Copulas are tools for modelling dependence of several random variables. The term copula was first used in the work of Sklar (1959) and is derived from the latin word copulare, to connect or to join. The main purpose of copulas is to describe the interrelation of several random variables. (Thorsten Schmidt, 2006). Copula is a function that joins the two distributions and known as dependence functions. Copula connect multivariate distribution function to its univariate marginal distribution. When we have two models having the problems relating to dependence, we can join that models becoming one model using marginal function. So, the dependency is taken care. It means that copula played an important role to join multivariate distributions to their one dimensional marginal distribution function

Sahrin, Sharainie

Yusof, Fadhilah

Universiti Teknologi Malaysia Institutional Repository

STUDENT COPULA METHOD IN RAINFALL DISTRIBUTION   Sharainie binti Sahrin, PM.Dr.Fadhilah Yusof    1.1 INTRODUCTION  Copulas are tools for modelling dependence of several random variables.  The term copula was first used in the work of Sklar (1959) and is derived from the latin word copulare, to connect or to join.  The main purpose of copulas is to describe the interrelation of several random variables.  (Thorsten Schmidt, 2006). Copula is a function that joins the two distributions and known as dependence functions. Copula connect multivariate distribution function to its univariate marginal distribution.  When we have two models having the problems relating to dependence, we can join that models becoming one model using marginal function. So, the dependency is taken care.  It means that copula played an important role to join multivariate distributions to their one dimensional marginal distribution function.  1.2 PROBLEM STATEMENT  This study will focus on expressing the bivariate distribution function from two univariate distribution functions. Considering these two distribution functions are independent, the multiplication of these two distributions can be done.  However, in this case, the dependency between these two distributions must be considered.  One of the way  allowing the dependency of these distributions by using copula method.    1.3 LITERATURE REVIEW  The Word Copula is a Latin noun that means ''A link, tie, bond''.  In 1959, the word Copula appeared for the  first time.   Copula played an important role to join multivariate distributions to their one dimensional marginal distribution function. When we are having the problems relating to dependence, we can join that models becoming one model using marginal function as we want dependency is taken care. Copula method  has been used for flood frequency analysis. This analysis was based on t-copula for Johor River, Malaysia. Student Copula was used to model the joint dependence of peak flow-volume, volume-duration and peak flow-duration.   (Salarpour et al., 2013). Copula also has been used to model bivariate rainfall distribution.  This model has been simulated  at two sites in the Murray-Darling Basin, Australia.  The selected sites are Hume and Beechworth.  In this study, the asymmetric t-copula, also known as skew-t is used to analyse the monthly rainfall data.(Zakariaet al., 2010)       1.4 METHODOLOGY  Joint Distribution Function   If Y1and Y2 are jointly continuous random variables with a joint density function given by f(y1,y2), then  1. f(y1,y2) ≥0 for all y1, y2 2. 1),( 2121  dydyyyf Joint distribution function is: ],[),( yYxXPyxH   (1.1)  Marginal distribution function: ][)(],[)( yYPyGxXPxF  (1.2)                 Gamma Distribution  The PDF an d CDF of the gamma distribution with continuous shape parameter ( ), continuous scale parameter (  ) and continuous location parameter   ( ) are as depicted below: )/)((exp)()()(1xxxf   (1.3)    )()()(/)(xxF       (1.4)    Where:  x  Parameter Estimation: Likelihood function  niininnii xxL1111exp)()(  (1.5)     The log-likelihood function iS    niniiixnnx1 1)(lnln)(ln)1(    (1.6)   nii nnxl10)(ln)(ln where )( is digamma function        (1.7)     (1.8)      0)()1(11nxl nii  (1.9) Use Newton-Raphson method. )()()(3211111fffJikkkkkkk     (1.10)  222222222222333222111lnlnlnlnlnlnlnlnln)()()()()()()()()(LLLLLLLLLfffffffffJik    (1.11) Weibull Distribution The PDF and CDF of the weibull distribution with continuous shape parameter ( ), continuous scale parameter ( ) and continuous location parameter    (  ) are as depicted below:       xxxf exp)(1  (1.12)  -x-exp1)(xF     (1.13) Where:  x  Parameter Estimation: Likelihood function    niinii xxL111exp  (1.14) 0)(21 niixnl Taking the logarithm for the likelihood function will get the log-likelihood function:         nininininiiiixxxnnXl1 1 1 1 1lnln)ln()ln(lnln);,,(                                                                                                   (1.15) The partial derivatives for the log-likelihood function with respect to  ,  and   are:    niiinii xxxnl11lnln=0            (1.16) 011 1   niniixnl (1.17) 01)1(111  niniiixxl  (1.18) )()()(3211111fffJikkkkkkk     (1.19)  222222222222333222111lnlnlnlnlnlnlnlnln)()()()()()()()()(LLLLLLLLLfffffffffJik  (1.20)   Log-normal Distribution The PDF and CDF of the log-normal distribution with shape parameter ( ), scale parameter (  ) and location parameter   ( ) are as depicted below   222)ln(exp2)(1)(xxxf  (1.21)   )ln()(xxF     (1.22) Where: x 0  Parameter Estimation: Likelihood function niniiin xxXL121212))(ln(exp)(21)|,,(                                                                                      (1.23) The log-likelihood function isniniiixxnnXL12122))(ln()ln(2lnln)|,,(ln                             (1.24) Differentiating the log-likelihood function and equating to zero will obtain the   maximum likelihood estimating equations.    niixL120])[ln(1ln    (1.25) nxnii 1)ln(      (1.26) 0])[ln(1ln123niixnL   (1.27) niixn 122 )(ln(1  (1.28) 0)ln(11ln1 12  nini iii xxxL(1.29) Maximum likelihood estimate for   by replacing eq.1.26and eq.1.28in the last equation, eq.1.39  to obtain an equation in ,eq.1.30.   0)ln()ln(1))(ln()ln(1121 1 121    ni iinininiiiini i xxnxnxxx (1.30) Kolmogorov-Smirnov (K-S) test  The K-S Test statistic is defined as:   )(,1)(max1iiNixFNiNixFD   (1.31) where ix  is sorted from smallest to largest, F the theoretical cumulative distribution and N is the number of sample size.The best fit distribution is selected based on the test statistic, D , where the distribution that gives the smallest value of  D  is known as the best fit distribution.   Evaluating the dependency The dependency can be evaluated using Kendall’s tau ( ) or Kendall’s tau coefficient as below:  =)1(21pairs) discordant of(number -pairs) concordant ofnumber (nn  (1.32) Where ),( 11 yx  and ),( 22 yx  are taken as concordant or dependent when: )( 21 xx  )( 21 yy  >0 and discordant or independent when: )( 21 xx  )( 21 yy  <0 Finding the Parameters. After getting the value of  , the value of   can be obtained using the formula below.Kendall’s tau for t copula as written below: )arcsin(2   11     Then,  make θ as the subject,      (1.33) In dependent t-test for paired samples, in which two samples are matched or paired, the degree of freedom used is n-1    t (student) copula = dydxvyxyxvvtut vv 21222)(2)()1(2(1exp)1(2111 (1.34) 11     2sinwhere (.)1vt  denotes the inverse function of the CDF (.)vt with v  as the degree of freedom , and   is between )(1 utv and )(1 vtvand controls the strength of the tails. x and y are the dependent variables.   Simulation of data i. Generate multivariate t-distributed random numbers which can be between   to   ii. Transform the multivariate data to uniform data using marginals of univariate t distribution. iii. Find t-copula values using the uniform data, correlation and degrees of freedom  Comparing The Observed Data And Simulated Data Afterward, the copula is compared by using Kolmogorov-Smirnov goodness of fit test at 5% significance level. If P-value is greater than 0.05 significance level, conclusion can be made that the two distributions are not significantly different at 5% significance level and vice versa.  1.5 RESULT AND DISCUSSION  Finding the Marginal Distribution Fx(x) and Fy(y) Table 4.1: Values of Parameters and Test Statistics   Malacca (x) Tangkak (y)  Parameters Test Statistics Rank Parameters Test Statistics Rank Lognormal 96.7071412.720143.0 0.14009 1 1.12224373.64.0 0.1125 3 Gamma 5.1273529.97415.7 0.14495 2 4.139832.1542383.3 0.10967 2 Weibull 5.152759.5268411.1 0.1454 3 145071.5016896.1 0.09936 1  i) Kendall’s tau,   Table 4.2: The Kendall’s tau Correlation between x = Station in Malacca and y=Station in Tangkak in 33 years Year Kendall’s tau, τ  1975 -0.0606 1976 0.5152 1977 0.3333              ii)       Find values of theta, θ The Kendall’s coefficient for the student t copula distribution is shown below, )arcsin(2                                                       (1.35) theta,  values, make the   To find the as the subject     Then substitute the value of Kendall’s tau in order to find value of   for every year.  The range of   values are shown to be from -1 to1.    Table 4.3: Values of θ in 33 Years Periods for Student t  Copula  1978 0.3698 1979 0.3333 1980 0.1212 1981 0.5758 1982 0.3636 1983 0.4848 1984 0.5152 1985 0.5152 1986 0.4242 1987 0.2901 1988 0.5152 1989 0.6364 1990 0.5455 1991 0.4545 1992 0.4848 1993 0.1818 1994 0.4545 1995 0.3206 1996 0.5758 1997 0.303 1998 0.5152 1999 0.3333 2000 0.6644 2001 0.3636 2002 0.4848 2003 0.2424 2004 0.303 2005 0.5649 2006 0.7273 2007 0.1985 Year Θ  2sin1975 -0.09505 1976 0.723787 1977 0.499955 1978 0.54876 1979 0.499955 1980 0.189233 1981 0.786094 1982 0.540593 1983 0.690024 1984 0.723787 1985 0.723787 1986 0.618107 1987 0.44008 1988 0.723787 1989 0.841284 1990 0.755796 1991 0.654807 1992 0.690024 1993 0.281705 1994 0.654807 1995 0.482579 1996 0.786094 1997 0.458184 1998 0.723787 1999 0.499955 2000 0.86424 2001 0.540593 2002 0.690024 2003 0.371627 2004 0.458184                                                        2005 0.775397 2006 0.90965 2007 0.306775 iii)  Find Degree of Freedom, v In dependent t-test for paired samples, in  which  two samples are matched or paired, the degree of freedom used is n-1. Since there are 12 samples for every year, thus, the degree of freedom is 11.    iv)       Find Real Values of C(u,v) By substituting the real values of u, v, θ and degree of freedom  for every year into Eq.1.34, the values of copula will be obtained.    dyvyxyxvvtut vv 21222)(2)()1(2(1exp)1(2111  The values of copula must be in range between 0 and 1.  Table 4.4: The Real Values of Student t Copula for 33 Years Period i Year Ci(u,v) 1 1975 0.7004 2 1976 0.4643 3 1977 0.0182 4 1978 0.3969 5 1979 0.4772 6 1980 0.077 7 1981 0.2471 8 1982 0.616 9 1983 0.9955 10 1984 0.8352 11 1985 0.0211 12 1986 0.6862 13 1987 0.7622 14 1988 0.1247 15 1989 0.6683 16 1990 0.2646 17 1991 0.4275 18 1992 0.1051                    Simulation experiment The procedure is divided into two parts: Part 1 – finding the theoretical copula values. Part 2 – comparing the analysis of empirical copula and theoretical copula.  i) Generate 100 simulated data for the multivariate-t distributed.  100 random numbers are generated. The degree of freedom is 11 using the formula of  n-1. The correlation coefficients is calculated using the Kendall’s tau of original data. The Kendall’s tau between distribution in Malacca and Tangkak is calculated using eq. (1.32) which gives the value 0.405. The PDF for multivariate-t distribution is given by    21 )()'(1||22))(,,(dvdtdvxPxPvvdvxPvf    where v is degree of freedom, d is dimensional random vector and   is the gamma function.  ii) Transform generated multivariate-t data to uniform data using   marginals of univariate t distribution 19 1993 0.823 20 1994 0.3194 21 1995 0.706 22 1996 0.0981 23 1997 0.8425 24 1998 0.3885 25 1999 0.3352 26 2000 0.5204 27 2001 0.6864 28 2002 0.1888 29 2003 0.5138 30 2004 0.0017 31 2005 0.4667 32 2006 0.0099 33 2007 0.0054  The PDF for univariate student’s t distribution is given by 2/)1(21)2/(21)( vvxvvvxf where  defines a gamma function and v is degrees of freedom. The Kendall’s tau between distribution in Malacca and Tangkak is calculated using eq. (1.32) gives the value of  0.405. Then, substitute the value of Kendall’s tau into eq. (1.33) to get the value of θ such as below:     2sin   405.02sin   594121.0  iii)  Eq (1.34) belowis used to calculate  t copula . The range of these copula values must be in between 0 and 1.  dydxvyxyxvvtut vv 21222)(2)()1(2(1exp)1(2111   Comparing The Analysis Of Empirical Copula And Theoretical Copula. In this case, use Kolmogorov-Smirnov goodness of fit test to compare the distributions of the empirical and theoretical copula.  The P-value is found to be greater than 0.05 significance level for this test, therefore the conclusion that the two distributions are not significantly different at 5% significance level can be made. For addition, 1000 and 5000 random numbers are generated to test the P-value. See table below:  Table 4.6: Goodness of fit test of theoretical and empirical copulas for the uniformised observed and generated data   Kolmogorov-Smirnov Number of simulated data 100 1000 5000 P-value 0.9278 0.8524 0.8214  Summary Two sets of data (two stations) were tested using Kolmogorov-Smirnov fit test to find the best fit marginal distributions. Lognormal provided the best fitted for station in Malacca and weibull was the best fitted for station in Tangkak. The parameters u,v,  and degree of freedom were calculated  in modelling the bivariate distribution. All calculated parameter values are within the acceptable range. Then, the bivariate joint distribution of rainfall data for student-t copula is computed. The simulation process of student-t copula has been carried out by using 100, 1000 and 5000 numbers of simulated data. All these simulations process give good results which their P-values are greater than 0.05 significance level.Therefore, every simulation distribution is not significantly different with the distribution of observed data.  1.6  CONCLUSION  Based on the Kolmogorov-Smirnov goodness-of-fit test, lognormal provided the best fitted distribution for station in Malacca and weibull was the best for station in Tangkak The distributions of observed data and simulated data were not significantly different at 5% significance level Kolmogorov-Smirnov goodness of fit test Results showed that all calculated  parameter  values were within the acceptable range and could be applied to compute the bivariate joint distribution of rainfall data for the student-t copula.      ACKNOWLEDGEMENT   The authors wish to thank Malaysia Meteorological Services for allowing us to do analysis on the climate data.   REFERENCES Defu, L., Shuqin, W. and Liping, W. (2002). Poisson-Gumbel Mixed Compound Distribution and its application.  Chinese Science Bulletin.  47(22): 1901-1906. Kelly, K.S. and Krzysztofowicz, R. (1997).  A bivariate meta-Gaussian density for use in hydrology.  Stochastic Hydrology and Hydrailics.  11(1):17-31. Nikoloulopoulos, A.K., H, Joe, H. and Haijun Li (2010). Vine copulas with asymmetric tail dependence and applications to financial return data. Computational Statistics & Data Analysis. 56(11), 3659-3673. Quesada-Molina, J.J., Rodr´ıguez-Lallena, J.A., and Ubeda-Flores, M. (2003).  What are copulas?  Monograf´ıas del Semin. Matem. Garc´ıa de Galdeano. 27(3), 499-506. Salarpour, M., Yusop, Z., Yusof, F., Shahid, S. and Jajarmizadeh, M. (2013).  Flood frequency analysis based on t-copula for Johor River, Malaysia.  Journal of Applied Sciences. 13(7), 1021-1028.  Salvadori, G. and De Michele, C. (2007).  On the Use of Copulas in Hydrology: Theory and Practice.  Journal of Hydrologic Engineering. 12(4), 369-380.  Schmidt, T. (2006). Copulas-From Theory to Applications in Finance. (1st ed.), London: Risk Books. Sklar, A. (1959).  Functions de repartition an dimensionset leursmarges.  (8th ed.), Paris: PublInst Stat. Univ.  Yue, S., Ourda, T.B.M.J., Bobee, B., Legendre, P., and Bruneau, P. (1999). The Gumbel mixed model for flood frequency analysis.  Journal of Hydrology.  226 (1-2): 88-100. Yue, S. (2000). The Gumbel mixed model applied to storm analysis.  Water Resources Management.  14(5): 377-389. Zakaria, R., Metcalfe, A.V., Howlett, P., Piantadosi, J. and Boland, J. (2010).  Using the skew-t  copula to model bivariate rainfall distribution.  ANZIAM (Australia and New Zealand Industrial and Applied Mathematics) Journal. 51, (c231-c246).            

Student copula method in rainfall distribution

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Student copula method in rainfall distribution

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