SUBROUTINE RUNS(X,N) C C PURPOSE--THIS SUBROUTINE PERFORMS A RUNS ANALYSIS C OF THE DATA IN THE INPUT VECTOR X. C THE ANALYSIS CONSISTS OF FIRST DETERMINING C THE OBSERVED NUMBER OF RUNS FROM THE DATA, C AND THEN COMPUTING C THE EXPECTED NUMBER OF RUNS, C THE STANDARD DEVIATION OF THE NUMBER OF RUNS, C AND THE RESULTING STANDARDIZED STATISTIC C FOR THE NUMBER OF RUNS FOR RUNS OF VARIOUS C LENGTHS. C THIS IS DONE FOR RUNS UP, RUNS DOWN, AND C RUNS UP AND DOWN. C THIS RUNS ANSLYSIS IS A USEFUL DISTRIBUTION-FREE C TEST OF THE RANDOMNESS OF A DATA SET. 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 OUTPUT--4 PAGES OF AUTOMATIC PRINTOUT C CONSISTING OF THE OBSERVED NUMBER, C EXPECTED NUMBER, STANDARD DEVIATION C AND RESULTING STANDARDIZED STATISTIC C FOR RUNS OF VARIOUS LENGTHS. C AND THE CUMULATIVE FREQUENCY. C PRINTING--YES. C RESTRICTIONS--THE MAXIMUM ALLOWABLE VALUE OF N C FOR THIS SUBROUTINE IS 15000. C OTHER DATAPAC SUBROUTINES NEEDED--NONE. C FORTRAN LIBRARY SUBROUTINES NEEDED--SQRT. C MODE OF INTERNAL OPERATIONS--SINGLE PRECISION. C LANGUAGE--ANSI FORTRAN. C REFERENCES--LEVENE AND WOLFOWITZ, ANNALS OF MATHEMATICAL C STATISTICS, 1944, PAGES 58-69; C ESPECIALLY PAGES 60, 63, AND 64. C REFERENCES--BRADLEY, DISTRIBUTION-FREE STATISTICAL TESTS, C 1968, CHAPTER 12, PAGES 271-282. 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--JUNE 1972. C UPDATED --NOVEMBER 1975. C UPDATED --FEBRUARY 1976. C C--------------------------------------------------------------------- C DIMENSION X(1) DIMENSION Y(15000) DIMENSION NRUL(16), NRDL(16), NRTL(16), NRULG(16), NRDLG(16) DIMENSION NRTLG(16) DIMENSION ENRUL(16),ENRTL(16),ENRULG(16),ENRTLG(16) DIMENSION SNRUL(16),SNRTL(16),SNRULG(16),SNRTLG(16) DIMENSION ZNRUL(16),ZNRDL(16),ZNRTL(16),ZNRULG(16),ZNRDLG(16) DIMENSION ZNRTLG(16) DIMENSION C1(15),C2(15),C3(15),C4(15) DIMENSION ANRUL(16),ANRDL(16),ANRTL(16) DIMENSION ANRULG(16),ANRDLG(16),ANRTLG(16) COMMON /BLOCK2/ WS(15000) EQUIVALENCE (Y(1),WS(1)) C DATA C1(1),C1(2),C1(3),C1(4),C1(5),C1(6),C1(7),C1(8),C1(9),C1(10), 1C1(11),C1(12),C1(13),C1(14),C1(15) 1/ .4236111111E+00, .1126675485E+00, .4191688713E-01, 1 .1076912487E-01, .2003959238E-02, .3023235799E-03, 1 .3911555473E-04, .4459038843E-05, .4551105210E-06, 1 .4207466837E-07, .3555930927E-08, .2768273257E-09, 1 .1997821524E-10, .1343876568E-11, .8465610177E-13/ DATA C2(1),C2(2),C2(3),C2(4),C2(5),C2(6),C2(7),C2(8),C2(9),C2(10), 1C2(11),C2(12),C2(13),C2(14),C2(15) 1/-.4819444444E+00, -.1628284832E+00, -.9690696649E-01, 1 -.3778106786E-01, -.9289228716E-02, -.1724429252E-02, 1 -.2638557888E-03, -.3466965096E-04, -.4004129153E-05, 1 -.4130382587E-06, -.3851876069E-07, -.3279103786E-08, 1 -.2568491117E-09, -.1863433868E-10, -.1259220466E-11/ DATA C3(1),C3(2),C3(3),C3(4),C3(5),C3(6),C3(7),C3(8),C3(9),C3(10), 1C3(11),C3(12),C3(13),C3(14),C3(15) 1/ .1777777778E+00, .7916666667E-01, .4738977072E-01, 1 .1274801587E-01, .2338606059E-02, .3461358734E-03, 1 .4407121770E-04, .4960020603E-05, .5010387575E-06, 1 .4592883352E-07, .3854170274E-08, .2982393839E-09, 1 .2141205844E-10, .1433843200E-11, .8996663214E-13/ DATA C4(1),C4(2),C4(3),C4(4),C4(5),C4(6),C4(7),C4(8),C4(9),C4(10), 1C4(11),C4(12),C4(13),C4(14),C4(15) 1/-.3222222222E+00, -.5972222222E-01, -.1130268959E+00, 1 -.4696428571E-01, -.1123273065E-01, -.2025170849E-02, 1 -.3029410411E-03, -.3912824548E-04, -.4459234519E-05, 1 -.4551128785E-06, -.4207469124E-07, -.3555931110E-08, 1 -.2768273269E-09, -.1997821525E-10, -.1343876568E-11/ C IPR=6 IUPPER=15000 C C CHECK THE INPUT ARGUMENTS FOR ERRORS C IF(N.LT.1.OR.N.GT.IUPPER)GOTO50 IF(N.EQ.1)GOTO55 HOLD=X(1) DO60I=2,N IF(X(I).NE.HOLD)GOTO90 60 CONTINUE WRITE(IPR, 9)HOLD GOTO90 50 WRITE(IPR,17)IUPPER WRITE(IPR,47)N RETURN 55 WRITE(IPR,18) RETURN 90 CONTINUE 9 FORMAT(1H ,109H***** NON-FATAL DIAGNOSTIC--THE FIRST INPUT ARGUME 1NT (A VECTOR) TO THE RUNS SUBROUTINE HAS ALL ELEMENTS = ,E15.8,6 1H *****) 17 FORMAT(1H , 98H***** FATAL ERROR--THE SECOND INPUT ARGUMENT TO THE 1 RUNS SUBROUTINE IS OUTSIDE THE ALLOWABLE (1,,I6,16H) INTERVAL * 1****) 18 FORMAT(1H ,100H***** FATAL ERROR-- THE SECOND INPUT ARGUME 1NT TO THE RUNS SUBROUTINE HAS THE VALUE 1 *****) 47 FORMAT(1H , 35H***** THE VALUE OF THE ARGUMENT IS ,I8 ,6H *****) C C-----START POINT----------------------------------------------------- C AN=N C C FORM THE SEQUENTIAL DIFFERENCE TABLE C NM1=N-1 DO100I=1,NM1 IP1=I+1 Y(I)=X(IP1)-X(I) 100 CONTINUE C C ZERO-OUT THE 6 'NUMBER OF RUNS' VECTORS C DO200I=1,16 NRUL(I)=0 NRDL(I)=0 NRTL(I)=0 NRULG(I)=0 NRDLG(I)=0 NRTLG(I)=0 200 CONTINUE C C DETERMINE THE NUMBER OF RUNS UP OF LENGTH EXACTLY I C AND THE NUMBER OF RUNS DOWN OF LENGTH EXACTLY I C DETERMINE THE LENGTH OF THE LONGEST RUN UP C AND THE LENGTH OF THE LONGEST RUN DOWN C LENUP=0 LENDN=0 MAXLNU=0 MAXLND=0 DO300I=1,NM1 IF(Y(I).EQ.0.0.AND.LENUP.GE.1)LENUP=LENUP+1 IF(Y(I).EQ.0.0.AND.LENDN.GE.1)LENDN=LENDN+1 IF(Y(I).EQ.0.0.AND.LENUP.EQ.0.AND.LENDN.EQ.0)LENUP=LENUP+1 IF(Y(I).GT.0.0.AND.LENDN.GE.1.AND.LENDN.LE.15)NRDL(LENDN)=NRDL(LEN 1DN)+1 IF(Y(I).GT.0.0.AND.LENDN.GE.1.AND.LENDN.GE.16)NRDL(16)=NRDL(16)+1 IF(Y(I).GT.0.0)LENDN=0 IF(Y(I).GT.0.0)LENUP=LENUP+1 IF(Y(I).LT.0.0.AND.LENUP.GE.1.AND.LENUP.LE.15)NRUL(LENUP)=NRUL(LEN 1UP)+1 IF(Y(I).LT.0.0.AND.LENUP.GE.1.AND.LENUP.GE.16)NRUL(16)=NRUL(16)+1 IF(Y(I).LT.0.0)LENUP=0 IF(Y(I).LT.0.0)LENDN=LENDN+1 IF(I.EQ.NM1. AND.LENDN.GE.1.AND.LENDN.LE.15)NRDL(LENDN)=NRDL(LEN 1DN)+1 IF(I.EQ.NM1. AND.LENDN.GE.1.AND.LENDN.GE.16)NRDL(16)=NRDL(16)+1 IF(I.EQ.NM1. AND.LENUP.GE.1.AND.LENUP.LE.15)NRUL(LENUP)=NRUL(LEN 1UP)+1 IF(I.EQ.NM1. AND.LENUP.GE.1.AND.LENUP.GE.16)NRUL(16)=NRUL(16)+1 IF(LENUP.GT.MAXLNU)MAXLNU=LENUP IF(LENDN.GT.MAXLND)MAXLND=LENDN 300 CONTINUE C C DETERMINE THE NUMBER OF RUNS TOTAL OF LENGTH EXACTLY I C AND THE LENGTH OF THE LONGEST RUN UP OR DOWN C DO400I=1,16 NRTL(I)=NRUL(I)+NRDL(I) 400 CONTINUE MAXLNT=MAXLNU IF(MAXLND.GT.MAXLNU)MAXLNT=MAXLND C C DETERMINE THE NUMBER OF RUNS UP OF LENGTH I OR MORE C AND THE NUMBER OF RUNS DOWN OF LENGTH I OR MORE C AND THE NUMBER OF RUNS TOTAL OF LENGTH I OR MORE C NRULG(16)=NRUL(16) NRDLG(16)=NRDL(16) NRTLG(16)=NRTL(16) DO500I=1,15 J=16-I JP1=J+1 NRULG(J)=NRULG(JP1)+NRUL(J) NRDLG(J)=NRDLG(JP1)+NRDL(J) NRTLG(J)=NRTLG(JP1)+NRTL(J) 500 CONTINUE C C DETERMINE THE NUMBER OF POSITIVE, ZERO, AND NEGATIVE ENTRIES C IN THE DIFFERENCE TABLE. IF RANDOM, THE NUMBER OF POSITIVE SHOULD BE C APPROXIMATELY EQUAL TO THE NUMBER OF NEGATIVE C NNEG=0 NZER=0 NPOS=0 DO800I=1,NM1 IF(Y(I).LT.0.0)NNEG=NNEG+1 IF(Y(I).EQ.0.0)NZER=NZER+1 IF(Y(I).GT.0.0)NPOS=NPOS+1 800 CONTINUE C C COMPUTE THE EXPECTED NUMBER OF RUNS UP OF LENGTH EXACTLY I = C THE EXPECTED NUMBER OF RUNS DOWN OF LENGTH EXACTLY I = C ONE HALF THE EXPECTED NUMBER OF RUNS TOTAL OF LENGTH EXACTLY I C DEN=6.0 DO2000I=1,15 AI=I ENRUL(I)=AN*(AI*AI+3.0*AI+1.0)-(AI*AI*AI+3.0*AI*AI-AI-4.0) DEN=DEN*(AI+3.0) ENRUL(I)=ENRUL(I)/DEN ENRTL(I)=2.0*ENRUL(I) 2000 CONTINUE C C COMPUTE THE EXPECTED NUMBER OF RUNS UP OF LENGTH I OR MORE = C THE EXPECTED NUMBER OF RUNS DOWN OF LENGTH I OR MORE = C ONE HALF THE EXPECTED NUMBER OF RUNS TOTAL OF LENGTH I OR MORE C DEN=2.0 DO2100I=1,15 AI=I ENRULG(I)=AN*(AI+1.0)-(AI*AI+AI-1.0) DEN=DEN*(AI+2.0) ENRULG(I)=ENRULG(I)/DEN ENRTLG(I)=2.0*ENRULG(I) 2100 CONTINUE C C COMPUTE THE STANDARD DEV. OF THE NUMBER OF RUNS UP OF LENGTH EXACTLY I = C THE STANDARD DEV. OF THE NUMBER OF RUNS DOWN OF LENGTH EXACTLY I = C SQRT(0.5)* THE STAND. DEV. OF THE NUMBER OF RUNS TOTAL OF LENGTH EXACTLY I C DO2500I=1,15 SNRTL(I)=SQRT(C1(I)*AN+C2(I)) SNRUL(I)=SQRT(0.5)*SNRTL(I) 2500 CONTINUE C C COMPUTE THE STAND. DEV. OF THE NUMBER OF RUNS UP OF LENGTH I OR MORE = C THE STAND. DEV. OF THE NUMBER OF RUNS DOWN OF LENGTH I OR MORE = C SQRT(0.5)* THE STAND. DEV. OF THE NUMBER OF RUNS TOTAL OF LENGTH I OR MORE C DO2600I=1,15 SNRTLG(I)=SQRT(C3(I)*AN+C4(I)) SNRULG(I)=SQRT(0.5)*SNRTLG(I) 2600 CONTINUE C C FORM Z STATISTICS C DO3100I=1,15 STAT=NRUL(I) ZNRUL(I)=(STAT -ENRUL(I))/SNRUL(I) STAT=NRDL(I) ZNRDL(I)=(STAT -ENRUL(I))/SNRUL(I) STAT=NRTL(I) ZNRTL(I)=(STAT -ENRTL(I))/SNRTL(I) STAT=NRULG(I) ZNRULG(I)=(STAT -ENRULG(I))/SNRULG(I) STAT=NRDLG(I) ZNRDLG(I)=(STAT -ENRULG(I))/SNRULG(I) STAT=NRTLG(I) ZNRTLG(I)=(STAT -ENRTLG(I))/SNRTLG(I) 3100 CONTINUE C DO3200I=1,15 ANRUL(I)=NRUL(I) ANRDL(I)=NRDL(I) ANRTL(I)=NRTL(I) ANRULG(I)=NRULG(I) ANRDLG(I)=NRDLG(I) ANRTLG(I)=NRTLG(I) 3200 CONTINUE C C WRITE EVERYTHING OUT C IMAX=15 WRITE(IPR,998) WRITE(IPR,4002) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4004) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4050I=1,IMAX WRITE(IPR,4060)I,ANRUL(I),ENRUL(I),SNRUL(I),ZNRUL(I) 4050 CONTINUE WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4064) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4070I=1,IMAX WRITE(IPR,4060)I,ANRULG(I),ENRULG(I),SNRULG(I),ZNRULG(I) 4070 CONTINUE WRITE(IPR,998) WRITE(IPR,4102) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4104) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4150I=1,IMAX WRITE(IPR,4060)I,ANRDL(I),ENRUL(I),SNRUL(I),ZNRDL(I) 4150 CONTINUE WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4164) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4170I=1,IMAX WRITE(IPR,4060)I,ANRDLG(I),ENRULG(I),SNRULG(I),ZNRDLG(I) 4170 CONTINUE WRITE(IPR,998) WRITE(IPR,4202) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4204) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4250I=1,IMAX WRITE(IPR,4060)I,ANRTL(I),ENRTL(I),SNRTL(I),ZNRTL(I) 4250 CONTINUE WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(6,999) WRITE(IPR,4264) WRITE(6,999) WRITE(6,999) WRITE(IPR,4006) WRITE(IPR,999) DO4270I=1,IMAX WRITE(IPR,4060)I,ANRTLG(I),ENRTLG(I),SNRTLG(I),ZNRTLG(I) 4270 CONTINUE WRITE(IPR,998) WRITE(IPR,4601)MAXLNU WRITE(IPR,4602)MAXLND WRITE(IPR,4603)MAXLNT WRITE(IPR,999) WRITE(IPR,4611)NPOS WRITE(IPR,4612)NNEG WRITE(IPR,4613)NZER 4002 FORMAT(1H ,48X,7HRUNS UP) 4102 FORMAT(1H ,48X,9HRUNS DOWN) 4202 FORMAT(1H ,40X,32HRUNS TOTAL = RUNS UP + RUNS DOWN) 4004 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS UP OF LENGTH EXACT 1LY I) 4104 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS DOWN OF LENGTH EXACT 1LY I) 4204 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS TOTAL OF LENGTH EXACT 1LY I) 4064 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS UP OF LENGTH I OR 1MORE) 4164 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS DOWN OF LENGTH I OR 1MORE) 4264 FORMAT(1H ,27X,52HSTATISTIC = NUMBER OF RUNS TOTAL OF LENGTH I OR 1MORE) 4006 FORMAT(1H ,105HI = LENGTH OF RUN VALUE OF STAT EXP( 1STAT) SD(STAT) (STAT-EXP(STAT))/SD(STAT)) 4060 FORMAT(1H ,4X,I4,13X,6X,F7.1,13X,F8.4,12X,F8.4,11X,F8.2) 4601 FORMAT(1H ,39HLENGTH OF THE LONGEST RUN UP = ,I5) 4602 FORMAT(1H ,39HLENGTH OF THE LONGEST RUN DOWN = ,I5) 4603 FORMAT(1H ,39HLENGTH OF THE LONGEST RUN UP OR DOWN = ,I5) 4611 FORMAT(1H ,33HNUMBER OF POSITIVE DIFFERENCES = ,I5) 4612 FORMAT(1H ,33HNUMBER OF NEGATIVE DIFFERENCES = ,I5) 4613 FORMAT(1H ,33HNUMBER OF ZERO DIFFERENCES = ,I5) 4010 FORMAT(1H ,2(I4,2X,F7.1,2X,F8.4,2X,F8.4,2X,F8.2,8X)) 4605 FORMAT(1H ,I6,2X,I6,2X,I6) 998 FORMAT(1H1) 999 FORMAT(1H ) RETURN END