solveLinear.C File Reference

Detailed Description

This macro shows several ways to perform a linear least-squares analysis .

To keep things simple we fit a straight line to 4 data points The first 4 methods use the linear algebra package to find x such that min \( (A x - b)^T (A x - b) \) where A and b are calculated with the data points and the functional expression :

1. Normal equations: Expanding the expression \( (A x - b)^T (A x - b) \) and taking the derivative wrt x leads to the "Normal Equations": \( A^T A x = A^T b \) where \( A^T A \) is a positive definite matrix. Therefore, a Cholesky decomposition scheme can be used to calculate its inverse . This leads to the solution \( x = (A^T A)^-1 A^T b \) . All this is done in routine NormalEqn . We made it a bit more complicated by giving the data weights . Numerically this is not the best way to proceed because effectively the condition number of \( A^T A \) is twice as large as that of A, making inversion more difficult
2. SVD One can show that solving \( A x = b \) for x with A of size \( (m x n) \) and \( m > n \) through a Singular Value Decomposition is equivalent to minimizing \( (A x - b)^T (A x - b) \) Numerically , this is the most stable method of all 5
3. Pseudo Inverse Here we calculate the generalized matrix inverse ("pseudo inverse") by solving \( A X = Unit \) for matrix \( X \) through an SVD . The formal expression for is \( X = (A^T A)^-1 A^T \) . Then we multiply it by \( b \) . Numerically, not as good as 2 and not as fast . In general it is not a good idea to solve a set of linear equations with a matrix inversion .
4. Pseudo Inverse , brute force The pseudo inverse is calculated brute force through a series of matrix manipulations . It shows nicely some operations in the matrix package, but is otherwise a big "no no" .
5. Least-squares analysis with Minuit An objective function L is minimized by Minuit, where \( L = sum_i { (y - c_0 -c_1 * x / e)^2 } \) Minuit will calculate numerically the derivative of L wrt c_0 and c_1 . It has not been told that these derivatives are linear in the parameters c_0 and c_1 . For ill-conditioned linear problems it is better to use the fact it is a linear fit as in 2 .

Another interesting thing is the way we assign data to the vectors and matrices through adoption . This allows data assignment without physically moving bytes around .

USAGE

This macro can be executed via CLING or via ACLIC

• via the interpreter, do

  root > .x solveLinear.C

• via ACLIC

  root > gSystem->Load("libMatrix");
  root > gSystem->Load("libGpad");
  root > .x solveLinear.C+

 
Perform the fit  y = c0 + c1 * x in four different ways
 - 1. solve through Normal Equations
 - 2. solve through SVD
 - 3. solve with pseudo inverse
 - 4. solve with pseudo inverse, calculated brute force
 - 5. Minuit through TGraph
 All solutions are the same within tolerance of 1e-12

 
#include "Riostream.h"
#include "TMatrixD.h"
#include "TVectorD.h"
#include "TGraphErrors.h"
#include "TDecompChol.h"
#include "TDecompSVD.h"
#include "TF1.h"
 
 
void solveLinear(Double_t eps = 1.e-12)
{
   cout << "Perform the fit  y = c0 + c1 * x in four different ways" << endl;
 
   const Int_t nrVar  = 2;
   const Int_t nrPnts = 4;
 
   Double_t ax[] = {0.0,1.0,2.0,3.0};
   Double_t ay[] = {1.4,1.5,3.7,4.1};
   Double_t ae[] = {0.5,0.2,1.0,0.5};
 
   // Make the vectors 'Use" the data : they are not copied, the vector data
   // pointer is just set appropriately
 
   TVectorD x; x.Use(nrPnts,ax);
   TVectorD y; y.Use(nrPnts,ay);
   TVectorD e; e.Use(nrPnts,ae);
 
   TMatrixD A(nrPnts,nrVar);
   TMatrixDColumn(A,0) = 1.0;
   TMatrixDColumn(A,1) = x;
 
   cout << " - 1. solve through Normal Equations" << endl;
 
   const TVectorD c_norm = NormalEqn(A,y,e);
 
   cout << " - 2. solve through SVD" << endl;
   // numerically  preferred method
 
   // first bring the weights in place
   TMatrixD Aw = A;
   TVectorD yw = y;
   for (Int_t irow = 0; irow < A.GetNrows(); irow++) {
      TMatrixDRow(Aw,irow) *= 1/e(irow);
      yw(irow) /= e(irow);
   }
 
   TDecompSVD svd(Aw);
   Bool_t ok;
   const TVectorD c_svd = svd.Solve(yw,ok);
 
   cout << " - 3. solve with pseudo inverse" << endl;
 
   const TMatrixD pseudo1  = svd.Invert();
   TVectorD c_pseudo1 = yw;
   c_pseudo1 *= pseudo1;
 
   cout << " - 4. solve with pseudo inverse, calculated brute force" << endl;
 
   TMatrixDSym AtA(TMatrixDSym::kAtA,Aw);
   const TMatrixD pseudo2 = AtA.Invert() * Aw.T();
   TVectorD c_pseudo2 = yw;
   c_pseudo2 *= pseudo2;
 
   cout << " - 5. Minuit through TGraph" << endl;
 
   TGraphErrors *gr = new TGraphErrors(nrPnts,ax,ay,0,ae);
   TF1 *f1 = new TF1("f1","pol1",0,5);
   gr->Fit("f1","Q");
   TVectorD c_graph(nrVar);
   c_graph(0) = f1->GetParameter(0);
   c_graph(1) = f1->GetParameter(1);
 
   // Check that all 4 answers are identical within a certain
   // tolerance . The 1e-12 is somewhat arbitrary . It turns out that
   // the TGraph fit is different by a few times 1e-13.
 
   Bool_t same = kTRUE;
   same &= VerifyVectorIdentity(c_norm,c_svd,0,eps);
   same &= VerifyVectorIdentity(c_norm,c_pseudo1,0,eps);
   same &= VerifyVectorIdentity(c_norm,c_pseudo2,0,eps);
   same &= VerifyVectorIdentity(c_norm,c_graph,0,eps);
   if (same)
      cout << " All solutions are the same within tolerance of " << eps << endl;
   else
      cout << " Some solutions differ more than the allowed tolerance of " << eps << endl;
}

Author

Eddy Offermann

Definition in file solveLinear.C.