C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C                                                                   C
C                                                                   C
C       CMS DSN  =  MIX1CMA CLUSPAC                                 C
C                                                                   C
C       THE PROGRAMS MIX* CLUSPAC IN ISOPAC ARE FOR CLUSTERING DATA C
C  BY ITERATIVE MAXIMIZATION OF THE MIXTURE-MODEL LIKELIHOOD        C
C                                                                   C
C                 N   K                                             C
C                --- --                                             C
C          L  =  | | >  P(C)*F(X(I)|C),                             C
C                | | --                                             C
C                I=1 C=1                                            C
C                                                                   C
C  WHERE                                                            C
C                                                                   C
C       N  =  NUMBER OF OBSERVATIONS ("SAMPLE SIZE"),               C
C       K  =  NUMBER OF CLUSTERS,                                   C
C       X(I) = I-TH OBSERVATION, I = 1,2,...,N,                     C
C       F(X|C) = VALUE AT X OF THE C-TH CLASS-CONDITIONAL           C
C                DENSITY FUNCTION (C=1,2,...,K)                     C
C  AND                                                              C
C       P(C) = PRIOR PROBABILITY OF CLASS C.                        C
C                                                                   C
C                                                                   C
C       REFERENCE FOR CLUSTERING BY MIXTURE MODEL:                  C
C                                                                   C
C       Wolfe, J. H. (1970).  Pattern clustering by multivariate    C
C  mixture analysis.  Multivariate Behavioral Research 5, 329-350.  C
C                                                                   C
C                                                                   C
C       THE "1" IN THE PROGRAM NAME "MIX1CMA ISOPAC" MEANS THAT     C
C  THE PROGRAM IS FOR UNIVARIATE (1-DIMENSIONAL) DATA               C
C  (DATA ON THE LINE); THE "CM" MEANS THAT A COMMON VARIANCE IS     C
C  ASSUMED ACROSS CLUSTERS; AND THE "A" MEANS THAT THERE IS         C
C  AUTOMATIC SETTING OF NUMBERS OF CLUSTERS AND INITIAL MEANS.      C
C                                                                   C
C                                                                   C
C          PROGRAMMED BY                                            C
C              DR. STANLEY L. SCLOVE                 312/996-2681   C
C              DEPARTMENT OF                                        C
C                 INFORMATION & DECISION SCIENCES         M/C 294   C
C              COLLEGE OF BUSINESS ADMINISTRATION                   C
C              UNIVERSITY OF ILLINOIS  AT CHICAGO                   C
C              BOX 4348                                             C
C              CHICAGO, IL  60680-4348                              C
C                                                                   C
C                                                                   C
C          VERSION 1.2   31-MAR-91                                  C
C                                                                   C
C     COPYRIGHT (C) 1991  STANLEY L. SCLOVE                         C
C                                                                   C
C                                                                   C
C                                                                   C
C          RESTRICTIONS (CAN BE MODIFIED):                          C
C     N, SAMPLE SIZE, AT MOST 999;                                  C
C     K, NUMBER OF CLUSTERS, AT MOST 29;                            C
C     ITER, MAXIMUM NUMBER OF ITERATIONS, 20.                       C
C                                                                   C
C                                                                   C
C                                                                   C
C     CONTROL CARDS:                                                C
C                                                                   C
C     (1) DATASET TITLE                                             C
C     (2) N, IN FORMAT (2X,I4)                                      C
C     (3) FMT, IN FORMAT (18A4), E.G.,  (1X,F4.1).                  C
C         ALLOW AT LEAST ONE BLANK IN FMT:  IT WILL ALSO BE USED    C
C         FOR OUTPUT, WHERE CC1 IS FOR CARRIAGE CONTROL.            C
C         ALLOW A CC FOR THE DECIMAL POINT ON OUTPUT,               C
C         WHETHER OR NOT THERE IS ONE ON INPUT.                     C
C     (4) DATA, IN FORMAT SPECIFIED BY FMT                          C
C                                                                   C
C                                                                   C
C                                                                   C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C                                                                   C
C                                                                   C
C                                                                   C
C                                                                   C
C
C
C     CONTROL CARDS:
C
C     (1) DATASET TITLE
C     (2) N, IN FORMAT (2X,I4)
C     (3) FMT, IN FORMAT (18A4), E.G.,  (1X,F4.1).
C         ALLOW AT LEAST ONE BLANK IN FMT:  IT WILL ALSO BE USED
C         FOR OUTPUT, WHERE CC1 IS FOR CARRIAGE CONTROL.
C         ALLOW A CC FOR THE DECIMAL POINT ON OUTPUT, WHETHER OR NOR
C         THERE IS ONE ON INPUT.
C     (4) DATA, IN FORMAT SPECIFIED BY FMT
C
C
C
C     WRITE PROGRAM INFORMATION.
C
C
C     READ SAMPLE SIZE, N.
C
C     READ DATA FORMAT.
C
C     READ DATA AND
C     COMPUTE STATISTICS OF WHOLE SAMPLE:
C
C     READ(5, * ) ( X(I), I = 1,N )
C
C       IF IT IS DESIRED TO PRINT OUT THE DATA, REMOVE THE "C"S
C     FROM CC1 IN THE APPROPRIATE FOLLOWING STATEMENTS:
C     WRITE DATA:
C     WRITE(6, 1009)
C1009 FORMAT(1X,'DATA:'/)
C     WRITE(6, FMT) (X(I), I=1, N)
C     WRITE SUMMARY STATISTICS FOR WHOLE SAMPLE:
C       NO. OF PARAMETERS FOR UNCLUSTERED SAMPLE IS:
C     1 MEAN + 1 VARIANCE = 2 PARAMETERS
C
C     A TABLE OF MODEL SELECTION CRITERIA-VALUES FOR VARIOUS K
C     WILL BE PRINTED AT THE END.  THE NEXT INSTRUCTIONS
C     SET UP THE FIRST ENTRIES FOR THAT TABLE.
C
C
C
C     SET CONSTANTS.
C       XJ(IC,K), K=1 TO 9, IC = 1 TO K
C       OPTIMAL CLASS PROBABILITIES - NORMAL DISTRIBUTION
C       JOHARI & SCLOVE (1975). "PARTITIONING A DISTRIBUTION."
C       COMMUNICATIONS IN STATISTICS.
C
C     PERFORM CLUSTERING FOR K = 2 TO 9 GROUPS.
C
C     COMPUTE INITIAL MEANS, EQUALLY SPACED THROUGH RANGE OF DATA,
C     INITIAL PRIOR PROBABILITIES (EQUAL) AND INITIAL VALUE
C     OF COMMON VARIANCE:
C     XC = IC
C       SET INITIAL VALUES OF PRIOR PROBABILITIES EQUAL TO THE
C     OPTIMAL VALUES FOR A NORMAL DISTRIBUTION:
C       TO SET INITIAL VALUES OF PRIOR PROBABILITIES EQUAL TO
C     BINOMIAL(X=IC-1;N=K-1,P=1/2).
C     FOR IC = 1,2,...,K,
C     REMOVE THE "C" FROM CC1 OF THE NEXT LINE:
C     P(IC) = GAMMA(XK)/(GAMMA(XC)*GAMMA(XK-XC+1)*(2**K))
C
C       FOR EQUAL INITIAL VALUES OF PRIOR PROBABILITIES,
C     REMOVE THE "C" FROM CC1 OF THE NEXT LINE:
C     P(IC) = 1.0/K .
C
C
C
C
C     WRITE INITIAL VALUES OF PRIOR PROBS:
C
C
C     WRITE INITIAL MEANS:
C
C     WRITE INITIAL VARIANCE:
C
C
C
C
C     STORE OLD CLUSTERING:
C     COMMENCE DISTANCE COMPUTATIONS.
C
C     COMPUTE POSTERIOR PROBABILITIES OF GROUP MEMBERSHIP:
C     IF ( DENOM(I) .EQ. 0.0 ) DENOM(I)=0.0001
C
C       COMPUTE NEW LABELS BY MAX POSTERIOR PROBABILITY:
C
C       WRITE NEW LABELS:
C
C       UPDATE CLUSTER PRIOR PROBABILITIES P(IC), MEANS XMEAN(IC) AND
C     VARIANCE VARHAT:
C       XNC(IC) WILL BE THE SUM OVER ALL N OBSERVATIONS OF THEIR
C     POSTERIOR PROBABILITIES OF MEMBERSHIP IN CLUSTER IC.
C
C       IF ONE OF THE K CLUSTERS BECOMES EMPTY, THE PROCEDURE
C     WILL GO ON TO THE NEXT VALUE OF K:
C     IF ( VAR(IC) .LE. 0.0 ) VAR(IC) = 0.0001
C
C       COUNT NUMBERS IN CLUSTERS:
C
C
C
C
C     F(I,IC) HAD NOT PREVIOUSLY BEEN DIVIDED BY SQRT(2*PI*VARHAT):
C
C
C
C     B(IC) IS BOUNDARY BETWEEN G-TH AND G+1-ST CLASSES.
C
C       IF PROCEDURE HAS NOT CONVERGED IN 20 ITERATIONS, IT WILL
C     GO ON TO THE NEXT VALUE OF K.
C
C       IF ONE OF THE K CLUSTERS BECOMES EMPTY, THE PROCEDURE
C     WILL GO ON TO THE NEXT VALUE OF K.
C
C
C
C     VARHAT  IS MLE OF VARIANCE.
C
C
C
C     COMPUTE MODEL-SELECTION CRITERIA:
C     NO. PARAMETERS = K MEANS + 1 VARIANCE + (K-1) PROBS.
C
C
C
C
C
C
C
C
C     GO ON TO NEXT VALUE OF K:
C
C     WRITE VALUES OF MSC'S FOR VARIOUS K:
C
C
C
C$DATA