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/*
* c++ implementation of Lowess weighted regression by
* Peter Glaus http://www.cs.man.ac.uk/~glausp/
*
*
* Based on fortran code by Cleveland downloaded from:
* http://netlib.org/go/lowess.f
* original author:
* wsc@research.belllabs.com Mon Dec 30 16:55 EST 1985
* W. S. Cleveland
* Bell Laboratories
* Murray Hill NJ 07974
*
* See original documentation below the code for details.
*
*/
#include<algorithm>
#include<cmath>
#include<fstream>
using namespace std;
#include "lowess.h"
#include "common.h"
void lowess(const vector<double> &x, const vector<double> &y, double f, long nsteps, vector<double> &ys){//{{{
vector<double> rw,res;
lowess(x,y,f,nsteps,0.,ys,rw,res);
}//}}}
void lowess(const vector<double> &x, const vector<double> &y, double f, long nsteps, double delta, vector<double> &ys, vector<double> &rw, vector<double>&res){ //{{{
long n=(long)x.size();
bool ok=false;
long nleft,nright, i, j, iter, last, m1, m2, ns;
double cut, cmad, r, d1, d2, c1, c9, alpha, denom;
if((n==0)((long)y.size()!=n)) return;
ys.resize(n);
rw.resize(n);
res.resize(n);
if(n==1){
ys[0]=y[0];
return;
}
// ns  at least 2, at most n
ns = max(min((long)(f*n),n),(long)2);
for(iter=0;iter<nsteps+1; iter++){
// robustnes iterations
nleft = 0;
nright = ns1;
// index of last estimated point
last = 1;
// index of current point
i=0;
do{
while(nright<n1){
// move <nleft,nright> right, while radius decreases
d1 = x[i]x[nleft];
d2 = x[nright+1]  x[i];
if(d1<=d2)break;
nleft++;
nright++;
}
// fit value at x[i]
lowest(x,y,x[i],ys[i],nleft,nright,res,iter>0,rw,ok);
if(!ok) ys[i]=y[i];
if(last<i1){
// interpolate skipped points
if(last<0){
warning("Lowess: out of range.\n");
}
denom = x[i]  x[last];
for(j=last+1;j<i;j++){
alpha = (x[j]x[last])/denom;
ys[j] = alpha * ys[i] + (1.0alpha)*ys[last];
}
}
last = i;
cut = x[last]+delta;
for(i=last+1;i<n;i++){
if(x[i]>cut)break;
if(x[i]==x[last]){
ys[i]=ys[last];
last=i;
}
}
i=max(last+1,i1);
}while(last<n1);
for(i=0;i<n;i++)
res[i] = y[i]ys[i];
if(iter==nsteps)break ;
for(i=0;i<n;i++)
rw[i]=abs(res[i]);
sort(rw.begin(),rw.end());
m1 = n/2+1;
m2 = nm1;
m1 ;
cmad = 3.0 *(rw[m1]+rw[m2]);
c9 = .999*cmad;
c1 = .001*cmad;
for(i=0;i<n;i++){
r = abs(res[i]);
if(r<=c1) rw[i]=1;
else if(r>c9) rw[i]=0;
else rw[i] = (1.0(r/cmad)*(r/cmad))*(1.0(r/cmad)*(r/cmad));
}
}
}//}}}
void lowest(const vector<double> &x, const vector<double> &y, double xs, double &ys, long nleft, long nright, vector<double> &w, bool userw, vector<double> &rw, bool &ok){//{{{
long n = (long)x.size();
long nrt, j;
double a, b, c, h, r, h1, h9, range;
range = x[n1]x[0];
h = max(xsx[nleft],x[nright]xs);
h9 = 0.999*h;
h1 = 0.001*h;
// sum of weights
a = 0;
for(j=nleft;j<n;j++){
// compute weights (pick up all ties on right)
w[j]=0.;
r = abs(x[j]xs);
if(r<=h9){
// small enough for nonzero weight
if(r>h1) w[j] = (1.0(r/h)*(r/h)*(r/h))*(1.0(r/h)*(r/h)*(r/h))*(1.0(r/h)*(r/h)*(r/h));
else w[j] = 1.;
if(userw) w[j] *= rw[j];
a += w[j];
}else if(x[j]>xs) break; // get out at first zero wt on right
}
nrt = j1;
// rightmost pt (may be greater than nright because of ties)
if(a<=0.) ok = false;
else{
// weighted least squares
ok = true;
// normalize weights
for(j=nleft;j<=nrt;j++)
w[j] /= a;
if(h>0.){
// use linear fit
a = 0.;
for(j=nleft;j<=nrt;j++)
a += w[j]*x[j]; // weighted centre of values
b = xsa;
c = 0;
for(j=nleft;j<=nrt;j++)
c += w[j]*(x[j]a)*(x[j]a);
if(sqrt(c)>0.001*range){
// points are spread enough to compute slope
b /= c;
for(j=nleft;j<=nrt;j++)
w[j] *= (1.0+b*(x[j]a));
}
}
ys = 0;
for(j=nleft;j<=nrt;j++)
ys += w[j]*y[j];
}
}//}}}
/* {{{ Documentation
* wsc@research.belllabs.com Mon Dec 30 16:55 EST 1985
* W. S. Cleveland
* Bell Laboratories
* Murray Hill NJ 07974
*
* outline of this file:
* lines 172 introduction
* 73177 documentation for lowess
* 178238 ratfor version of lowess
* 239301 documentation for lowest
* 302350 ratfor version of lowest
* 351end test driver and fortran version of lowess and lowest
*
* a multivariate version is available by "send dloess from a"
*
* COMPUTER PROGRAMS FOR LOCALLY WEIGHTED REGRESSION
*
* This package consists of two FORTRAN programs for
* smoothing scatterplots by robust locally weighted
* regression, or lowess. The principal routine is LOWESS
* which computes the smoothed values using the method
* described in The Elements of Graphing Data, by William S.
* Cleveland (Wadsworth, 555 Morego Street, Monterey,
* California 93940).
*
* LOWESS calls a support routine, LOWEST, the code for
* which is included. LOWESS also calls a routine SORT, which
* the user must provide.
*
* To reduce the computations, LOWESS requires that the
* arrays X and Y, which are the horizontal and vertical
* coordinates, respectively, of the scatterplot, be such that
* X is sorted from smallest to largest. The user must
* therefore use another sort routine which will sort X and Y
* according to X.
* To summarize the scatterplot, YS, the fitted values,
* should be plotted against X. No graphics routines are
* available in the package and must be supplied by the user.
*
* The FORTRAN code for the routines LOWESS and LOWEST has
* been generated from higher level RATFOR programs
* (B. W. Kernighan, ``RATFOR: A Preprocessor for a Rational
* Fortran,'' Software Practice and Experience, Vol. 5 (1975),
* which are also included.
*
* The following are data and output from LOWESS that can
* be used to check your implementation of the routines. The
* notation (10)v means 10 values of v.
*
*
*
*
* X values:
* 1 2 3 4 5 (10)6 8 10 12 14 50
*
* Y values:
* 18 2 15 6 10 4 16 11 7 3 14 17 20 12 9 13 1 8 5 19
*
*
* YS values with F = .25, NSTEPS = 0, DELTA = 0.0
* 13.659 11.145 8.701 9.722 10.000 (10)11.300 13.000 6.440 5.596
* 5.456 18.998
*
* YS values with F = .25, NSTEPS = 0 , DELTA = 3.0
* 13.659 12.347 11.034 9.722 10.511 (10)11.300 13.000 6.440 5.596
* 5.456 18.998
*
* YS values with F = .25, NSTEPS = 2, DELTA = 0.0
* 14.811 12.115 8.984 9.676 10.000 (10)11.346 13.000 6.734 5.744
* 5.415 18.998
*
*
*
*
* LOWESS
*
*
*
* Calling sequence
*
* CALL LOWESS(X,Y,N,F,NSTEPS,DELTA,YS,RW,RES)
*
* Purpose
*
* LOWESS computes the smooth of a scatterplot of Y against X
* using robust locally weighted regression. Fitted values,
* YS, are computed at each of the values of the horizontal
* axis in X.
*
* Argument description
*
* X = Input; abscissas of the points on the
* scatterplot; the values in X must be ordered
* from smallest to largest.
* Y = Input; ordinates of the points on the
* scatterplot.
* N = Input; dimension of X,Y,YS,RW, and RES.
* F = Input; specifies the amount of smoothing; F is
* the fraction of points used to compute each
* fitted value; as F increases the smoothed values
* become smoother; choosing F in the range .2 to
* .8 usually results in a good fit; if you have no
* idea which value to use, try F = .5.
* NSTEPS = Input; the number of iterations in the robust
* fit; if NSTEPS = 0, the nonrobust fit is
* returned; setting NSTEPS equal to 2 should serve
* most purposes.
* DELTA = input; nonnegative parameter which may be used
* to save computations; if N is less than 100, set
* DELTA equal to 0.0; if N is greater than 100 you
* should find out how DELTA works by reading the
* additional instructions section.
* YS = Output; fitted values; YS(I) is the fitted value
* at X(I); to summarize the scatterplot, YS(I)
* should be plotted against X(I).
* RW = Output; robustness weights; RW(I) is the weight
* given to the point (X(I),Y(I)); if NSTEPS = 0,
* RW is not used.
* RES = Output; residuals; RES(I) = Y(I)YS(I).
*
*
* Other programs called
*
* LOWEST
* SSORT
*
* Additional instructions
*
* DELTA can be used to save computations. Very roughly the
* algorithm is this: on the initial fit and on each of the
* NSTEPS iterations locally weighted regression fitted values
* are computed at points in X which are spaced, roughly, DELTA
* apart; then the fitted values at the remaining points are
* computed using linear interpolation. The first locally
* weighted regression (l.w.r.) computation is carried out at
* X(1) and the last is carried out at X(N). Suppose the
* l.w.r. computation is carried out at X(I). If X(I+1) is
* greater than or equal to X(I)+DELTA, the next l.w.r.
* computation is carried out at X(I+1). If X(I+1) is less
* than X(I)+DELTA, the next l.w.r. computation is carried out
* at the largest X(J) which is greater than or equal to X(I)
* but is not greater than X(I)+DELTA. Then the fitted values
* for X(K) between X(I) and X(J), if there are any, are
* computed by linear interpolation of the fitted values at
* X(I) and X(J). If N is less than 100 then DELTA can be set
* to 0.0 since the computation time will not be too great.
* For larger N it is typically not necessary to carry out the
* l.w.r. computation for all points, so that much computation
* time can be saved by taking DELTA to be greater than 0.0.
* If DELTA = Range (X)/k then, if the values in X were
* uniformly scattered over the range, the full l.w.r.
* computation would be carried out at approximately k points.
* Taking k to be 50 often works well.
*
* Method
*
* The fitted values are computed by using the nearest neighbor
* routine and robust locally weighted regression of degree 1
* with the tricube weight function. A few additional features
* have been added. Suppose r is FN truncated to an integer.
* Let h be the distance to the rth nearest neighbor
* from X(I). All points within h of X(I) are used. Thus if
* the rth nearest neighbor is exactly the same distance as
* other points, more than r points can possibly be used for
* the smooth at X(I). There are two cases where robust
* locally weighted regression of degree 0 is actually used at
* X(I). One case occurs when h is 0.0. The second case
* occurs when the weighted standard error of the X(I) with
* respect to the weights w(j) is less than .001 times the
* range of the X(I), where w(j) is the weight assigned to the
* jth point of X (the tricube weight times the robustness
* weight) divided by the sum of all of the weights. Finally,
* if the w(j) are all zero for the smooth at X(I), the fitted
* value is taken to be Y(I).
*
*
*
*
* subroutine lowess(x,y,n,f,nsteps,delta,ys,rw,res)
* real x(n),y(n),ys(n),rw(n),res(n)
* logical ok
* if (n<2){ ys(1) = y(1); return }
* ns = max0(min0(ifix(f*float(n)),n),2) # at least two, at most n points
* for(iter=1; iter<=nsteps+1; iter=iter+1){ # robustness iterations
* nleft = 1; nright = ns
* last = 0 # index of prev estimated point
* i = 1 # index of current point
* repeat{
* while(nright<n){
* # move nleft, nright to right if radius decreases
* d1 = x(i)x(nleft)
* d2 = x(nright+1)x(i)
* # if d1<=d2 with x(nright+1)==x(nright), lowest fixes
* if (d1<=d2) break
* # radius will not decrease by move right
* nleft = nleft+1
* nright = nright+1
* }
* call lowest(x,y,n,x(i),ys(i),nleft,nright,res,iter>1,rw,ok)
* # fitted value at x(i)
* if (!ok) ys(i) = y(i)
* # all weights zero  copy over value (all rw==0)
* if (last<i1) { # skipped points  interpolate
* denom = x(i)x(last) # nonzero  proof?
* for(j=last+1; j<i; j=j+1){
* alpha = (x(j)x(last))/denom
* ys(j) = alpha*ys(i)+(1.0alpha)*ys(last)
* }
* }
* last = i # last point actually estimated
* cut = x(last)+delta # x coord of close points
* for(i=last+1; i<=n; i=i+1){ # find close points
* if (x(i)>cut) break # i one beyond last pt within cut
* if(x(i)==x(last)){ # exact match in x
* ys(i) = ys(last)
* last = i
* }
* }
* i=max0(last+1,i1)
* # back 1 point so interpolation within delta, but always go forward
* } until(last>=n)
* do i = 1,n # residuals
* res(i) = y(i)ys(i)
* if (iter>nsteps) break # compute robustness weights except last time
* do i = 1,n
* rw(i) = abs(res(i))
* call sort(rw,n)
* m1 = 1+n/2; m2 = nm1+1
* cmad = 3.0*(rw(m1)+rw(m2)) # 6 median abs resid
* c9 = .999*cmad; c1 = .001*cmad
* do i = 1,n {
* r = abs(res(i))
* if(r<=c1) rw(i)=1. # near 0, avoid underflow
* else if(r>c9) rw(i)=0. # near 1, avoid underflow
* else rw(i) = (1.0(r/cmad)**2)**2
* }
* }
* return
* end
*
*
*
*
* LOWEST
*
*
*
* Calling sequence
*
* CALL LOWEST(X,Y,N,XS,YS,NLEFT,NRIGHT,W,USERW,RW,OK)
*
* Purpose
*
* LOWEST is a support routine for LOWESS and ordinarily will
* not be called by the user. The fitted value, YS, is
* computed at the value, XS, of the horizontal axis.
* Robustness weights, RW, can be employed in computing the
* fit.
*
* Argument description
*
*
* X = Input; abscissas of the points on the
* scatterplot; the values in X must be ordered
* from smallest to largest.
* Y = Input; ordinates of the points on the
* scatterplot.
* N = Input; dimension of X,Y,W, and RW.
* XS = Input; value of the horizontal axis at which the
* smooth is computed.
* YS = Output; fitted value at XS.
* NLEFT = Input; index of the first point which should be
* considered in computing the fitted value.
* NRIGHT = Input; index of the last point which should be
* considered in computing the fitted value.
* W = Output; W(I) is the weight for Y(I) used in the
* expression for YS, which is the sum from
* I = NLEFT to NRIGHT of W(I)*Y(I); W(I) is
* defined only at locations NLEFT to NRIGHT.
* USERW = Input; logical variable; if USERW is .TRUE., a
* robust fit is carried out using the weights in
* RW; if USERW is .FALSE., the values in RW are
* not used.
* RW = Input; robustness weights.
* OK = Output; logical variable; if the weights for the
* smooth are all 0.0, the fitted value, YS, is not
* computed and OK is set equal to .FALSE.; if the
* fitted value is computed OK is set equal to
*
*
* Method
*
* The smooth at XS is computed using (robust) locally weighted
* regression of degree 1. The tricube weight function is used
* with h equal to the maximum of XSX(NLEFT) and X(NRIGHT)XS.
* Two cases where the program reverts to locally weighted
* regression of degree 0 are described in the documentation
* for LOWESS.
*
*
*
*
* subroutine lowest(x,y,n,xs,ys,nleft,nright,w,userw,rw,ok)
* real x(n),y(n),w(n),rw(n)
* logical userw,ok
* range = x(n)x(1)
* h = amax1(xsx(nleft),x(nright)xs)
* h9 = .999*h
* h1 = .001*h
* a = 0.0 # sum of weights
* for(j=nleft; j<=n; j=j+1){ # compute weights (pick up all ties on right)
* w(j)=0.
* r = abs(x(j)xs)
* if (r<=h9) { # small enough for nonzero weight
* if (r>h1) w(j) = (1.0(r/h)**3)**3
* else w(j) = 1.
* if (userw) w(j) = rw(j)*w(j)
* a = a+w(j)
* }
* else if(x(j)>xs)break # get out at first zero wt on right
* }
* nrt=j1 # rightmost pt (may be greater than nright because of ties)
* if (a<=0.0) ok = FALSE
* else { # weighted least squares
* ok = TRUE
* do j = nleft,nrt
* w(j) = w(j)/a # make sum of w(j) == 1
* if (h>0.) { # use linear fit
* a = 0.0
* do j = nleft,nrt
* a = a+w(j)*x(j) # weighted center of x values
* b = xsa
* c = 0.0
* do j = nleft,nrt
* c = c+w(j)*(x(j)a)**2
* if(sqrt(c)>.001*range) {
* # points are spread out enough to compute slope
* b = b/c
* do j = nleft,nrt
* w(j) = w(j)*(1.0+b*(x(j)a))
* }
* }
* ys = 0.0
* do j = nleft,nrt
* ys = ys+w(j)*y(j)
* }
* return
* end
*
}}}*/
