! Last change: DOS 3 Aug 2000 6:38 pm ! *** copyright 2000 *** ! *** filename wlcx2s.f95 *** John F. Monahan ** ! ********************** program wlcx2s ! DEMONSTRATION OF WLCX2S -- WHICH COMPUTES THE DISTRIBUTION ! OF THE TWO SAMPLE WILCOXON STATISTIC USING FFT ! implicit none complex, dimension(256) :: phi complex, dimension(20) :: psi complex eitak real, dimension(20) :: ties real, parameter :: twopi = 6.2831853 real flq,t integer q,lq,i,j,jj,k,m,N ! 20 format(' Probability Dist of Wilcoxon Two Sample Statistic', & & /'(unnormalized) with m=',i2,' n+m=N=',i2,', and using Q=',i4,& & //' j real imaginary') 21 format(i4,2f12.6) ! open( unit=6, file='wlcx2s.out' ) ! ! follow Pagano and Trichler ! try case m=4, n=8, so N=12 m = 4 N = 12 ! need q > mn + m(m+1)/2 + 1 or roughly mN q = 64 lq = 6 flq = real(q) ! do case without ties -- it's easier for now do i = 1,N ties(i) = real(i) end do ! loop on i ! do i = 1,q t = -twopi*real(i-1)/flq ! get characteristic function at this t ! ! initialize psi(1) = cmplx( cos(t*ties(1)), sin(t*ties(1)) ) do k = 2,N psi(k) = ( 0., 0. ) eitak = cmplx( cos(t*ties(k)), sin(t*ties(k)) ) do jj = 2,k j = k + 2 - jj ! recursion formula psi(j) = psi(j) + psi(j-1)*eitak end do ! loop on jj psi(1) = psi(1) + eitak end do ! loop on k ! we want psi(m) phi(i) = psi(m) !w if( i .eq. 1 ) write(6,21) (j,psi(j),j=1,N) end do ! loop on i !w write(6,21) (i-1,phi(i),i=1,q) ! invert characteristic function call fft2n(phi,lq,-1.) ! write out results write(6,20) m,N,q do i = 1,q write(6,21) i-1,phi(i) end do ! loop on i stop end program wlcx2s SUBROUTINE FFT2N(A,N,SGN) ! ! SUBROUTINE FFT2N -- DISCRETE FOURIER TRANSFORM ! ! A COMPLEX VECTOR INPUT: VECTOR TO BE TRANSFORMED ! OUTPUT: TRANSFORMED VECTOR ! N INTEGER INPUT: LOG(BASE 2) OF LENGTH OF VECTOR A ! LENGTH OF A IS 2 ** N ! SGN REAL INPUT: INDICATES FORWARD OR INVERSE TRANSFO ! SGN = 1. THEN FORWARD TRANSFORM ! SGN = -1. THEN INVERSE TRANSFORM ! ! DISCRETE FOURIER TRANSFORM OF A VECTOR A(J) IS GIVEN BY ! NN-1 ! T( A ) = SUM A(J) EXP( - I SGN 2 PI J K / NN ) ! J K=0 ! ! WHERE I = SQRT( -1 ) ! NN = 2 ** N ! PI = 3.1415923565... ! ! NOTE THAT THE INDEXING OF A IS SHIFTED BY ONE -- ! A(0) STORED IN A(1), ... , A( NN-1 ) STORED IN A( NN ) ! ! J F MONAHAN (NOV,1984) DEPT OF STAT, N C S U, RALEIGH, N C 27695-8203 ! RECODED DECEMBER 1999, APRIL 2000 FOR FORTRAN 95 ! ! THIS IS A 'POWER OF 2' ALGORITHM BASED ON THE PRESENTATION IN ! ! A. V. AHO, J. E. HOPCROFT, AND J. D. ULLMAN (1974) THE DESIGN AND ! ANALYSIS OF COMPUTER ALGORITHMS, ADDISON-WESLEY, READING. ! ! ! BEGIN WITH DECLARATIONS IMPLICIT NONE INTEGER, INTENT(IN) :: N COMPLEX, DIMENSION(1:N), INTENT(IN OUT) :: A REAL, INTENT(IN) :: SGN REAL C,S COMPLEX, DIMENSION(25) :: WK COMPLEX WS,ALEFT,ARIGHT LOGICAL CARRY,NEW LOGICAL, DIMENSION(25) :: RBITS INTEGER I,J,L,LM,LL,M,MM,M2,NN ! ! COMPUTE WK(I) = EXP( - 2 PI IMAG SGN / 2**I ) ! ! SPECIAL VALUES FIRST WK(1) = CMPLX( -1., 0. ) WK(2) = CMPLX( 0., -SGN ) ! SKIP IF N IS ONLY 1 OR 2 IF( N .GT. 2 ) THEN ! AVOID SINES AND COSINES -- JUST USE SQUARE ROOTS C = 0. S = 1. DO I = 3,N C = SQRT( ( 1. + C ) / 2. ) S = S / ( 2.*C ) WK(I) = CMPLX( C, -SGN*S ) END DO ! LOOP ON I END IF ! ( N .GT. 2 ) ! ! THIS IS THE MAIN LOOP OF THE ALGORITHM ! DO MM = 1,N ! CONVERT TO REVERSE LOOP M = N - MM ! M STEPS DOWN FROM N-1 TO 0 ! ! INITIALIZE DO I = 1,N RBITS(I) = .TRUE. END DO ! LOOP ON I ! M2 M2 = 2**M ! LM CONTROLS THE NUMBER OF SUBLOOPS LM = 2**(MM-1) ! DO LL = 1,LM ! L IS THE MAJOR INDEX VARIABLE L = 2*M2*(LL-1) ! INITIALIZE TO GET W ** REV( L / M2 ) WS = CMPLX(1., 0.) CARRY = .TRUE. ! START AT 2 (SHIFT) SINCE L/M2 IS ALWAYS EVEN DO I = 2,N NEW = CARRY .OR. RBITS(I) CARRY = CARRY .AND. RBITS(I) RBITS(I) = NEW .AND. ( .NOT. CARRY ) IF( RBITS(I) ) WS = WS * WK(I) END DO ! LOOP ON I ! WS IS NOW W ** REV( L / M2 ) ! NOW DO THE REAL COMPUTATION DO J = 1,M2 ALEFT = A( L + J ) ARIGHT = WS*A( L + J + M2 ) A( L + J ) = ALEFT + ARIGHT A( L + J + M2 ) = ALEFT - ARIGHT END DO ! LOOP ON J ! DONE WITH LOOP ON L END DO ! LOOP ON LL ! DONE WITH MAIN LOOP ON M END DO ! LOOP ON MM ! ! SCRAMBLE THE NUMBERS -- SWITCH A(J) AND A( REV(J) ) ! BUT DON'T DO IT TWICE ! INITIALIZE DO I = 1,N RBITS(I) = .TRUE. END DO ! LOOP ON I NN = 2 ** N ! LOOP THROUGH ALL THE NUMBERS DO LL = 1,NN ! DO THE BIT REVERSAL LM = 0 M2 = NN CARRY = .TRUE. ! START AT 1 HERE -- USE ALL THE NUMBERS DO I = 1,N M2 = M2 / 2 NEW = CARRY .OR. RBITS(I) CARRY = CARRY .AND. RBITS(I) RBITS(I) = NEW .AND. ( .NOT. CARRY ) IF( RBITS(I) ) LM = LM + M2 END DO ! LOOP ON I ! DON'T SWITCH TWICE IF( LL-1 .LT. LM ) THEN ALEFT = A(LL) ARIGHT = A(LM+1) A(LL) = ARIGHT A(LM+1) = ALEFT END IF ! ( LL-1 .LT. LM ) END DO ! LOOP ON LL ! IF INVERSE TRANSFORM, REMEMBER TO DIVIDE BY NN IF( SGN .GT. 0. ) RETURN C = REAL(NN) DO I = 1,NN A(I) = A(I) / C END DO ! LOOP ON I RETURN END SUBROUTINE FFT2N ! *** end of filename wlcx2s.f95 *********************