottng plot formulas and ther applcaton n IX Determne crtcal ponts of r usng PPCC test for varous statstcal dstrbutons [10]. Cook conducted a statstcal dstrbuton (step III). study about the effect of samplng error on plottng X End. poston formula n extreme values. Hs study was about normal dstrbuton [11]. Ths algorthm was appled for the followng In ths paper, the Chow formula was used to apply statstcal dstrbuton: the PPCC test for the Normal, Log-normal, PearsonIII, Log-pearsonIII and Gumbel dstrbutons. The Chow Normal and Log-Normal dstrbuton formula s as followng [1]: Pearson type III and Log-Pearson type III dstrbuton X TR = XMEAN + K.S (1) Gumbel dstrbuton Whch X TR s amount of varable wth return perod Detaled of calculaton for each dstrbuton s TR, XMEAN s mean of varable X, K s frequency factor presented n the followng subsectons. and S s standard devaton of X. In ths equaton, K s a functon of probablty occurrence. There are dfferent Normal and Log-nomral Dstrbutons: To determne relatons for calculaton of emprcal probablty approprate plottng poston formula for normal occurrence whch are lsted n Table 1 [1]. Therefore, the dstrbuton, the followng steps were consdered: selecton of the best relatonshp s very mportant. The crtcal ponts (or sgnfcance levels) of dstrbutons of r I Random data (x ) of length 1,000,000 were generated (correlaton coeffcent) were obtaned by usng the n nterval [0,1]. emprcal samplng procedure. II Data were dvded nto samples of length n: 10, 0, 0, 40, 50, 100. For example, there wll be 100000 classes MATERIALS AND METHODS for sample of length length n=10. III The normal dstrbuton was ftted on data (step ) If the sample to be tested s actually dstrbuted as usng the followng equatons [6]: hypotheszed, one would expect the plot of the ordered observatons versus the order statcs means or medans.515517+0.8085r 0.0108r 1 + If x 0.5 r x r - = N = to be approxmately lnear. Thus the product moment x 1 1.4788r 0.18969r 0.00108r + + + correlaton coeffcent whch measures the degree of lnear assocaton between two random varables s an () approprate test statc. Fllben s PPCC test s a smply.515517+0.8085r 0.0108r 1 If x 0.5 r x -[r - < = N = formalzaton of a technque used by statstcal (1-x ) 1 1.4788r 0.18969r 0.00108r + + + hydrologsts for many decades; that s, t determnes the lnearty of a probablty plot. In ths study, n order to () select approprate plottng poston formula for Normal, Log-Normal, PearsonIII, Log-pearsonIII and Gumbel Whch x s generated data n step 1, r s a coeffcent dstrbutons the followng steps were used: and x s normalzed value of x. N 118
World Appl. Sc. J., 15 (8): 1181-1185, 011 Table 1: Plottng Poston Formulas Formula Year Descrpton Calforna 19 Hazen 190 Webul 199 Blom 1954 Chegodayov 1955 Tukey 196 Grngorton 196 IV V m P(x X)= N m-1 P(x X)= N m P(x X)= N+1 m-0.75 P(x X)= N+0.5 m-0. P(x X)= N+0.4 m-1 P(x X)= N+1 m+0.1 P(x X)= N-0.44 Data (step ) n each class were arranged n the descendng order. Probablty of each data n each class was calculated usng the formulas were lsted n the Table 1. Correspondng frequency factors were calculated usng followng equaton for the normal dstrbuton [6]: 1.515517+0.8085w+0.0108w If P 0.5 w = KN = w- P 1+ 1.4788w + 0.18969w + 0.00108w (4) Table : Crtcal ponts of r whch r s the Normal Probablty Plot Correlaton Coeffcent -------------------------------------------------------------------- Class Length 99% 98% 95% 90% 10 0.99579 0.9997 0.99174 0.989 0 0.99640 0.9950 0.9980 0.9964 0 0.9960 0.99071 0.99514 0.99445 40 0.99717 0.99671 0.99641 0.9955 50 0.99710 0.99669 0.9965 0.99574 100 0.9981 0.9981 0.9981 0.9971 Pearson Type III and Log-Pearson Type III Dstrbutons: Sx mentoned steps n subsecton Normal ang Log- Normal were used for pearson type III dstrbuton, too. However, to ft Pearson III dstrbuton on generated data n step, the followng equaton was used [6]: g g xp = {[ x N - + 1] -1} g 6 6 Whch x N s normalzed data, g s skewness coeffcent and x P s transformed data to pearson dstrbuton. The amount of g for Pearson III dstrbuton ftted data s often n nterval [-4,4]; thus, x P were calculated for g= -4,-,-,-1,1,, and 4. Another dfference s n step 5. Frequency factor for pearsoniii dstrbuton s calculated as followng equaton [6]: (6) 1.515517+0.8085+0.0108w If P < 0.5 w = KN = - [w- (1-P) 1+ 1.4788w+0.18969w + 0.00108w Whch P s probablty, w s a coeffcent and K s N frequency factor for normal dstrbuton. I (5) g g Kp =- {[ KN- + 1] -1} g 6 6 Correlaton coeffcents were calculated between each class n steps and 7. These coeffcents (for example, f n=10 then the number of r equals 100000) logarthm of pearson dstrbuton ftted data have pearson were used for two purposes: dstrbuton. Fnal results are presented n Tables to 10. II The maxmum values of r were appled to select approprate plottng poston formula. III The average of r correspondng to approprate probablty formula was calculated as crtcal ponts. The obtaned results for normal dstrbuton can be developed to Log-normal dstrbuton; because the logarthm of normalzed data has normal dstrbuton. Fnal results are presented n Table. Whch K Pand K Nare frequency factor of pearson and normal dstrbuton respectvely, g s skewness coeffcent. Amounts of K were calculated for g= -4,-,-,- P 1,1,, and 4. The obtaned results for normal dstrbuton can be developed to Log-normal dstrbuton; because the Gumbel Dstrbuton: All steps mentoned n subsecton Normal and Log-Normal are appled for the Gumbel dstrbuton, too. There are two dfferences n steps and 5. The followng equaton s used to ft the Gumbel dstrbuton on generated data n step [6]: 6 1 xg = - [0.577 + Ln Ln ] 1-x (7) (8) 118
World Appl. Sc. J., 15 (8): 1181-1185, 011 Table : Crtcal ponts of r whch r s the PearsonIII and log-pearson Table 8: Crtcal ponts of r whch r s the PearsonIII and log-pearson Probablty Plot Correlaton Coeffcent for g= -4 Probablty Plot Correlaton Coeffcent for g= 10 0.9774 0.96566 0.94971 0.9065 10 0.990 0.9899 0.98505 0.97867 0 0.94778 0.9751 0.91506 0.89667 0 0.98846 0.98689 0.97666 0.96986 0 0.941 0.951 0.8961 0.87157 0 0.9858 0.97914 0.97000 0.9677 40 0.9165 0.91016 0.88895 0.8695 40 0.98114 0.9755 0.9646 0.95560 50 0.91180 0.9007 0.874 0.85708 50 0.974 0.96959 0.9650 0.95511 100 0.9019 0.85785 0.8568 0.8448 100 0.94968 0.94655 0.9477 0.9961 Table 4: Crtcal ponts of r whch r s the PearsonIII and log-pearson Table 9: Crtcal ponts of r whch r s the PearsonIII and log-pearson Probablty Plot Correlaton Coeffcent for g= - Probablty Plot Correlaton Coeffcent for g= 10 0.9857 0.97975 0.970 0.9670 10 0.98614 0.97895 0.96867 0.95674 0 0.96914 0.9660 0.94870 0.9484 0 0.969 0.96099 0.949 0.94 0 0.961 0.959 0.9998 0.91817 0 0.9518 0.945 0.9488 0.914 40 0.94817 0.9491 0.99 0.91704 40 0.9445 0.98 0.915 0.90044 50 0.987 0.9748 0.915 0.90567 50 0.9619 0.9456 0.90805 0.8985 100 0.977 0.907 0.89670 0.88494 100 0.8875 0.8861 0.87964 0.87460 Table 5: Crtcal ponts of r whch r s the PearsonIII and log-pearson Table 10: Crtcal ponts of r whch r s the PearsonIII and log-pearson Probablty Plot Correlaton Coeffcent for g= - Probablty Plot Correlaton Coeffcent for g=4 10 0.99170 0.98977 0.9854 0.9804 10 0.9777 0.966 0.9498 0.917 0 0.98671 0.9859 0.97857 0.97145 0 0.95185 0.965 0.90548 0.88849 0 0.98545 0.9870 0.97706 0.96667 0 0.91641 0.90808 0.8879 0.8694 40 0.98004 0.97770 0.97 0.968 40 0.91160 0.90040 0.8767 0.85456 50 0.97649 0.97419 0.96458 0.9589 50 0.8898 0.8861 0.870 0.85144 100 0.97108 0.95994 0.95716 0.94596 100 0.8847 0.876 0.817 0.8568 Table 6: Crtcal ponts of r whch r s the PearsonIII and log-pearson Table 11: Crtcal ponts of r whch r s the Gumbel Probablty Plot Probablty Plot Correlaton Coeffcent for g= -1 Correlaton Coeffcent 10 0.9946 0.99 0.989 0.98601 10 0.99400 0.9980 0.99060 0.9870 0 0.9944 0.9945 0.99075 0.9885 0 0.99480 0.99450 0.990 0.99070 0 0.9940 0.9986 0.9901 0.98974 0 0.99640 0.99600 0.9940 0.9950 40 0.9969 0.99445 0.9941 0.9894 40 0.99710 0.99580 0.99450 0.990 50 0.99451 0.9996 0.995 0.9898 50 0.99840 0.99670 0.99540 0.99450 100 0.99 0.99 0.98947 0.98785 100 0.99890 0.99750 0.99700 0.99640 Table 7: Crtcal ponts of r whch r s the PearsonIII and log-pearson Probablty Plot Correlaton Coeffcent for g=1 -------------------------------------------------------------------- Class Length 99% 98% 95% 90% 10 0.990 0.9915 0.98896 0.9861 0 0.99405 0.998 0.99100 0.98781 0 0.99460 0.9954 0.99140 0.98950 40 0.99597 0.9957 0.99150 0.98810 50 0.9954 0.9941 0.99160 0.98811 100 0.99058 0.98940 0.98681 0.9850 Whch x s random generated data and x G s correspondng value of x wth gumbel dstrbuton. Also, the followng equaton s used to calculate frequency factor n the Gumbel dstrbuton n step 5 [6]: 6 1 KG = - [0.577 + Ln Ln ] (9) 1-P Whch P s occurrence probablty.the fnal results are presented n Table 11. 1184
World Appl. Sc. J., 15 (8): 1181-1185, 011 RESULTS AND DISCUSSION. Vogel, R.M., 1986. The probablty plot correlaton coeffcent test for the normal, lognormal and Gumbel The probablty plot correlaton coeffcent test s an dstrbutonal hypothess, Water Resources Res., attractve and useful tool for testng the normal, log- (4): 587-590. normal, pearson III, log-pearson III and Gumbel 4. Looney, S.W. and T.R. Gulledge, 1985. Use the dstrbutons. In ths paper, seven plottng poston correlaton coeffcent wth normal probablty formulas were used for mentoned dstrbutons. The plots. The Amercan Statstcan, 9(1): 78-79. PPCC test was appled for samples of length 10, 0, 0, 5. Vogel, R.M. and C.N. Kroll, 1989. Low-flow 40, 50 and 100. frequency analyss usng probablty plot correlaton The results show that the PPCC test s flexble, coeffcents, J. Water Resources Plannng and because t s not lmted to any sample sze. In addton, Management, 115(): 8-57. ths test was developed for normalty ntally, whereas 6. Vogel, R.M. and D.E. McMartn, 1991. Probablty t s extendble readly to non-normal hypotheses. plot goodness-of-ft and skewness estmaton Dfferent plottng poston formulas cause dfferent procedures for the Pearson type dstrbuton, Water results. Therefore, approprate plottng poston formula Resources Res., 7(1): 149-158. plays mportant role to ft dstrbutons on random data. 7. Chowdhury, J.D., J.R. Stednger and L.H. Lu, 1991. Blom, Grngorton and Webul plottng poston Goodness-of-ft tests for regonal generalzed extreme formulas are recommended for the Normal (Log-Normal), value flood dstrbutons. Water Resources Res., Pearson III(log-pearson) and Gumble dstrbutons, 7(7): 1765-1776. respectvely. 8. Heo, J., Y. Kho, Y.H. Shn, S. Km and T. Km, 007. Vogel and Kroll recommendaton usng Blom plottng Regresson Equatons of Probablty Plot Correlaton poston formula for Pearson III and Log-Pearson III s Coeffcent Test Statstcs from Several Probablty rejected. Dstrbutons. J. Hydrol., 55(1-4): 1-15. Crtcal ponts were calculated for samples of dfferent 9. Km, S., H. Shn, T. K and J.H. Heo, 010. Dervaton length. They are applcable as at least value to ft of the probablty plot correlaton coeffcent test statstcal dstrbuton on a sample data of certan length. statstcs for the generalzed logstc dstrbuton. Internatonal Workshop of advances n statstcal CONCLUSION hydrology, May -5, Taormna, Italy. 10. Goda, Y., 010. Plottng poston estmator for the The selecton of approprate plottng poston L-moment method and quantle confdence nterval formula and the calculaton of crtcal ponts for the for the GEV, GPA and webull dstrbutona appled Normal, log-normal, Pearson III, log-pearson III and for extreme analyss. Coastal Engneerng Journal, Gumbel dstrbutons were studed n ths paper. Fllben s 5(): 111-149. PPCC test was used for ths purpose. The test was 11. Cook, N., 011. Comments on Plottng Postons n appled for g [-4,4] for the PearsonIII and log-pearsoniii Extreme Value Analyss. J. Appled Meteorology and dstrbuton. The results were n the agreement of the Clmatol., 50: 55-66. other studes wth the excepton of the PearsonIII and log- 1. Vto, I., M. Forentno, A. Goa and S. Manfreda, pearsoniii dstrbutons. Webul plottng poston formula 010. Best Ft and Selecton of Theoretcal Flood was recommended for these two dstrbutons unlke Frequency Dstrbutons Based on Dfferent Runoff Vogel and Kroll recommendaton. Generaton Mechansms, Water, : 9-56. REFERENCES 1. Raghunath, H.M., 007. Hydrology. Prncples. Analyss. Desgn, New Age Internatonal Press, New Delh, pp: 7-6.. Fllben, J.J., 1975. The Probablty Plot Correlaton Coeffcent Test for Normalty. Technometrcs, 17(1): 111-117. 1185