PARPLT Subroutine

public subroutine PARPLT(X, N, Gamma)

NAME

parplt(3f) - [M_datapac:LINE_PLOT] generate a Pareto probability plot

SYNOPSIS

   SUBROUTINE PARPLT(X,N,Gamma)

DESCRIPTION

PARPLT(3f) generates a Pareto probability plot (with tail length
parameter value = GAMMA).

The prototype pareto distribution used herein is defined for all X
equal to or greater than 1, and has the probability density function

    f(X) = GAMMA / (X**(GAMMA+1)).

As used herein, a probability plot for a distribution is a plot
of the ordered observations versus the order statistic medians for
that distribution.

The Pareto probability plot is useful in graphically testing the
composite (that is, location and scale parameters need not be
specified) hypothesis that the underlying distribution from which
the data have been randomly drawn is the pareto distribution with
tail length parameter value = gamma.

If the hypothesis is true, the probability plot should be near-linear.

A measure of such linearity is given by the calculated probability
plot correlation coefficient.

OPTIONS

 X   description of parameter
 Y   description of parameter

EXAMPLES

Sample program:

program demo_parplt
use M_datapac, only : parplt
implicit none
! call parplt(x,y)
end program demo_parplt

Results:

AUTHOR

The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.

MAINTAINER

John Urban, 2022.05.31

LICENSE

CC0-1.0

REFERENCES

  • Filliben, ‘Techniques for Tail Length Analysis’, Proceedings of the Eighteenth Conference on the Design of Experiments in Army Research Development and Testing (Aberdeen, Maryland, October, 1972), pages 425-450.
  • Hahn and Shapiro, Statistical Methods in Engineering, 1967, pages 260-308.
  • Johnson and Kotz, Continuous Univariate Distributions–1, 1970, pages 233-249.

Arguments

Type IntentOptional Attributes Name
real(kind=wp), dimension(:) :: X
integer :: N
real(kind=wp) :: Gamma

Common Blocks

CAUPLT (subroutine)
CHSCDF (subroutine)
CODE (subroutine)
DECOMP (subroutine)
DEMOD (subroutine)
DEXPLT (subroutine)
EV1PLT (subroutine)
EV2PLT (subroutine)
EXPPLT (subroutine)
EXTREM (subroutine)
EXTREM (subroutine)
FREQ (subroutine)
GAMPLT (subroutine)
GEOPLT (subroutine)
HFNPLT (subroutine)
INVXWX (subroutine)
LAMPLT (subroutine)
LGNPLT (subroutine)
LOGPLT (subroutine)
MEDIAN (subroutine)
norout (subroutine)
NORPLT (subroutine)
PLOTU (subroutine)
POIPLT (subroutine)
RUNS (subroutine)
SAMPP (subroutine)
SPCORR (subroutine)
TAIL (subroutine)
TPLT (subroutine)
TRIM (subroutine)
UNIPLT (subroutine)
WEIB (subroutine)
WEIPLT (subroutine)
WIND (subroutine)
"> common /BLOCK2_real32/

Type Attributes Name Initial
real :: WS(15000)

Source Code

SUBROUTINE PARPLT(X,N,Gamma)
REAL(kind=wp) :: an, cc, Gamma, hold, pp0025, pp025, pp975, pp9975,   q, sum1, sum2, sum3, tau, W, wbar, WS, X, Y, ybar, yint
REAL(kind=wp) :: yslope
INTEGER       :: i, iupper, N
!
!     INPUT ARGUMENTS--X      = THE  VECTOR OF
!                                (UNSORTED OR SORTED) OBSERVATIONS.
!                     --N      = THE INTEGER NUMBER OF OBSERVATIONS
!                                IN THE VECTOR X.
!                     --GAMMA  = THE  VALUE OF THE
!                                TAIL LENGTH PARAMETER.
!                                GAMMA SHOULD BE POSITIVE.
!     OUTPUT--A ONE-page PARETO PROBABILITY PLOT.
!     PRINTING--YES.
!     RESTRICTIONS--THE MAXIMUM ALLOWABLE VALUE OF N
!                   FOR THIS SUBROUTINE IS 7500.
!                 --GAMMA SHOULD BE POSITIVE.
!     OTHER DATAPAC   SUBROUTINES NEEDED--SORT, UNIMED, PLOT.
!     FORTRAN LIBRARY SUBROUTINES NEEDED--SQRT.
!     MODE OF INTERNAL OPERATIONS--.
!     ORIGINAL VERSION--NOVEMBER  1975.
!     UPDATED         --FEBRUARY  1976.
!---------------------------------------------------------------------
      DIMENSION X(:)
      DIMENSION Y(7500) , W(7500)
      COMMON /BLOCK2_real32/ WS(15000)
      EQUIVALENCE (Y(1),WS(1))
      EQUIVALENCE (W(1),WS(7501))
!
      iupper = 7500
!
!     CHECK THE INPUT ARGUMENTS FOR ERRORS
!
      IF ( N<1 .OR. N>iupper ) THEN
         WRITE (G_IO,99001) iupper
99001    FORMAT (' ',                                                   &
     &'***** FATAL ERROR--THE SECOND INPUT ARGUMENT TO THE PARPLT SUBROU&
     &TINE IS OUTSIDE THE ALLOWABLE (1,',I0,') INTERVAL *****')
         WRITE (G_IO,99002) N
99002    FORMAT (' ','***** THE VALUE OF THE ARGUMENT IS ',I0,' *****')
         RETURN
      ELSEIF ( N==1 ) THEN
         WRITE (G_IO,99003)
99003    FORMAT (' ',                                                   &
     &'***** NON-FATAL DIAGNOSTIC--THE SECOND INPUT ARGUMENT TO THE PARP&
     &LT SUBROUTINE HAS THE VALUE 1 *****')
         RETURN
      ELSE
         IF ( Gamma<=0.0_wp ) THEN
            WRITE (G_IO,99004)
99004       FORMAT (' ',                                                &
     &'***** FATAL ERROR--THE THIRD  INPUT ARGUMENT TO THE PARPLT SUBROU&
     &TINE IS NON-POSITIVE *****')
            WRITE (G_IO,99005) Gamma
99005       FORMAT (' ','***** THE VALUE OF THE ARGUMENT IS ',E15.8,    &
     &              ' *****')
            RETURN
         ELSE
            hold = X(1)
            DO i = 2 , N
               IF ( X(i)/=hold ) GOTO 50
            ENDDO
            WRITE (G_IO,99006) hold
99006       FORMAT (' ',                                                &
     &'***** NON-FATAL DIAGNOSTIC--THE FIRST  INPUT ARGUMENT (A VECTOR) &
     &TO THE PARPLT SUBROUTINE HAS ALL ELEMENTS = ',E15.8,' *****')
            RETURN
         ENDIF
!
!-----START POINT-----------------------------------------------------
!
 50      an = N
!
!     SORT THE DATA
!
         CALL SORT(X,N,Y)
!
!     GENERATE UNIFORM ORDER STATISTIC MEDIANS
!
         CALL UNIMED(N,W)
!
!     COMPUTE PARETO DISTRIBUTION ORDER STATISTIC MEDIANS
!
         DO i = 1 , N
            W(i) = (1.0_wp-W(i))**(-1.0_wp/Gamma)
         ENDDO
!
!     PLOT THE ORDERED OBSERVATIONS VERSUS ORDER STATISTICS MEDIANS.
!     COMPUTE THE TAIL LENGTH MEASURE OF THE DISTRIBUTION.
!     WRITE OUT THE TAIL LENGTH MEASURE OF THE DISTRIBUTION
!     AND THE SAMPLE SIZE.
!
         CALL PLOT(Y,W,N)
         q = 0.9975_wp
         pp9975 = (1.0_wp-q)**(-1.0_wp/Gamma)
         q = 0.0025_wp
         pp0025 = (1.0_wp-q)**(-1.0_wp/Gamma)
         q = 0.975_wp
         pp975 = (1.0_wp-q)**(-1.0_wp/Gamma)
         q = 0.025_wp
         pp025 = (1.0_wp-q)**(-1.0_wp/Gamma)
         tau = (pp9975-pp0025)/(pp975-pp025)
         WRITE (G_IO,99007) Gamma , tau , N
!
99007    FORMAT (' ',                                                   &
     &           'PARETO PROBABILITY PLOT WITH EXPONENT PARAMETER = ',  &
     &           E17.10,1X,'(TAU = ',E15.8,')',11X,                     &
     &           'THE SAMPLE SIZE N = ',I0)
!
!     COMPUTE THE PROBABILITY PLOT CORRELATION COEFFICIENT.
!     COMPUTE LOCATION AND SCALE ESTIMATES
!     FROM THE INTERCEPT AND SLOPE OF THE PROBABILITY PLOT.
!     THEN WRITE THEM OUT.
!
         sum1 = 0.0_wp
         sum2 = 0.0_wp
         DO i = 1 , N
            sum1 = sum1 + Y(i)
            sum2 = sum2 + W(i)
         ENDDO
         ybar = sum1/an
         wbar = sum2/an
         sum1 = 0.0_wp
         sum2 = 0.0_wp
         sum3 = 0.0_wp
         DO i = 1 , N
            sum1 = sum1 + (Y(i)-ybar)*(Y(i)-ybar)
            sum2 = sum2 + (Y(i)-ybar)*(W(i)-wbar)
            sum3 = sum3 + (W(i)-wbar)*(W(i)-wbar)
         ENDDO
         cc = sum2/SQRT(sum3*sum1)
         yslope = sum2/sum3
         yint = ybar - yslope*wbar
         WRITE (G_IO,99008) cc , yint , yslope
99008    FORMAT (' ','PROBABILITY PLOT CORRELATION COEFFICIENT = ',F8.5,&
     &           5X,'ESTIMATED INTERCEPT = ',E15.8,3X,                  &
     &           'ESTIMATED SLOPE = ',E15.8)
      ENDIF
!
END SUBROUTINE PARPLT