EV2PLT Subroutine

public subroutine EV2PLT(X, N, Gamma)

NAME

ev2plt(3f) - [M_datapac:LINE_PLOT] generate a extreme value type 2
(Frechet) probability plot

SYNOPSIS

 SUBROUTINE EV2PLT(X,N,Gamma)

DESCRIPTION

ev2plt(3f) generates a extreme value type 2 probability plot (with
tail length parameter value = gamma).

the prototype extreme value type 2 distribution used n herein
is defined for all non-negative x, and has the probability density
function

    f(x) = gamma * (x**(-gamma-1)) * exp(-(x**(-gamma))).

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 extreme value type 2 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 extreme value type
2 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_ev2plt
use M_datapac, only : ev2plt
implicit none
! call ev2plt(x,y)
end program demo_ev2plt

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 272-295.

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)
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)
PARPLT (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_real64/

Type Attributes Name Initial
real :: WS(15000)

Source Code

SUBROUTINE EV2PLT(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 EXTREME VALUE TYPE 2 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, LOG.
!     MODE OF INTERNAL OPERATIONS--.
!
!---------------------------------------------------------------------
!
      DIMENSION X(:)
      DIMENSION Y(7500) , W(7500)
      COMMON /BLOCK2_real64/ 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 EV2PLT 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 EV2P&
     &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 EV2PLT 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 EV2PLT 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 EXREME VALUE TYPE 2 DISTRIBUTION ORDER STATISTIC MEDIANS
!
         DO i = 1 , N
            W(i) = (-LOG(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 = .9975_wp
         pp9975 = (-LOG(q))**(-1.0_wp/Gamma)
         q = .0025_wp
         pp0025 = (-LOG(q))**(-1.0_wp/Gamma)
         q = .975_wp
         pp975 = (-LOG(q))**(-1.0_wp/Gamma)
         q = .025_wp
         pp025 = (-LOG(q))**(-1.0_wp/Gamma)
         tau = (pp9975-pp0025)/(pp975-pp025)
         WRITE (G_IO,99007) Gamma , tau , N
!
99007    FORMAT (' ',                                                   &
     & 'EXTREME VALUE TYPE 2 (CAUCHY TYPE) PROB. PLOT WITH EXP. PAR. = '&
     & ,E17.10,1X,'(TAU = ',E15.8,')',1X,'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 EV2PLT