SUBROUTINE POIPLT(X,N,ALAMBA) C C PURPOSE--THIS SUBROUTINE GENERATES A POISSON C PROBABILITY PLOT C (WITH SINGLE PRECISION C TAIL LENGTH PARAMETER = ALAMBA). C THE PROTOTYPE POISSON DISTRIBUTION USED C HEREIN HAS MEAN = ALAMBA C AND STANDARD DEVIATION = SQRT(ALAMBA). C THIS DISTRIBUTION IS DEFINED FOR C ALL DISCRETE NON-NEGATIVE INTEGER X--X = 0, 1, 2, ... . C THIS DISTRIBUTION HAS THE PROBABILITY FUNCTION C F(X) = EXP(-ALAMBA) * ALAMBA**X / X!. C THE POISSON DISTRIBUTION IS THE C DISTRIBUTION OF THE NUMBER OF EVENTS C IN THE INTERVAL (0,ALAMBA) WHEN C THE WAITING TIME BETWEEN EVENTS C IS EXPONENTIALLY DISTRIBUTED C WITH MEAN = 1 AND STANDARD DEVIATION = 1. C THE PROTOTYPE DISTRIBUTION RESTRICTIONS OF C DISCRETENESS AND NON-NEGATIVENESS C MENTIONED ABOVE DO NOT CARRY OVER TO THE C INPUT VECTOR X OF OBSERVATIONS TO BE ANALYZED. C THE INPUT OBSERVATIONS IN X MAY BE DISCRETE, CONTINUOUS, C NON-NEGATIVE, OR NEGATIVE. C AS USED HEREIN, A PROBABILITY PLOT FOR A DISTRIBUTION C IS A PLOT OF THE ORDERED OBSERVATIONS VERSUS C THE ORDER STATISTIC MEDIANS FOR THAT DISTRIBUTION. C THE POISSON PROBABILITY PLOT IS USEFUL IN C GRAPHICALLY TESTING THE COMPOSITE (THAT IS, C LOCATION AND SCALE PARAMETERS NEED NOT BE SPECIFIED) C HYPOTHESIS THAT THE UNDERLYING DISTRIBUTION C FROM WHICH THE DATA HAVE BEEN RANDOMLY DRAWN C IS THE POISSON DISTRIBUTION C WITH TAIL LENGTH PARAMETER VALUE = ALAMBA. C IF THE HYPOTHESIS IS TRUE, THE PROBABILITY PLOT C SHOULD BE NEAR-LINEAR. C A MEASURE OF SUCH LINEARITY IS GIVEN BY THE C CALCULATED PROBABILITY PLOT CORRELATION COEFFICIENT. C INPUT ARGUMENTS--X = THE SINGLE PRECISION VECTOR OF C (UNSORTED OR SORTED) OBSERVATIONS. C --N = THE INTEGER NUMBER OF OBSERVATIONS C IN THE VECTOR X. C --ALAMBA = THE SINGLE PRECISION VALUE OF THE C TAIL LENGTH PARAMETER. C ALAMBA SHOULD BE POSITIVE. C OUTPUT--A ONE-PAGE POISSON PROBABILITY PLOT. C PRINTING--YES. C RESTRICTIONS--THE MAXIMUM ALLOWABLE VALUE OF N C FOR THIS SUBROUTINE IS 5000. C --ALAMBA SHOULD BE POSITIVE. C OTHER DATAPAC SUBROUTINES NEEDED--SORT, UNIMED, PLOT, C CHSCDF, NORPPF. C FORTRAN LIBRARY SUBROUTINES NEEDED--SQRT. C MODE OF INTERNAL OPERATIONS--SINGLE PRECISION. C LANGUAGE--ANSI FORTRAN. C COMMENT--FOR LARGE VALUES OF ALAMBA (IN EXCESS OF 500.) C THIS SUBROUTINE USES THE NORMAL APPROXIMATION TO C THE POISSON. THIS IS DONE TO SAVE EXECUTION TIME C WHICH INCREASES AS A FUNCTION OF ALAMBA AND WOULD C BE EXCESSIVE FOR LARGE VALUES OF ALAMBA. C REFERENCES--FILLIBEN, 'TECHNIQUES FOR TAIL LENGTH ANALYSIS', C PROCEEDINGS OF THE EIGHTEENTH CONFERENCE C ON THE DESIGN OF EXPERIMENTS IN ARMY RESEARCH C DEVELOPMENT AND TESTING (ABERDEEN, MARYLAND, C OCTOBER, 1972), PAGES 425-450. C --HAHN AND SHAPIRO, STATISTICAL METHODS IN ENGINEERING, C 1967, PAGES 260-308. C --JOHNSON AND KOTZ, DISCRETE C DISTRIBUTIONS, 1969, PAGES 87-121. C WRITTEN BY--JAMES J. FILLIBEN C STATISTICAL ENGINEERING LABORATORY (205.03) C NATIONAL BUREAU OF STANDARDS C WASHINGTON, D. C. 20234 C PHONE: 301-921-2315 C ORIGINAL VERSION--NOVEMBER 1974. C UPDATED --AUGUST 1975. C UPDATED --SEPTEMBER 1975. C UPDATED --NOVEMBER 1975. C UPDATED --FEBRUARY 1976. C C--------------------------------------------------------------------- C DIMENSION X(1) DIMENSION Y(5000),W(5000) DIMENSION Z(5000) COMMON /BLOCK2/ WS(15000) EQUIVALENCE (Y(1),WS(1)),(W(1),WS(5001)) EQUIVALENCE (Z(1),WS(10001)) C IPR=6 IUPPER=5000 C C CHECK THE INPUT ARGUMENTS FOR ERRORS C IF(N.LT.1.OR.N.GT.IUPPER)GOTO50 IF(N.EQ.1)GOTO55 IF(ALAMBA.LE.0.0)GOTO60 HOLD=X(1) DO65I=2,N IF(X(I).NE.HOLD)GOTO90 65 CONTINUE WRITE(IPR, 9)HOLD RETURN 50 WRITE(IPR,17)IUPPER WRITE(IPR,47)N RETURN 55 WRITE(IPR,18) RETURN 60 WRITE(IPR,25) WRITE(IPR,46)ALAMBA RETURN 90 CONTINUE 9 FORMAT(1H ,109H***** NON-FATAL DIAGNOSTIC--THE FIRST INPUT ARGUME 1NT (A VECTOR) TO THE POIPLT SUBROUTINE HAS ALL ELEMENTS = ,E15.8,6 1H *****) 17 FORMAT(1H , 98H***** FATAL ERROR--THE SECOND INPUT ARGUMENT TO THE 1 POIPLT SUBROUTINE IS OUTSIDE THE ALLOWABLE (1,,I6,16H) INTERVAL * 1****) 18 FORMAT(1H ,100H***** NON-FATAL DIAGNOSTIC--THE SECOND INPUT ARGUME 1NT TO THE POIPLT SUBROUTINE HAS THE VALUE 1 *****) 25 FORMAT(1H , 91H***** FATAL ERROR--THE THIRD INPUT ARGUMENT TO THE 1 POIPLT SUBROUTINE IS NON-POSITIVE *****) 46 FORMAT(1H , 35H***** THE VALUE OF THE ARGUMENT IS ,E15.8,6H *****) 47 FORMAT(1H , 35H***** THE VALUE OF THE ARGUMENT IS ,I8 ,6H *****) C C-----START POINT----------------------------------------------------- C AN=N CUTOFF=500.0 C C SORT THE DATA C CALL SORT(X,N,Y) C C GENERATE UNIFORM ORDER STATISTIC MEDIANS C CALL UNIMED(N,W) C C COMPUTE POISSON ORDER STATISTIC MEDIANS. C IF THE INPUT ALAMBA VALUE IS LARGE (IN EXCESS OF C CUTOFF VALUE OF 500.0), THEN USE THE NORMAL APPROXIMATION C TO THE POISSON. C IF(ALAMBA.LE.CUTOFF)GOTO150 SQALAM=SQRT(ALAMBA) DO170I=1,N CALL NORPPF(W(I),W(I)) W(I)=ALAMBA+W(I)*SQALAM 170 CONTINUE GOTO1550 150 CONTINUE C C DETERMINE WHICH UNIFORM ORDER STATISTIC MEDIAN IS ASSOCIATED WITH C THE CLOSEST INTEGER TO ALAMBA. C DO100I=1,N Z(I)=-1.0 100 CONTINUE C ILAMBA=ALAMBA+0.5 ARG1=2.0*ALAMBA IARG2=2*(ILAMBA+1) CALL CHSCDF(ARG1,IARG2,CDF) CDF=1.0-CDF DO200J=1,N IF(W(J).GT.CDF)GOTO250 200 CONTINUE 250 JM1=J-1 Z(JM1)=ILAMBA C C FILL IN THE POISSON ORDER STATISTIC MEDIANS BELOW ALAMBA C IMAX=6.0*SQRT(ALAMBA) DO500I=1,IMAX K=ILAMBA-I IF(K.LT.0)GOTO750 IARG2=2*(K+1) CALL CHSCDF(ARG1,IARG2,CDF) CDF=1.0-CDF DO600J=1,N IF(W(J).GT.CDF)GOTO650 600 CONTINUE 650 JM1=J-1 IF(JM1.LE.0)GOTO750 IF(Z(JM1).LT.-0.5)Z(JM1)=K 500 CONTINUE C C FILL IN THE POISSON ORDER STATISTIC MEDIANS ABOVE ALAMBA C 750 DO800I=1,IMAX K=ILAMBA+I IARG2=2*(K+1) CALL CHSCDF(ARG1,IARG2,CDF) CDF=1.0-CDF DO900J=1,N IF(W(J).GT.CDF)GOTO950 900 CONTINUE Z(N)=K GOTO1050 950 JM1=J-1 IF(Z(JM1).LT.-0.5)Z(JM1)=K 800 CONTINUE C C FILL IN THE EMPTY HOLES IN THE POISSON ORDER STATISTIC MEDIAN C Z MATRIX WITH THE PROPER VALUES. C THEN FOR SAKE OF CONSISTENCY WITH OTHER DATAPAC C PROBABILITY PLOT SUBROUTINES, COPY THE Z VECTOR C INTO THE W VECTOR. C 1050 HOLD=Z(N) DO1200IREV=1,N I=N-IREV+1 IF(Z(I).GE.-0.5)HOLD=Z(I) IF(Z(I).LT.-0.5)Z(I)=HOLD 1200 CONTINUE DO1300I=1,N W(I)=Z(I) 1300 CONTINUE C C PLOT THE ORDERED OBSERVATIONS VERSUS ORDER STATISTICS MEDIANS. C WRITE OUT THE SAMPLE SIZE. C 1550 CALL PLOT(Y,W,N) WRITE(IPR,2105)ALAMBA,N C C COMPUTE THE PROBABILITY PLOT CORRELATION COEFFICIENT. C COMPUTE LOCATION AND SCALE ESTIMATES C FROM THE INTERCEPT AND SLOPE OF THE PROBABILITY PLOT. C THEN WRITE THEM OUT. C SUM1=0.0 SUM2=0.0 DO2200I=1,N SUM1=SUM1+Y(I) SUM2=SUM2+W(I) 2200 CONTINUE YBAR=SUM1/AN WBAR=SUM2/AN SUM1=0.0 SUM2=0.0 SUM3=0.0 DO2300I=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) 2300 CONTINUE CC=SUM2/SQRT(SUM3*SUM1) YSLOPE=SUM2/SUM3 YINT=YBAR-YSLOPE*WBAR WRITE(IPR,2305)CC,YINT,YSLOPE C 2105 FORMAT(1H ,42HPOISSON PROBABILITY PLOT WITH PARAMETER = ,9X, 1E17.10,1X,8X,11X,20HTHE SAMPLE SIZE N = ,I7) 2305 FORMAT(1H ,43HPROBABILITY PLOT CORRELATION COEFFICIENT = ,F8.5,5X, 122HESTIMATED INTERCEPT = ,E15.8,3X,18HESTIMATED SLOPE = ,E15.8) C RETURN END