TRobustEstimator.cxx

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// @(#)root/physics:$Id$
// Author: Anna Kreshuk  08/10/2004
 
/*************************************************************************
 * Copyright (C) 1995-2004, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
/** \class TRobustEstimator
    \ingroup Physics
Minimum Covariance Determinant Estimator - a Fast Algorithm
invented by Peter J.Rousseeuw and Katrien Van Dreissen
"A Fast Algorithm for the Minimum covariance Determinant Estimator"
Technometrics, August 1999, Vol.41, NO.3
 
What are robust estimators?
"An important property of an estimator is its robustness. An estimator
is called robust if it is insensitive to measurements that deviate
from the expected behaviour. There are 2 ways to treat such deviating
measurements: one may either try to recognise them and then remove
them from the data sample; or one may leave them in the sample, taking
care that they do not influence the estimate unduly. In both cases robust
estimators are needed...Robust procedures compensate for systematic errors
as much as possible, and indicate any situation in which a danger of not being
able to operate reliably is detected."
R.Fruhwirth, M.Regler, R.K.Bock, H.Grote, D.Notz
"Data Analysis Techniques for High-Energy Physics", 2nd edition
 
What does this algorithm do?
It computes a highly robust estimator of multivariate location and scatter.
Then, it takes those estimates to compute robust distances of all the
data vectors. Those with large robust distances are considered outliers.
Robust distances can then be plotted for better visualization of the data.
 
How does this algorithm do it?
The MCD objective is to find h observations(out of n) whose classical
covariance matrix has the lowest determinant. The MCD estimator of location
is then the average of those h points and the MCD estimate of scatter
is their covariance matrix. The minimum(and default) h = (n+nvariables+1)/2
so the algorithm is effective when less than (n+nvar+1)/2 variables are outliers.
The algorithm also allows for exact fit situations - that is, when h or more
observations lie on a hyperplane. Then the algorithm still yields the MCD location T
and scatter matrix S, the latter being singular as it should be. From (T,S) the
program then computes the equation of the hyperplane.
 
How can this algorithm be used?
In any case, when contamination of data is suspected, that might influence
the classical estimates.
Also, robust estimation of location and scatter is a tool to robustify
other multivariate techniques such as, for example, principal-component analysis
and discriminant analysis.
 
Technical details of the algorithm:
 
1. The default h = (n+nvariables+1)/2, but the user may choose any integer h with
   (n+nvariables+1)/2<=h<=n. The program then reports the MCD's breakdown value
   (n-h+1)/n. If you are sure that the dataset contains less than 25% contamination
   which is usually the case, a good compromise between breakdown value and
   efficiency is obtained by putting h=[.75*n].
2. If h=n,the MCD location estimate is the average of the whole dataset, and
   the MCD scatter estimate is its covariance matrix. Report this and stop
3. If nvariables=1 (univariate data), compute the MCD estimate by the exact
   algorithm of Rousseeuw and Leroy (1987, pp.171-172) in O(nlogn)time and stop
4. From here on, h<n and nvariables>=2.
   1. If n is small:
      - repeat (say) 500 times:
        - construct an initial h-subset, starting from a random (nvar+1)-subset
        - carry out 2 C-steps (described in the comments of CStep function)
      - for the 10 results with lowest det(S):
        - carry out C-steps until convergence
      - report the solution (T, S) with the lowest det(S)
   2. If n is larger (say, n>600), then
      - construct up to 5 disjoint random subsets of size nsub (say, nsub=300)
      - inside each subset repeat 500/5 times:
         - construct an initial subset of size hsub=[nsub*h/n]
         - carry out 2 C-steps
         - keep the best 10 results (Tsub, Ssub)
      - pool the subsets, yielding the merged set (say, of size nmerged=1500)
      - in the merged set, repeat for each of the 50 solutions (Tsub, Ssub)
         - carry out 2 C-steps
         - keep the 10 best results
      - in the full dataset, repeat for those best results:
         - take several C-steps, using n and h
         - report the best final result (T, S)
5. To obtain consistency when the data comes from a multivariate normal
   distribution, covariance matrix is multiplied by a correction factor
6. Robust distances for all elements, using the final (T, S) are calculated
   Then the very final mean and covariance estimates are calculated only for
   values, whose robust distances are less than a cutoff value (0.975 quantile
   of chi2 distribution with nvariables degrees of freedom)
*/
 
#include "TRobustEstimator.h"
#include "TMatrixDSymEigen.h"
#include "TRandom.h"
#include "TMath.h"
#include "TDecompChol.h"
 
ClassImp(TRobustEstimator);
 
const Double_t kChiMedian[50]= {
         0.454937, 1.38629, 2.36597, 3.35670, 4.35146, 5.34812, 6.34581, 7.34412, 8.34283,
         9.34182, 10.34, 11.34, 12.34, 13.34, 14.34, 15.34, 16.34, 17.34, 18.34, 19.34,
        20.34, 21.34, 22.34, 23.34, 24.34, 25.34, 26.34, 27.34, 28.34, 29.34, 30.34,
        31.34, 32.34, 33.34, 34.34, 35.34, 36.34, 37.34, 38.34, 39.34, 40.34,
        41.34, 42.34, 43.34, 44.34, 45.34, 46.34, 47.34, 48.34, 49.33};
 
const Double_t kChiQuant[50]={
         5.02389, 7.3776,9.34840,11.1433,12.8325,
        14.4494,16.0128,17.5346,19.0228,20.4831,21.920,23.337,
        24.736,26.119,27.488,28.845,30.191,31.526,32.852,34.170,
        35.479,36.781,38.076,39.364,40.646,41.923,43.194,44.461,
        45.722,46.979,48.232,49.481,50.725,51.966,53.203,54.437,
        55.668,56.896,58.120,59.342,60.561,61.777,62.990,64.201,
        65.410,66.617,67.821,69.022,70.222,71.420};
 
////////////////////////////////////////////////////////////////////////////////
///this constructor should be used in a univariate case:
///first call this constructor, then - the EvaluateUni(..) function
 
TRobustEstimator::TRobustEstimator(){
}
 
////////////////////////////////////////////////////////////////////////////////
///constructor
 
TRobustEstimator::TRobustEstimator(Int_t nvectors, Int_t nvariables, Int_t hh)
   :fMean(nvariables),
    fCovariance(nvariables),
    fInvcovariance(nvariables),
    fCorrelation(nvariables),
    fRd(nvectors),
    fSd(nvectors),
    fOut(1),
    fHyperplane(nvariables),
    fData(nvectors, nvariables)
{
   if ((nvectors<=1)||(nvariables<=0)){
      Error("TRobustEstimator","Not enough vectors or variables");
      return;
   }
   if (nvariables==1){
      Error("TRobustEstimator","For the univariate case, use the default constructor and EvaluateUni() function");
      return;
   }
 
   fN=nvectors;
   fNvar=nvariables;
   if (hh<(fN+fNvar+1)/2){
      if (hh>0)
         Warning("TRobustEstimator","chosen h is too small, default h is taken instead");
      fH=(fN+fNvar+1)/2;
   } else
      fH=hh;
 
   fVarTemp=0;
   fVecTemp=0;
   fExact=0;
}
 
////////////////////////////////////////////////////////////////////////////////
///adds a column to the data matrix
///it is assumed that the column has size fN
///variable fVarTemp keeps the number of columns l
///already added
 
void TRobustEstimator::AddColumn(Double_t *col)
{
   if (fVarTemp==fNvar) {
      fNvar++;
      fCovariance.ResizeTo(fNvar, fNvar);
      fInvcovariance.ResizeTo(fNvar, fNvar);
      fCorrelation.ResizeTo(fNvar, fNvar);
      fMean.ResizeTo(fNvar);
      fHyperplane.ResizeTo(fNvar);
      fData.ResizeTo(fN, fNvar);
   }
   for (Int_t i=0; i<fN; i++) {
      fData(i, fVarTemp)=col[i];
   }
   fVarTemp++;
}
 
////////////////////////////////////////////////////////////////////////////////
///adds a vector to the data matrix
///it is supposed that the vector is of size fNvar
 
void TRobustEstimator::AddRow(Double_t *row)
{
   if(fVecTemp==fN) {
      fN++;
      fRd.ResizeTo(fN);
      fSd.ResizeTo(fN);
      fData.ResizeTo(fN, fNvar);
   }
   for (Int_t i=0; i<fNvar; i++)
      fData(fVecTemp, i)=row[i];
 
   fVecTemp++;
}
 
////////////////////////////////////////////////////////////////////////////////
///Finds the estimate of multivariate mean and variance
 
void TRobustEstimator::Evaluate()
{
   Double_t kEps=1e-14;
 
   if (fH==fN){
      Warning("Evaluate","Chosen h = #observations, so classic estimates of location and scatter will be calculated");
      Classic();
      return;
   }
 
   Int_t i, j, k;
   Int_t ii, jj;
   Int_t nmini = 300;
   Int_t k1=500;
   Int_t nbest=10;
   TMatrixD sscp(fNvar+1, fNvar+1);
   TVectorD vec(fNvar);
 
   Int_t *index = new Int_t[fN];
   Double_t *ndist = new Double_t[fN];
   Double_t det;
   Double_t *deti=new Double_t[nbest];
   for (i=0; i<nbest; i++)
      deti[i]=1e16;
 
   for (i=0; i<fN; i++)
      fRd(i)=0;
   ////////////////////////////
   //for small n
   ////////////////////////////
   if (fN<nmini*2) {
      //for storing the best fMeans and covariances
 
      TMatrixD mstock(nbest, fNvar);
      TMatrixD cstock(fNvar, fNvar*nbest);
 
      for (k=0; k<k1; k++) {
         CreateSubset(fN, fH, fNvar, index, fData, sscp, ndist);
         //calculate the mean and covariance of the created subset
         ClearSscp(sscp);
         for (i=0; i<fH; i++) {
            for(j=0; j<fNvar; j++)
               vec(j)=fData[index[i]][j];
            AddToSscp(sscp, vec);
         }
         Covar(sscp, fMean, fCovariance, fSd, fH);
         det = fCovariance.Determinant();
         if (det < kEps) {
            fExact = Exact(ndist);
            delete [] index;
            delete [] ndist;
            delete [] deti;
            return;
         }
         //make 2 CSteps
         det = CStep(fN, fH, index, fData, sscp, ndist);
         if (det < kEps) {
            fExact = Exact(ndist);
            delete [] index;
            delete [] ndist;
            delete [] deti;
            return;
         }
         det = CStep(fN, fH, index, fData, sscp, ndist);
         if (det < kEps) {
            fExact = Exact(ndist);
            delete [] index;
            delete [] ndist;
            delete [] deti;
            return;
         } else {
            Int_t maxind=TMath::LocMax(nbest, deti);
            if(det<deti[maxind]) {
               deti[maxind]=det;
               for(ii=0; ii<fNvar; ii++) {
                  mstock(maxind, ii)=fMean(ii);
                  for(jj=0; jj<fNvar; jj++)
                     cstock(ii, jj+maxind*fNvar)=fCovariance(ii, jj);
               }
            }
         }
      }
 
      //now for nbest best results perform CSteps until convergence
 
      for (i=0; i<nbest; i++) {
         for(ii=0; ii<fNvar; ii++) {
            fMean(ii)=mstock(i, ii);
            for (jj=0; jj<fNvar; jj++)
               fCovariance(ii, jj)=cstock(ii, jj+i*fNvar);
         }
 
         det=1;
         while (det>kEps) {
            det=CStep(fN, fH, index, fData, sscp, ndist);
            if(TMath::Abs(det-deti[i])<kEps)
               break;
            else
               deti[i]=det;
         }
         for(ii=0; ii<fNvar; ii++) {
            mstock(i,ii)=fMean(ii);
            for (jj=0; jj<fNvar; jj++)
               cstock(ii,jj+i*fNvar)=fCovariance(ii, jj);
         }
      }
 
      Int_t detind=TMath::LocMin(nbest, deti);
      for(ii=0; ii<fNvar; ii++) {
         fMean(ii)=mstock(detind,ii);
 
         for(jj=0; jj<fNvar; jj++)
            fCovariance(ii, jj)=cstock(ii,jj+detind*fNvar);
      }
 
      if (deti[detind]!=0) {
         //calculate robust distances and throw out the bad points
         Int_t nout = RDist(sscp);
         Double_t cutoff=kChiQuant[fNvar-1];
 
         fOut.Set(nout);
 
         j=0;
         for (i=0; i<fN; i++) {
            if(fRd(i)>cutoff) {
               fOut[j]=i;
               j++;
            }
         }
 
      } else {
         fExact=Exact(ndist);
      }
      delete [] index;
      delete [] ndist;
      delete [] deti;
      return;
 
   }
   /////////////////////////////////////////////////
  //if n>nmini, the dataset should be partitioned
  //partitioning
  ////////////////////////////////////////////////
   Int_t indsubdat[5];
   Int_t nsub;
   for (ii=0; ii<5; ii++)
      indsubdat[ii]=0;
 
   nsub = Partition(nmini, indsubdat);
 
   Int_t sum=0;
   for (ii=0; ii<5; ii++)
      sum+=indsubdat[ii];
   Int_t *subdat=new Int_t[sum];
   //printf("allocates subdat[ %d ]\n", sum);
   // init the subdat matrix
   for (int iii = 0; iii < sum; ++iii) subdat[iii] = -999;
   RDraw(subdat, nsub, indsubdat);
   for (int iii = 0; iii < sum; ++iii) {
      if (subdat[iii] < 0 || subdat[iii] >= fN ) {
         Error("Evaluate","subdat index is invalid subdat[%d] = %d",iii, subdat[iii] );
         R__ASSERT(0);
      }
   }
   //now the indexes of selected cases are in the array subdat
   //matrices to store best means and covariances
   Int_t nbestsub=nbest*nsub;
   TMatrixD mstockbig(nbestsub, fNvar);
   TMatrixD cstockbig(fNvar, fNvar*nbestsub);
   TMatrixD hyperplane(nbestsub, fNvar);
   for (i=0; i<nbestsub; i++) {
      for(j=0; j<fNvar; j++)
         hyperplane(i,j)=0;
   }
   Double_t *detibig = new Double_t[nbestsub];
   Int_t maxind;
   maxind=TMath::LocMax(5, indsubdat);
   TMatrixD dattemp(indsubdat[maxind], fNvar);
 
   Int_t k2=Int_t(k1/nsub);
   //construct h-subsets and perform 2 CSteps in subgroups
 
   for (Int_t kgroup=0; kgroup<nsub; kgroup++) {
      //printf("group #%d\n", kgroup);
      Int_t ntemp=indsubdat[kgroup];
      Int_t temp=0;
      for (i=0; i<kgroup; i++)
         temp+=indsubdat[i];
      Int_t par;
 
 
      for(i=0; i<ntemp; i++) {
         for (j=0; j<fNvar; j++) {
            dattemp(i,j)=fData[subdat[temp+i]][j];
         }
      }
      Int_t htemp=Int_t(fH*ntemp/fN);
 
      for (i=0; i<nbest; i++)
         deti[i]=1e16;
 
      for(k=0; k<k2; k++) {
         CreateSubset(ntemp, htemp, fNvar, index, dattemp, sscp, ndist);
         ClearSscp(sscp);
         for (i=0; i<htemp; i++) {
            for(j=0; j<fNvar; j++) {
               vec(j)=dattemp(index[i],j);
            }
            AddToSscp(sscp, vec);
         }
         Covar(sscp, fMean, fCovariance, fSd, htemp);
         det = fCovariance.Determinant();
         if (det<kEps) {
            par =Exact2(mstockbig, cstockbig, hyperplane, deti, nbest, kgroup, sscp,ndist);
            if(par==nbest+1) {
 
               delete [] detibig;
               delete [] deti;
               delete [] subdat;
               delete [] ndist;
               delete [] index;
               return;
            } else
               deti[par]=det;
         } else {
            det = CStep(ntemp, htemp, index, dattemp, sscp, ndist);
            if (det<kEps) {
               par=Exact2(mstockbig, cstockbig, hyperplane, deti, nbest, kgroup, sscp, ndist);
               if(par==nbest+1) {
 
                  delete [] detibig;
                  delete [] deti;
                  delete [] subdat;
                  delete [] ndist;
                  delete [] index;
                  return;
               } else
                  deti[par]=det;
            } else {
               det=CStep(ntemp,htemp, index, dattemp, sscp, ndist);
               if(det<kEps){
                  par=Exact2(mstockbig, cstockbig, hyperplane, deti, nbest, kgroup, sscp,ndist);
                  if(par==nbest+1) {
 
                     delete [] detibig;
                     delete [] deti;
                     delete [] subdat;
                     delete [] ndist;
                     delete [] index;
                     return;
                  } else {
                     deti[par]=det;
                  }
               } else {
                  maxind=TMath::LocMax(nbest, deti);
                  if(det<deti[maxind]) {
                     deti[maxind]=det;
                     for(i=0; i<fNvar; i++) {
                        mstockbig(nbest*kgroup+maxind,i)=fMean(i);
                        for(j=0; j<fNvar; j++) {
                           cstockbig(i,nbest*kgroup*fNvar+maxind*fNvar+j)=fCovariance(i,j);
 
                        }
                     }
                  }
 
               }
            }
         }
 
         maxind=TMath::LocMax(nbest, deti);
         if (deti[maxind]<kEps)
            break;
      }
 
 
      for(i=0; i<nbest; i++) {
         detibig[kgroup*nbest + i]=deti[i];
 
      }
 
   }
 
   //now the arrays mstockbig and cstockbig store nbest*nsub best means and covariances
   //detibig stores nbest*nsub their determinants
   //merge the subsets and carry out 2 CSteps on the merged set for all 50 best solutions
 
   TMatrixD datmerged(sum, fNvar);
   for(i=0; i<sum; i++) {
      for (j=0; j<fNvar; j++)
         datmerged(i,j)=fData[subdat[i]][j];
   }
   //  printf("performing calculations for merged set\n");
   Int_t hmerged=Int_t(sum*fH/fN);
 
   Int_t nh;
   for(k=0; k<nbestsub; k++) {
      //for all best solutions perform 2 CSteps and then choose the very best
      for(ii=0; ii<fNvar; ii++) {
         fMean(ii)=mstockbig(k,ii);
         for(jj=0; jj<fNvar; jj++)
            fCovariance(ii, jj)=cstockbig(ii,k*fNvar+jj);
      }
      if(detibig[k]==0) {
         for(i=0; i<fNvar; i++)
            fHyperplane(i)=hyperplane(k,i);
         CreateOrtSubset(datmerged,index, hmerged, sum, sscp, ndist);
 
      }
      det=CStep(sum, hmerged, index, datmerged, sscp, ndist);
      if (det<kEps) {
         nh= Exact(ndist);
         if (nh>=fH) {
            fExact = nh;
 
            delete [] detibig;
            delete [] deti;
            delete [] subdat;
            delete [] ndist;
            delete [] index;
            return;
         } else {
            CreateOrtSubset(datmerged, index, hmerged, sum, sscp, ndist);
         }
      }
 
      det=CStep(sum, hmerged, index, datmerged, sscp, ndist);
      if (det<kEps) {
         nh=Exact(ndist);
         if (nh>=fH) {
            fExact = nh;
            delete [] detibig;
            delete [] deti;
            delete [] subdat;
            delete [] ndist;
            delete [] index;
            return;
         }
      }
      detibig[k]=det;
      for(i=0; i<fNvar; i++) {
         mstockbig(k,i)=fMean(i);
         for(j=0; j<fNvar; j++) {
            cstockbig(i,k*fNvar+j)=fCovariance(i, j);
         }
      }
   }
   //now for the subset with the smallest determinant
   //repeat CSteps until convergence
   Int_t minind=TMath::LocMin(nbestsub, detibig);
   det=detibig[minind];
   for(i=0; i<fNvar; i++) {
      fMean(i)=mstockbig(minind,i);
      fHyperplane(i)=hyperplane(minind,i);
      for(j=0; j<fNvar; j++)
         fCovariance(i, j)=cstockbig(i,minind*fNvar + j);
   }
   if(det<kEps)
      CreateOrtSubset(fData, index, fH, fN, sscp, ndist);
   det=1;
   while (det>kEps) {
      det=CStep(fN, fH, index, fData, sscp, ndist);
      if(TMath::Abs(det-detibig[minind])<kEps) {
         break;
      } else {
         detibig[minind]=det;
      }
   }
   if(det<kEps) {
      Exact(ndist);
      fExact=kTRUE;
   }
   Int_t nout = RDist(sscp);
   Double_t cutoff=kChiQuant[fNvar-1];
 
   fOut.Set(nout);
 
   j=0;
   for (i=0; i<fN; i++) {
      if(fRd(i)>cutoff) {
         fOut[j]=i;
         j++;
      }
   }
 
   delete [] detibig;
   delete [] deti;
   delete [] subdat;
   delete [] ndist;
   delete [] index;
   return;
}
 
////////////////////////////////////////////////////////////////////////////////
///for the univariate case
///estimates of location and scatter are returned in mean and sigma parameters
///the algorithm works on the same principle as in multivariate case -
///it finds a subset of size hh with smallest sigma, and then returns mean and
///sigma of this subset
 
void TRobustEstimator::EvaluateUni(Int_t nvectors, Double_t *data, Double_t &mean, Double_t &sigma, Int_t hh)
{
   if (hh==0)
      hh=(nvectors+2)/2;
   Double_t faclts[]={2.6477,2.5092,2.3826,2.2662,2.1587,2.0589,1.9660,1.879,1.7973,1.7203,1.6473};
   Int_t *index=new Int_t[nvectors];
   TMath::Sort(nvectors, data, index, kFALSE);
 
   Int_t nquant;
   nquant=TMath::Min(Int_t(Double_t(((hh*1./nvectors)-0.5)*40))+1, 11);
   Double_t factor=faclts[nquant-1];
 
   Double_t *aw=new Double_t[nvectors];
   Double_t *aw2=new Double_t[nvectors];
   Double_t sq=0;
   Double_t sqmin=0;
   Int_t ndup=0;
   Int_t len=nvectors-hh;
   Double_t *slutn=new Double_t[len];
   for(Int_t i=0; i<len; i++)
      slutn[i]=0;
   for(Int_t jint=0; jint<len; jint++) {
      aw[jint]=0;
      for (Int_t j=0; j<hh; j++) {
         aw[jint]+=data[index[j+jint]];
         if(jint==0)
            sq+=data[index[j]]*data[index[j]];
      }
      aw2[jint]=aw[jint]*aw[jint]/hh;
 
      if(jint==0) {
         sq=sq-aw2[jint];
         sqmin=sq;
         slutn[ndup]=aw[jint];
 
      } else {
         sq=sq - data[index[jint-1]]*data[index[jint-1]]+
            data[index[jint+hh]]*data[index[jint+hh]]-
            aw2[jint]+aw2[jint-1];
         if(sq<sqmin) {
            ndup=0;
            sqmin=sq;
            slutn[ndup]=aw[jint];
 
         } else {
            if(sq==sqmin) {
               ndup++;
               slutn[ndup]=aw[jint];
            }
         }
      }
   }
 
   slutn[0]=slutn[Int_t((ndup)/2)]/hh;
   Double_t bstd=factor*TMath::Sqrt(sqmin/hh);
   mean=slutn[0];
   sigma=bstd;
   delete [] aw;
   delete [] aw2;
   delete [] slutn;
   delete [] index;
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the breakdown point of the algorithm
 
Int_t TRobustEstimator::GetBDPoint()
{
   Int_t n;
   n=(fN-fH+1)/fN;
   return n;
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the chi2 quantiles
 
Double_t TRobustEstimator::GetChiQuant(Int_t i) const
{
   if (i < 0 || i >= 50) return 0;
   return kChiQuant[i];
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the covariance matrix
 
void TRobustEstimator::GetCovariance(TMatrixDSym &matr)
{
   if (matr.GetNrows()!=fNvar || matr.GetNcols()!=fNvar){
      Warning("GetCovariance","provided matrix is of the wrong size, it will be resized");
      matr.ResizeTo(fNvar, fNvar);
   }
   matr=fCovariance;
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the correlation matrix
 
void TRobustEstimator::GetCorrelation(TMatrixDSym &matr)
{
   if (matr.GetNrows()!=fNvar || matr.GetNcols()!=fNvar) {
      Warning("GetCorrelation","provided matrix is of the wrong size, it will be resized");
      matr.ResizeTo(fNvar, fNvar);
   }
   matr=fCorrelation;
}
 
////////////////////////////////////////////////////////////////////////////////
///if the points are on a hyperplane, returns this hyperplane
 
const TVectorD* TRobustEstimator::GetHyperplane() const
{
   if (fExact==0) {
      Error("GetHyperplane","the data doesn't lie on a hyperplane!\n");
      return 0;
   } else {
      return &fHyperplane;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///if the points are on a hyperplane, returns this hyperplane
 
void TRobustEstimator::GetHyperplane(TVectorD &vec)
{
   if (fExact==0){
      Error("GetHyperplane","the data doesn't lie on a hyperplane!\n");
      return;
   }
   if (vec.GetNoElements()!=fNvar) {
      Warning("GetHyperPlane","provided vector is of the wrong size, it will be resized");
      vec.ResizeTo(fNvar);
   }
   vec=fHyperplane;
}
 
////////////////////////////////////////////////////////////////////////////////
///return the estimate of the mean
 
void TRobustEstimator::GetMean(TVectorD &means)
{
   if (means.GetNoElements()!=fNvar) {
      Warning("GetMean","provided vector is of the wrong size, it will be resized");
      means.ResizeTo(fNvar);
   }
   means=fMean;
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the robust distances (helps to find outliers)
 
void TRobustEstimator::GetRDistances(TVectorD &rdist)
{
   if (rdist.GetNoElements()!=fN) {
      Warning("GetRDistances","provided vector is of the wrong size, it will be resized");
      rdist.ResizeTo(fN);
   }
   rdist=fRd;
}
 
////////////////////////////////////////////////////////////////////////////////
///returns the number of outliers
 
Int_t TRobustEstimator::GetNOut()
{
   return fOut.GetSize();
}
 
////////////////////////////////////////////////////////////////////////////////
///update the sscp matrix with vector vec
 
void TRobustEstimator::AddToSscp(TMatrixD &sscp, TVectorD &vec)
{
   Int_t i, j;
   for (j=1; j<fNvar+1; j++) {
      sscp(0, j) +=vec(j-1);
      sscp(j, 0) = sscp(0, j);
   }
   for (i=1; i<fNvar+1; i++) {
      for (j=1; j<fNvar+1; j++) {
         sscp(i, j) += vec(i-1)*vec(j-1);
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///clear the sscp matrix, used for covariance and mean calculation
 
void TRobustEstimator::ClearSscp(TMatrixD &sscp)
{
   for (Int_t i=0; i<fNvar+1; i++) {
      for (Int_t j=0; j<fNvar+1; j++) {
         sscp(i, j)=0;
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///called when h=n. Returns classic covariance matrix
///and mean
 
void TRobustEstimator::Classic()
{
   TMatrixD sscp(fNvar+1, fNvar+1);
   TVectorD temp(fNvar);
   ClearSscp(sscp);
   for (Int_t i=0; i<fN; i++) {
      for (Int_t j=0; j<fNvar; j++)
         temp(j)=fData(i, j);
      AddToSscp(sscp, temp);
   }
   Covar(sscp, fMean, fCovariance, fSd, fN);
   Correl();
 
}
 
////////////////////////////////////////////////////////////////////////////////
///calculates mean and covariance
 
void TRobustEstimator::Covar(TMatrixD &sscp, TVectorD &m, TMatrixDSym &cov, TVectorD &sd, Int_t nvec)
{
   Int_t i, j;
   Double_t f;
   for (i=0; i<fNvar; i++) {
      m(i)=sscp(0, i+1);
      sd[i]=sscp(i+1, i+1);
      f=(sd[i]-m(i)*m(i)/nvec)/(nvec-1);
      if (f>1e-14) sd[i]=TMath::Sqrt(f);
      else sd[i]=0;
      m(i)/=nvec;
   }
   for (i=0; i<fNvar; i++) {
      for (j=0; j<fNvar; j++) {
         cov(i, j)=sscp(i+1, j+1)-nvec*m(i)*m(j);
      cov(i, j)/=nvec-1;
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///transforms covariance matrix into correlation matrix
 
void TRobustEstimator::Correl()
{
   Int_t i, j;
   Double_t *sd=new Double_t[fNvar];
   for(j=0; j<fNvar; j++)
      sd[j]=1./TMath::Sqrt(fCovariance(j, j));
   for(i=0; i<fNvar; i++) {
      for (j=0; j<fNvar; j++) {
         if (i==j)
            fCorrelation(i, j)=1.;
         else
            fCorrelation(i, j)=fCovariance(i, j)*sd[i]*sd[j];
      }
   }
   delete [] sd;
}
 
////////////////////////////////////////////////////////////////////////////////
///creates a subset of htotal elements from ntotal elements
///first, p+1 elements are drawn randomly(without repetitions)
///if their covariance matrix is singular, more elements are
///added one by one, until their covariance matrix becomes regular
///or it becomes clear that htotal observations lie on a hyperplane
///If covariance matrix determinant!=0, distances of all ntotal elements
///are calculated, using formula d_i=Sqrt((x_i-M)*S_inv*(x_i-M)), where
///M is mean and S_inv is the inverse of the covariance matrix
///htotal points with smallest distances are included in the returned subset.
 
void TRobustEstimator::CreateSubset(Int_t ntotal, Int_t htotal, Int_t p, Int_t *index, TMatrixD &data, TMatrixD &sscp, Double_t *ndist)
{
   Double_t kEps = 1e-14;
   Int_t i, j;
   Bool_t repeat=kFALSE;
   Int_t nindex=0;
   Int_t num;
   for(i=0; i<ntotal; i++)
      index[i]=ntotal+1;
 
   for (i=0; i<p+1; i++) {
      num=Int_t(gRandom->Uniform(0, 1)*(ntotal-1));
      if (i>0){
         for(j=0; j<=i-1; j++) {
            if(index[j]==num)
            repeat=kTRUE;
         }
      }
      if(repeat==kTRUE) {
         i--;
         repeat=kFALSE;
      } else {
         index[i]=num;
         nindex++;
      }
   }
 
   ClearSscp(sscp);
 
   TVectorD vec(fNvar);
   Double_t det;
   for (i=0; i<p+1; i++) {
      for (j=0; j<fNvar; j++) {
         vec[j]=data[index[i]][j];
 
      }
      AddToSscp(sscp, vec);
   }
 
   Covar(sscp, fMean, fCovariance, fSd, p+1);
   det=fCovariance.Determinant();
   while((det<kEps)&&(nindex < htotal)) {
    //if covariance matrix is singular,another vector is added until
    //the matrix becomes regular or it becomes clear that all
    //vectors of the group lie on a hyperplane
      repeat=kFALSE;
      do{
         num=Int_t(gRandom->Uniform(0,1)*(ntotal-1));
         repeat=kFALSE;
         for(i=0; i<nindex; i++) {
            if(index[i]==num) {
               repeat=kTRUE;
               break;
            }
         }
      }while(repeat==kTRUE);
 
      index[nindex]=num;
      nindex++;
      //check if covariance matrix is singular
      for(j=0; j<fNvar; j++)
         vec[j]=data[index[nindex-1]][j];
      AddToSscp(sscp, vec);
      Covar(sscp, fMean, fCovariance, fSd, nindex);
      det=fCovariance.Determinant();
   }
 
   if(nindex!=htotal) {
      TDecompChol chol(fCovariance);
      fInvcovariance = chol.Invert();
 
      TVectorD temp(fNvar);
      for(j=0; j<ntotal; j++) {
         ndist[j]=0;
         for(i=0; i<fNvar; i++)
            temp[i]=data[j][i] - fMean(i);
         temp*=fInvcovariance;
         for(i=0; i<fNvar; i++)
            ndist[j]+=(data[j][i]-fMean(i))*temp[i];
      }
      KOrdStat(ntotal, ndist, htotal-1,index);
   }
 
}
 
////////////////////////////////////////////////////////////////////////////////
///creates a subset of hmerged vectors with smallest orthogonal distances to the hyperplane
///hyp[1]*(x1-mean[1])+...+hyp[nvar]*(xnvar-mean[nvar])=0
///This function is called in case when less than fH samples lie on a hyperplane.
 
void TRobustEstimator::CreateOrtSubset(TMatrixD &dat,Int_t *index, Int_t hmerged, Int_t nmerged, TMatrixD &sscp, Double_t *ndist)
{
   Int_t i, j;
      TVectorD vec(fNvar);
   for (i=0; i<nmerged; i++) {
      ndist[i]=0;
      for(j=0; j<fNvar; j++) {
         ndist[i]+=fHyperplane[j]*(dat[i][j]-fMean[j]);
         ndist[i]=TMath::Abs(ndist[i]);
      }
   }
   KOrdStat(nmerged, ndist, hmerged-1, index);
   ClearSscp(sscp);
   for (i=0; i<hmerged; i++) {
      for(j=0; j<fNvar; j++)
         vec[j]=dat[index[i]][j];
      AddToSscp(sscp, vec);
   }
   Covar(sscp, fMean, fCovariance, fSd, hmerged);
}
 
////////////////////////////////////////////////////////////////////////////////
///from the input htotal-subset constructs another htotal subset with lower determinant
///
///As proven by Peter J.Rousseeuw and Katrien Van Driessen, if distances for all elements
///are calculated, using the formula:d_i=Sqrt((x_i-M)*S_inv*(x_i-M)), where M is the mean
///of the input htotal-subset, and S_inv - the inverse of its covariance matrix, then
///htotal elements with smallest distances will have covariance matrix with determinant
///less or equal to the determinant of the input subset covariance matrix.
///
///determinant for this htotal-subset with smallest distances is returned
 
Double_t TRobustEstimator::CStep(Int_t ntotal, Int_t htotal, Int_t *index, TMatrixD &data, TMatrixD &sscp, Double_t *ndist)
{
   Int_t i, j;
   TVectorD vec(fNvar);
   Double_t det;
 
   TDecompChol chol(fCovariance);
   fInvcovariance = chol.Invert();
 
   TVectorD temp(fNvar);
   for(j=0; j<ntotal; j++) {
      ndist[j]=0;
      for(i=0; i<fNvar; i++)
         temp[i]=data[j][i]-fMean[i];
      temp*=fInvcovariance;
      for(i=0; i<fNvar; i++)
         ndist[j]+=(data[j][i]-fMean[i])*temp[i];
   }
 
   //taking h smallest
   KOrdStat(ntotal, ndist, htotal-1, index);
   //writing their mean and covariance
   ClearSscp(sscp);
   for (i=0; i<htotal; i++) {
      for (j=0; j<fNvar; j++)
         temp[j]=data[index[i]][j];
      AddToSscp(sscp, temp);
   }
   Covar(sscp, fMean, fCovariance, fSd, htotal);
   det = fCovariance.Determinant();
   return det;
}
 
////////////////////////////////////////////////////////////////////////////////
///for the exact fit situations
///returns number of observations on the hyperplane
 
Int_t TRobustEstimator::Exact(Double_t *ndist)
{
   Int_t i, j;
 
   TMatrixDSymEigen eigen(fCovariance);
   TVectorD eigenValues=eigen.GetEigenValues();
   TMatrixD eigenMatrix=eigen.GetEigenVectors();
 
   for (j=0; j<fNvar; j++) {
      fHyperplane[j]=eigenMatrix(j,fNvar-1);
   }
   //calculate and return how many observations lie on the hyperplane
   for (i=0; i<fN; i++) {
      ndist[i]=0;
      for(j=0; j<fNvar; j++) {
         ndist[i]+=fHyperplane[j]*(fData[i][j]-fMean[j]);
         ndist[i]=TMath::Abs(ndist[i]);
      }
   }
   Int_t nhyp=0;
 
   for (i=0; i<fN; i++) {
      if(ndist[i] < 1e-14) nhyp++;
   }
   return nhyp;
 
}
 
////////////////////////////////////////////////////////////////////////////////
///This function is called if determinant of the covariance matrix of a subset=0.
///
///If there are more then fH vectors on a hyperplane,
///returns this hyperplane and stops
///else stores the hyperplane coordinates in hyperplane matrix
 
Int_t TRobustEstimator::Exact2(TMatrixD &mstockbig, TMatrixD &cstockbig, TMatrixD &hyperplane,
                             Double_t *deti, Int_t nbest, Int_t kgroup,
                             TMatrixD &sscp, Double_t *ndist)
{
   Int_t i, j;
 
   TVectorD vec(fNvar);
   Int_t maxind = TMath::LocMax(nbest, deti);
   Int_t nh=Exact(ndist);
   //now nh is the number of observation on the hyperplane
   //ndist stores distances of observation from this hyperplane
   if(nh>=fH) {
      ClearSscp(sscp);
      for (i=0; i<fN; i++) {
         if(ndist[i]<1e-14) {
            for (j=0; j<fNvar; j++)
               vec[j]=fData[i][j];
            AddToSscp(sscp, vec);
         }
      }
      Covar(sscp, fMean, fCovariance, fSd, nh);
 
      fExact=nh;
      return nbest+1;
 
   } else {
      //if less than fH observations lie on a hyperplane,
      //mean and covariance matrix are stored in mstockbig
      //and cstockbig in place of the previous maximum determinant
      //mean and covariance
      for(i=0; i<fNvar; i++) {
         mstockbig(nbest*kgroup+maxind,i)=fMean(i);
         hyperplane(nbest*kgroup+maxind,i)=fHyperplane(i);
         for(j=0; j<fNvar; j++) {
            cstockbig(i,nbest*kgroup*fNvar+maxind*fNvar+j)=fCovariance(i,j);
         }
      }
      return maxind;
   }
}
 
 
////////////////////////////////////////////////////////////////////////////////
///divides the elements into approximately equal subgroups
///number of elements in each subgroup is stored in indsubdat
///number of subgroups is returned
 
Int_t TRobustEstimator::Partition(Int_t nmini, Int_t *indsubdat)
{
   Int_t nsub;
   if ((fN>=2*nmini) && (fN<=(3*nmini-1))) {
      if (fN%2==1){
         indsubdat[0]=Int_t(fN*0.5);
      indsubdat[1]=Int_t(fN*0.5)+1;
      } else
         indsubdat[0]=indsubdat[1]=Int_t(fN/2);
      nsub=2;
   }
   else{
      if((fN>=3*nmini) && (fN<(4*nmini -1))) {
         if(fN%3==0){
            indsubdat[0]=indsubdat[1]=indsubdat[2]=Int_t(fN/3);
         } else {
            indsubdat[0]=Int_t(fN/3);
            indsubdat[1]=Int_t(fN/3)+1;
            if (fN%3==1) indsubdat[2]=Int_t(fN/3);
            else indsubdat[2]=Int_t(fN/3)+1;
         }
         nsub=3;
      }
      else{
         if((fN>=4*nmini)&&(fN<=(5*nmini-1))){
            if (fN%4==0) indsubdat[0]=indsubdat[1]=indsubdat[2]=indsubdat[3]=Int_t(fN/4);
            else {
               indsubdat[0]=Int_t(fN/4);
               indsubdat[1]=Int_t(fN/4)+1;
               if(fN%4==1) indsubdat[2]=indsubdat[3]=Int_t(fN/4);
               if(fN%4==2) {
                  indsubdat[2]=Int_t(fN/4)+1;
                  indsubdat[3]=Int_t(fN/4);
               }
               if(fN%4==3) indsubdat[2]=indsubdat[3]=Int_t(fN/4)+1;
            }
            nsub=4;
         } else {
            for(Int_t i=0; i<5; i++)
               indsubdat[i]=nmini;
            nsub=5;
         }
      }
   }
   return nsub;
}
 
////////////////////////////////////////////////////////////////////////////////
///Calculates robust distances.Then the samples with robust distances
///greater than a cutoff value (0.975 quantile of chi2 distribution with
///fNvar degrees of freedom, multiplied by a correction factor), are given
///weiht=0, and new, reweighted estimates of location and scatter are calculated
///The function returns the number of outliers.
 
Int_t TRobustEstimator::RDist(TMatrixD &sscp)
{
   Int_t i, j;
   Int_t nout=0;
 
   TVectorD temp(fNvar);
   TDecompChol chol(fCovariance);
   fInvcovariance = chol.Invert();
 
 
   for (i=0; i<fN; i++) {
      fRd[i]=0;
      for(j=0; j<fNvar; j++) {
         temp[j]=fData[i][j]-fMean[j];
      }
      temp*=fInvcovariance;
      for(j=0; j<fNvar; j++) {
         fRd[i]+=(fData[i][j]-fMean[j])*temp[j];
      }
   }
 
   Double_t med;
   Double_t chi = kChiMedian[fNvar-1];
 
   med=TMath::Median(fN, fRd.GetMatrixArray());
   med/=chi;
   fCovariance*=med;
   TDecompChol chol2(fCovariance);
   fInvcovariance = chol2.Invert();
 
   for (i=0; i<fN; i++) {
      fRd[i]=0;
      for(j=0; j<fNvar; j++) {
         temp[j]=fData[i][j]-fMean[j];
   }
 
      temp*=fInvcovariance;
      for(j=0; j<fNvar; j++) {
         fRd[i]+=(fData[i][j]-fMean[j])*temp[j];
      }
   }
 
   Double_t cutoff = kChiQuant[fNvar-1];
 
   ClearSscp(sscp);
   for(i=0; i<fN; i++) {
      if (fRd[i]<=cutoff) {
         for(j=0; j<fNvar; j++)
            temp[j]=fData[i][j];
         AddToSscp(sscp,temp);
      } else {
         nout++;
      }
   }
 
   Covar(sscp, fMean, fCovariance, fSd, fN-nout);
   return nout;
}
 
////////////////////////////////////////////////////////////////////////////////
///Draws ngroup nonoverlapping subdatasets out of a dataset of size n
///such that the selected case numbers are uniformly distributed from 1 to n
 
void TRobustEstimator::RDraw(Int_t *subdat, Int_t ngroup, Int_t *indsubdat)
{
   Int_t jndex = 0;
   Int_t nrand;
   Int_t i, k, m, j;
   for (k=1; k<=ngroup; k++) {
      for (m=1; m<=indsubdat[k-1]; m++) {
         nrand = Int_t(gRandom->Uniform(0, 1) * double(fN-jndex))+1;
         //printf("nrand = %d - jndex %d\n",nrand,jndex);
         jndex++;
         if (jndex==1) {
            subdat[0]=nrand-1;  // in case nrand is equal to fN
         } else {
            subdat[jndex-1]=nrand+jndex-2;
            for (i=1; i<=jndex-1; i++) {
               if(subdat[i-1] > nrand+i-2) {
                  for(j=jndex; j>=i+1; j--) {
                     subdat[j-1]=subdat[j-2];
                  }
                  //printf("subdata[] i = %d - nrand %d\n",i,nrand);
                  subdat[i-1]=nrand+i-2;
                  break;  //breaking the loop for(i=1...
               }
            }
         }
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
///because I need an Int_t work array
 
Double_t TRobustEstimator::KOrdStat(Int_t ntotal, Double_t *a, Int_t k, Int_t *work){
   Bool_t isAllocated = kFALSE;
   const Int_t kWorkMax=100;
   Int_t i, ir, j, l, mid;
   Int_t arr;
   Int_t *ind;
   Int_t workLocal[kWorkMax];
   Int_t temp;
 
 
   if (work) {
      ind = work;
   } else {
      ind = workLocal;
      if (ntotal > kWorkMax) {
         isAllocated = kTRUE;
         ind = new Int_t[ntotal];
      }
   }
 
   for (Int_t ii=0; ii<ntotal; ii++) {
      ind[ii]=ii;
   }
   Int_t rk = k;
   l=0;
   ir = ntotal-1;
   for(;;) {
      if (ir<=l+1) { //active partition contains 1 or 2 elements
         if (ir == l+1 && a[ind[ir]]<a[ind[l]])
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}
         Double_t tmp = a[ind[rk]];
         if (isAllocated)
            delete [] ind;
         return tmp;
      } else {
         mid = (l+ir) >> 1; //choose median of left, center and right
         {temp = ind[mid]; ind[mid]=ind[l+1]; ind[l+1]=temp;}//elements as partitioning element arr.
         if (a[ind[l]]>a[ind[ir]])  //also rearrange so that a[l]<=a[l+1]
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}
 
         if (a[ind[l+1]]>a[ind[ir]])
            {temp=ind[l+1]; ind[l+1]=ind[ir]; ind[ir]=temp;}
 
         if (a[ind[l]]>a[ind[l+1]])
                {temp = ind[l]; ind[l]=ind[l+1]; ind[l+1]=temp;}
 
         i=l+1;        //initialize pointers for partitioning
         j=ir;
         arr = ind[l+1];
         for (;;) {
            do i++; while (a[ind[i]]<a[arr]);
            do j--; while (a[ind[j]]>a[arr]);
            if (j<i) break;  //pointers crossed, partitioning complete
               {temp=ind[i]; ind[i]=ind[j]; ind[j]=temp;}
         }
         ind[l+1]=ind[j];
         ind[j]=arr;
         if (j>=rk) ir = j-1; //keep active the partition that
         if (j<=rk) l=i;      //contains the k_th element
      }
   }
}