WEIPLT Subroutine

public subroutine WEIPLT(X, N, Gamma)

NAME

weiplt(3f) - [M_datapac:LINE_PLOT] generate a Weibull probability plot
(line printer graph)

SYNOPSIS

   SUBROUTINE WEIPLT(X,N,Gamma)

DESCRIPTION

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

The prototype weibull distribution used herein is defined for all
positive 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 Weibull 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 Weibull 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_weiplt
use M_datapac, only : weiplt
implicit none
! call weiplt(x,y)
end program demo_weiplt

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 250-271.

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)
PARPLT (subroutine)
PLOTU (subroutine)
POIPLT (subroutine)
RUNS (subroutine)
SAMPP (subroutine)
SPCORR (subroutine)
TAIL (subroutine)
TPLT (subroutine)
TRIM (subroutine)
UNIPLT (subroutine)
WEIB (subroutine)
WIND (subroutine)
"> common /BLOCK2_real64/

Type Attributes Name Initial
real :: WS(15000)

Source Code

SUBROUTINE WEIPLT(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 WEIBULL PROBABILITY PLOT.

!     RESTRICTIONS--THE MAXIMUM ALLOWABLE VALUE OF N
!                   FOR THIS SUBROUTINE IS 7500.
!
!---------------------------------------------------------------------
!
      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 WEIPLT 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 WEIP&
     &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 WEIPLT 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 WEIPLT 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 WEIBULL DISTRIBUTION ORDER STATISTIC MEDIANS
!
         DO i = 1 , N
            W(i) = (-LOG(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 = (-LOG(1.0_wp-q))**(1.0_wp/Gamma)
         q = 0.0025_wp
         pp0025 = (-LOG(1.0_wp-q))**(1.0_wp/Gamma)
         q = 0.975_wp
         pp975 = (-LOG(1.0_wp-q))**(1.0_wp/Gamma)
         q = 0.025_wp
         pp025 = (-LOG(1.0_wp-q))**(1.0_wp/Gamma)
         tau = (pp9975-pp0025)/(pp975-pp025)
         WRITE (G_IO,99007) Gamma , tau , N
!
99007    FORMAT (' ',                                                   &
     &           'WEIBULL 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 WEIPLT