class TUnfoldDensity: public TUnfoldSys

  TUnfoldDensity : public TUnfoldSys : public TUnfold

  TUnfold is used to decompose a measurement y into several sources x
  given the measurement uncertainties and a matrix of migrations A

  More details are described with the documentation of TUnfold.

  For most applications, it is best to use TUnfoldDensity
  instead of using TUnfoldSys or TUnfold

  If you use this software, please consider the following citation
       S.Schmitt, JINST 7 (2012) T10003 [arXiv:1205.6201]

  More documentation and updates are available on
      http://www.desy.de/~sschmitt


  As compared to TUnfold, TUndolfDensity adds the following functionality
    * background subtraction (see documentation of TUnfoldSys)
    * error propagation (see documentation of TUnfoldSys)
    * regularisation schemes respecting the bin widths
    * support for complex, multidimensional input distributions

  Complex binning schemes are imposed on the measurements y and
  on the result vector x with the help of the class TUnfoldBinning
  The components of x or y are part of multi-dimensional distributions.
  The bin widths along the relevant directions in these distributions
  are used to calculate bin densities (number of events divided by bin width)
  or to calculate derivatives taking into account the proper distance of
  adjacent bin centers

  Complex binning schemes

  in literature on unfolding, the "standard" test case is a
  one-dimensional distribution without underflow or overflow bins.
  The migration matrix is almost diagonal.

  This "standard" case is rarely realized for real problems.

  Often one has to deal with multi-dimensional input distributions.
  In addition, there are underflow and overflow bins
  or other background bins, possibly determined with the help of auxillary
  measurements

  In TUnfoldDensity, such complex binning schemes are handled with the help
  of the class TUnfoldBinning. For each vector there is a tree
  structure. The tree nodes hold multi-dimensiopnal distributions

  For example, the "measurement" tree could have two leaves, one for
  the primary distribution and one for auxillary measurements

  Similarly, the "truth" tree could have two leaves, one for the
  signal and one for the background.

  each of the leaves may then have a multi-dimensional distribution.

  The class TUnfoldBinning takes care to map all bins of the
  "measurement" to the one-dimensional vector y.
  Similarly, the "truth" bins are mapped to the vector x.

  Choice of the regularisation

  In TUnfoldDensity, two methods are implemented to determine tau**2
    (1)  ScanLcurve()  locate the tau where the L-curve plot has a "kink"
      this function is implemented in the TUnfold class
    (2)  ScanTau() finds the solution such that some variable
           (e.g. global correlation coefficient) is minimized
      this function is implemented in the TUnfoldDensity class,
      such that the variable could be made depend on the binning scheme

  Each of the algorithms has its own advantages and disadvantages

  The algorithm (1) does not work if the input data are too similar to the
  MC prediction, that is unfolding with tau=0 gives a least-square sum
  of zero. Typical no-go cases of the L-curve scan are:
    (a) the number of measurements is too small (e.g. ny=nx)
    (b) the input data have no statistical fluctuations
         [identical MC events are used to fill the matrix of migrations
          and the vector y]

  The algorithm (2) only works if the variable does have a real minimum
  as a function of tau.
  If global correlations are minimized, the situation is as follows:
  The matrix of migration typically introduces negative correlations.
   The area constraint introduces some positive correlation.
   Regularisation on the "size" introduces no correlation.
   Regularisation on 1st or 2nd derivatives adds positive correlations.
   For this reason, "size" regularisation does not work well with
   the tau-scan: the higher tau, the smaller rho, but there is no minimum.
   In contrast, the tau-scan is expected to work well with 1st or 2nd
   derivative regularisation, because at some point the negative
   correlations from migrations are approximately cancelled by the
   positive correlations from the regularisation conditions.

  whichever algorithm is used, the output has to be checked:
  (1) The L-curve should have approximate L-shape
       and the final choice of tau should not be at the very edge of the
       scanned region
  (2) The scan result should have a well-defined minimum and the
       final choice of tau should sit right in the minimum