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Dataplot Vol 1 Auxiliary Chapter

KAPPDF

Name:
    KAPPDF (LET)
Type:
    Library Function
Purpose:
    Compute the kappa probability density function with shape parameters h and k.
Description:
    The general form of the kappa distribution has the following probability density function:

      f(x;k,h,xi,alpha) = (1/alpha)*[1 - k*(x-xi)/alpha]**((1/k)-1)*
[F(x;k,h,xi,alpha)]**(1-h) alpha > 0

    with k and h denoting the shape parameters and xi and alpha denoting the location and scale parameters, respectively, and where F is the kappa cumulative distribution function.

    The upper bound of x is

      x < xi + alpha*(1 - h**(-k)       if k >  0
 x < infinity                      if k <= 0

    The lower bound of x is

      x > xi + alpha*(1 - h**(-k))/k    if h >  0
 x > xi alpha/k                    if h <= 0, k <  0
 x > -infinity                     if h <= 0, k >= 0

    The standard form of the distribution is defined as xi = 0 and alpha = 1.

    The cases h = -1, h = 0, and h = 1 reduce to the Hosking generalized logistic distribution, the generalized extreme value distribution, and the generalized Pareto distribution, respectively. According to Hosking and Wallis, the most useful values of the shape parameters are when h >= -1.

Syntax:
    LET <y> = KAPPDF(<x>,<k>,<h>,<xi>,<alpha>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <k> is a number, parameter, or variable that specifies the first shape parameter;
                <h> is a number, parameter, or variable that specifies the second shape parameter;
                <xi> is a number, parameter, or variable that specifies the location parameter;
                <alpha> is a number, parameter, or variable that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed kappa pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The <xi> and <alpha> parameters are optional.

Examples:
    LET A = KAPPDF(3,0.5,2,0,1.5)
    LET X2 = KAPPDF(X1,K,H)
Note:
    Kappa random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET K = <value>>
      LET H = <value>>
      LET Y = KAPPA RANDOM NUMBERS FOR I = 1 1 N>
      KAPPA PROBABILITY PLOT Y>
      KAPPA PROBABILITY PLOT Y2 X2>
      KAPPA PROBABILITY PLOT Y3 XLOW XHIGH>
      KAPPA KOLMOGOROV SMIRNOV GOODNESS OF FIT Y>
      KAPPA CHI-SQUARE GOODNESS OF FIT Y2 X2>
      KAPPA CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH>

    The following commands can be used to estimate the k and h shape parameter for the kappa distribution:

      LET K1 = <value>
      LET K2 = <value>
      LET H1 = <value>
      LET H2 = <value>
      KAPPA PPCC PLOT Y
      KAPPA PPCC PLOT Y2 X2
      KAPPA PPCC PLOT Y3 XLOW XHIGH
      KAPPA KS PLOT Y
      KAPPA KS PLOT Y2 X2
      KAPPA KS PLOT Y3 XLOW XHIGH

    The default values for K1 and K2 are -5 and 5, respectively. The default values for THETA1 and THETA2 are -2 and 5, respectively.

    The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1).

    The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the ppcc plot and the ks plot.

    The parameters of the kappa distribution can be estimated by the method of L-moments using the command

      KAPPA MAXIMUM LIKELIHOOD Y

    Dataplot uses Hoskings code for computing the L-moments parameter estimates. Hoskings report and associated Fortran code can be downloaded from the Statlib archive at

Default:
    None
Synonyms:
    None
Related Commands:
    KAPCDF = Compute the kappa cumulative distribution function.
    KAPPPF = Compute the kappa percent point function.
    MIEPDF = Compute Miekle's beta-kappa probability density function.
    GEVPDF = Compute the generalized extreme value probability density function.
    GEPPDF = Compute the generalized Pareto probability density function.
    GL5PDF = Compute the Hosking's generalized logistic probability density function.
Reference:
    Hosking and Wallis (1997), "Regional Frequency Analysis", Cambridge University Press, Appendix A10.

    J. R. M. Hosking (2000), "Research Report: Fortran Routines for use with the Method of L-Moments", IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598.

    Hoskings (1990), "L-moments: Analysis and Estimation of Distribution using Linear Combinations of Order Statistics", Journal of the Royal Statistical Society, Series B, 52, pp. 105-124.

Applications:
    Distributional Modeling
Implementation Date:
    2008/5
Program 1:
     
    LET KP = DATA -0.5  0.1  0.5  1.0
    LET H1 = -0.5
    LET H2 = 0.1
    LET H3 = 1
    LET H4 = 2
    .
    MULTIPLOT 2 2
    MULTIPLOT CORNER COORDINATES 0 0 95 95
    MULTIPLOT SCALE FACTOR 2
    TITLE CASE ASIS
    TITLE OFFSET 2
    X3LABEL
    LINE COLOR BLACK RED BLUE GREEN
    .
    LOOP FOR KK = 1 1 4
       LET K = KP(KK)
       LET LL1 = KAPPPF(0.05,K,H1)
       LET UL1 = KAPPPF(0.95,K,H1)
       LET LL2 = KAPPPF(0.05,K,H2)
       LET UL2 = KAPPPF(0.95,K,H2)
       LET LL3 = KAPPPF(0.05,K,H3)
       LET UL3 = KAPPPF(0.95,K,H3)
       LET LL4 = KAPPPF(0.05,K,H4)
       LET UL4 = KAPPPF(0.95,K,H4)
       TITLE K = ^K, H = -0.5, 0.1, 1, 2
       PLOT KAPPDF(X,K,H1) FOR X = LL1  0.01  UL1  AND
       PLOT KAPPDF(X,K,H2) FOR X = LL2  0.01  UL2  AND
       PLOT KAPPDF(X,K,H3) FOR X = LL3  0.01  UL3  AND
       PLOT KAPPDF(X,K,H4) FOR X = LL4  0.01  UL4
    END OF LOOP
    END OF MULTIPLOT
    .
    JUSTIFICATION CENTER
    MOVE 50 97
    TEXT Kappa PDF Functions
        

    plot generated by sample program

Program 2:
     
    let k = 0.3
    let h = 0.8
    let ksav = k
    let hsav = h
    .
    let xlow  = kapppf(0.001,k,h)
    let xhigh = kapppf(0.999,k,h)
    .
    x3label
    .
    let y = kappa rand numb for i = 1 1 200
    let y = 5*y
    .
    x3label
    title automatic
    title case asis
    .
    kappa ppcc plot y
    let k = shape1
    let h = shape2
    just center
    move 50 6
    text k = ^k, h = ^h
    move 50 2
    text ksav = ^ksav, hsav = ^hsav
    pause
    .
    char x
    line blank
    kappa prob plot y
    move 50 6
    text ppa0 = ^ppa0, ppa1 = ^ppa1
    move 50 2
    text ppcc = ^ppcc
    char blank
    line solid
    pause
    .
    relative hist y
    limits  freeze
    pre-erase off
    title
    let a1 = minimum y
    let a2 = maximum y
    line color blue red
    let ll = kapppf(0.01,k,h,ppa0,ppa1)
    let ul = kapppf(0.99,k,h,ppa0,ppa1)
    plot kappdf(x,k,h,ppa0,ppa1) for x = ll 0.01 ul
    line color black
    limits
    pre-erase on
    title automatic
    pause
    .
    let ksloc = ppa0
    let ksscale = ppa1
    kappa kolm smir goodness of fit y
    pause
    .
    kappa mle y
    .
    let k = kml
    let h = hml
    let ksloc = ximl
    let ksscale = alphaml
    relative hist y
    limits  freeze
    pre-erase off
    let a1 = minimum y
    let a2 = maximum y
    line color blue red
    plot kappdf(x,k,h,ksloc,ksscale) for x = a1 0.01 a2
    line color black
    limits
    pre-erase on
    pause
    .
    kappa kolm smir goodness of fit y
        

    Outputs for PPCC fitting.

    plot generated by sample program

    plot generated by sample program

        
                      KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
     
    NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
    ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
    DISTRIBUTION:            KAPPA
       NUMBER OF OBSERVATIONS              =      200
     
    TEST:
    KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3728781E-01
     
       ALPHA LEVEL         CUTOFF              CONCLUSION
               10%       0.086*              ACCEPT H0
                         0.085**
                5%       0.096*              ACCEPT H0
                         0.095**
                1%       0.115*              ACCEPT H0
                         0.114**
     
        *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
       ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )
        
        

    Outputs for L-moments fitting.

        
                KAPPA PARAMETER ESTIMATION:
     
    SUMMARY STATISTICS:
    NUMBER OF OBSERVATIONS                     =      200
    SAMPLE MEAN                                =    3.826996
    SAMPLE STANDARD DEVIATION                  =    3.380150
    SAMPLE MINIMUM                             =   -1.124539
    SAMPLE MAXIMUM                             =    13.35444
     
    L-MOMENTS:
    FIRST SAMPLE L-MOMENT                      =    3.826996
    SECOND SAMPLE L-MOMENT                     =    1.901070
    THIRD SAMPLE L-MOMENT                      =   0.1587699
    FOURTH SAMPLE L-MOMENT                      =   0.7563092E-01
     
    ESTIMATE OF LOCATION                       =   0.2250852
    ESTIMATE OF SCALE                          =    5.308738
    ESTIMATE OF K                              =   0.3297350
    ESTIMATE OF H                              =   0.7777997
        
        
    plot generated by sample program
        
                      KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
     
    NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
    ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
    DISTRIBUTION:            KAPPA
       NUMBER OF OBSERVATIONS              =      200
     
    TEST:
    KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3728781E-01
     
       ALPHA LEVEL         CUTOFF              CONCLUSION
               10%       0.086*              ACCEPT H0
                         0.085**
                5%       0.096*              ACCEPT H0
                         0.095**
                1%       0.115*              ACCEPT H0
                         0.114**
     
        *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
       ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )
     
        
        

Date created: 7/15/2009
Last updated: 7/15/2009
Please email comments on this WWW page to alan.heckert@nist.gov.