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C
C SUBROUTINE MACE(P,N,X,Y,W,L,DELRSQ,NS,TX,TY,RSQ,IERR,M,Z)
C------------------------------------------------------------------
C
C ESTIMATE MULTIPLE OPTIMAL TRANSFORMATIONS FOR REGRESSION AND
C CORRELATION BY ALTERNATING CONDITIONAL EXPECTATION ESTIMATES.
C
C VERSION 3/28/85.
C
C BREIMAN AND FRIEDMAN, JOURNAL OF THE AMERICAN STATISTICAL
C ASSOCIATION (SEPTEMBER, 1985)
C
C CODED AND COPYWRITE (C) 1985 BY:
C
C JEROME H. FRIEDMAN
C DEPARTMENT OF STATISTICS
C AND
C STANFORD LINEAR ACCELERATOR CENTER
C STANFORD UNIVERSITY
C
C ALL RIGHTS RESERVED.
C
C
C INPUT:
C
C N : NUMBER OF OBSERVATIONS.
C P : NUMBER OF PREDICTOR VARIABLES FOR EACH OBSERVATION.
C X(P,N) : PREDICTOR DATA MATRIX.
C Y(N) : RESPONSE VALUES FOR THE OBSERVATIONS.
C MISSING VALUES ARE SIGNIFIED BY A VALUE (RESPONSE OR
C PREDICTOR) GREATER THAN OR EQUAL TO BIG.
C (SEE BELOW - DEFAULT, BIG = 1.0E20)
C W(N) : WEIGHTS FOR THE OBSERVATIONS.
C L(P+1) : FLAG FOR EACH VARIABLE.
C L(1) THROUGH L(P) : PREDICTOR VARIABLES.
C L(P+1) : RESPONSE VARIABLE.
C L(I)=0 => ITH VARIABLE NOT TO BE USED.
C L(I)=1 => ITH VARIABLE ASSUMES ORDERABLE VALUES.
C L(I)=2 => ITH VARIABLE ASSUMES CIRCULAR (PERIODIC) VALUES
C IN THE RANGE (0.0,1.0) WITH PERIOD 1.0.
C L(I)=3 => ITH VARIABLE TRANSFORMATION IS TO BE MONOTONE.
C L(I)=4 => ITH VARIABLE TRANSFORMATION IS TO BE LINEAR.
C L(I)=5 => ITH VARIABLE ASSUMES CATEGORICAL (UNORDERABLE) VALUES.
C DELRSQ : TERMINATION THRESHOLD. ITERATION STOPS WHEN
C RSQ CHANGES LESS THAN DELRSQ IN NTERM
C CONSECUTIVE ITERATIONS (SEE BELOW - DEFAULT, NTERM=3).
C NS : NUMBER OF EIGENSOLUTIONS (SETS OF TRANSFORMATIONS).
C
C OUTPUT:
C
C TX(N,P,NS) : PREDICTOR TRANSFORMATIONS.
C TX(J,I,K) = TRANSFORMED VALUE OF ITH PREDICTOR FOR JTH OBS
C FOR KTH EIGENSOLUTION.
C TY(N,NS) = RESPONSE TRANSFORMATIONS.
C TY(J,K) = TRANSFORMED RESPONSE VALUE FOR JTH OBSERVATION
C FOR KTH EIGENSOLUTION.
C RSQ(NS) = FRACTION OF VARIANCE(TY<Y>)
C P
C EXPLAINED BY SUM TX(I)<X(I)> FOR EACH EIGENSOLUTION.
C I=1
C IERR : ERROR FLAG.
C IERR = 0 : NO ERRORS DETECTED.
C IERR > 0 : ERROR DETECTED - SEE FORMAT STATEMENTS BELOW.
C
C SCRATCH:
C
C M(N,P+1), Z(N,12) : INTERNAL WORKING STORAGE.
C
C NOTE: MACE USES AN ITERATIVE PROCEDURE FOR SOLVING THE OPTIMIZATION
C PROBLEM. DEFAULT STARTING TRANSFORMATIONS ARE TY(J,K)=Y(J),
C TX(J,I,K)=X(I,J) : J=1,N, I=1,P, K=1,NS. OTHER STARTING TRANSFORMATIONS
C CAN BE SPECIFIED (IF DESIRED) FOR EITHER THE RESPONSE AND/OR ANY OF
C THE PREDICTOR VARIABLES. THIS IS SIGNALED BY NEGATING THE
C CORRESPONDING L(I) VALUE AND STORING THE STARTING TRANSFORMED
C VALUES IN THE CORRESPONDING ARRAY (TY(J,K), TX(J,I,K)) BEFORE
C CALLING MACE.
C
C------------------------------------------------------------------
C
C
C SUBROUTINE MODEL(P,N,Y,W,L,TX,TY,F,T,M,Z)
C--------------------------------------------------------------------
C
C COMPUTES RESPONSE PREDICTIVE FUNCTION F FOR THE MODEL YHAT = F(T),
C WHERE
C P
C F(T) = E(Y : T), T = SUM TX<I> ( X<I> )
C I=1
C USING THE X TRANSFORMATIONS TX CONSTRUCTED BY SUBROUTINE ACE.
C IF Y IS A CATEGORICAL VARIABLE (CLASSIFICATION) THEN
C -1
C F(T) = TY (T).
C INPUT:
C
C P,N,Y,W,L : SAME INPUT AS FOR SUBROUTINE ACE.
C TX,TY,M,Z : OUTPUT FROM SUBROUTINE ACE.
C
C OUTPUT:
C
C F(N),T(N) : INPUT FOR SUBROUTINE ACEMOD.
C
C NOTE: THIS SUBROUTINE MUST BE CALLED BEFORE SUBROUTINE ACEMOD.
C
C-------------------------------------------------------------------
C
C SUBROUTINE ACEMOD(V,P,N,X,L,TX,F,T,M,YHAT)
C--------------------------------------------------------------------
C
C COMPUTES RESPONSE Y ESTIMATES FROM THE MODEL
C
C YHAT = F ( T( V ) )
C
C USING THE X TRANSFORMATIONS TX CONSTRUCTED BY SUBROUTINE ACE AND
C THE PREDICTOR FUNCTION (F,T) CONSTRUCTED BY SUBROUTINE MODEL.
C
C INPUT:
C
C V(P) : VECTOR OF PREDICTOR VALUES.
C P,N,X,L : SAME INPUT AS FOR SUBROUTINE ACE.
C TX,M : OUTPUT FROM SUBROUTINE ACE.
C F,T : OUTPUT FROM SUBROUTINE MODEL.
C
C OUTPUT:
C
C YHAT : ESTIMATED RESPONSE VALUE FOR V.
C
C NOTE: THIS SUBROUTINE MUST NOT BE CALLED BEFORE SUBROUTINE MODEL.
C
C-------------------------------------------------------------------
C
C
C BLOCK DATA
C COMMON /PARMS/ ITAPE,MAXIT,NTERM,SPAN,ALPHA,BIG
C
C------------------------------------------------------------------
C
C THESE PROCEDURE PARAMETERS CAN BE CHANGED IN THE CALLING ROUTINE
C BY DEFINING THE ABOVE LABELED COMMON AND RESETTING THE VALUES WITH
C EXECUTABLE STATEMENTS.
C
C ITAPE : FORTRAN FILE NUMBER FOR PRINTER OUTPUT.
C (ITAPE.LE.0 => NO PRINTER OUTPUT.)
C MAXIT : MAXIMUM NUMBER OF ITERATIONS.
C NTERM : NUMBER OF CONSECUTIVE ITERATIONS FOR WHICH
C RSQ MUST CHANGE LESS THAN DELCOR FOR CONVERGENCE.
C SPAN, ALPHA : SUPER SMOOTHER PARAMETERS (SEE BELOW).
C BIG : A LARGE REPRESENTABLE FLOATING POINT NUMBER.
C
C------------------------------------------------------------------
C
subroutine supsmu (n,x,y,w,iper,span,alpha,smo,sc)
Note, although the following comments are extracted
from a relatively old version, the arguements are the same
as those in all subsequent versions. Pat Neville, 26 Sept 85
c------------------------------------------------------------------
c
c super smoother (friedman and stuetzle, 1984).
c
c version 3/10/84
c
c coded by: j. h. friedman
c department of statistics and
c stanford linear accelerator center
c stanford university
c stanford ca. 94305
c
c input:
c n : number of observations (x,y - pairs).
c x(n) : ordered abscissa values.
c y(n) : corresponding ordinate (response) values.
c w(n) : weight for each (x,y) observation.
c iper : periodic variable flag.
c iper=1 => x is ordered interval variable.
c iper=2 => x is a periodic variable with values
c in the range (0.0,1.0) and peroid 1.0.
c span : smoother span (fraction of observations in window).
c span=0.0 => automatic (variable) span selection.
c alpha : controles high frequency (small span) penality
c used with automatic span selection (base tone control).
c (alpha.le.0.0 or alpha.gt.10.0 => no effect.)
c output:
c smo(n) : smoothed ordinate (response) values.
c scratch:
c sc(n,7) : internal working storage.
c
c note:
c for small samples (n < 40) or if there are substantial serial
c correlations between obserations close in x - value, then
c a prespecified fixed span smoother (span > 0) should be
c used. reasonable span values are 0.3 to 0.5.
c
c------------------------------------------------------------------
subroutine sort (v,a,ii,jj)
c------------------------------------------------------------------
c
c applies to a and v the permutation vector which sorts v into
c increasing order. only elements from ii to jj are considered.
c arrays iu(k) and il(k) permit sorting up to 2**(k+1)-1 elements
c
c this is a modification of cacm algorithm #347 by r. c. singleton,
c which is a modified hoare quicksort.
c
c------------------------------------------------------------------
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