TMVA::MethodLikelihood

Class Description

/* Likelihood analysis ("non-parametric approach")

Also implemented is a "diagonalized likelihood approach", which improves over the uncorrelated likelihood ansatz by transforming linearly the input variables into a diagonal space, using the square-root of the covariance matrix

The method of maximum likelihood is the most straightforward, and certainly among the most elegant multivariate analyser approaches. We define the likelihood ratio, R_L, for event i, by:

Here the signal and background likelihoods, L_S, L_B, are products of the corresponding probability densities, p_S, p_B, of the N_var discriminating variables used in the MVA:
and accordingly for L_B. In practise, TMVA uses polynomial splines to estimate the probability density functions (PDF) obtained from the distributions of the training variables.

Note that in TMVA the output of the likelihood ratio is transformed by

to avoid the occurrence of heavy peaks at R_L=0,1. Decorrelated (or "diagonalized") Likelihood

The biggest drawback of the Likelihood approach is that it assumes that the discriminant variables are uncorrelated. If it were the case, it can be proven that the discrimination obtained by the above likelihood ratio is optimal, ie, no other method can beat it. However, in most practical applications of MVAs correlations are present.

Linear correlations, measured from the training sample, can be taken into account in a straightforward manner through the square-root of the covariance matrix. The square-root of a matrix C is the matrix C′ that multiplied with itself yields C: C=C′C′. We compute the square-root matrix (SQM) by means of diagonalising (D) the covariance matrix:

and the linear transformation of the linearly correlated into the uncorrelated variables space is then given by multiplying the measured variable tuple by the inverse of the SQM. Note that these transformations are performed for both signal and background separately, since the correlation pattern is not the same in the two samples.

The above diagonalisation is complete for linearly correlated, Gaussian distributed variables only. In real-world examples this is not often the case, so that only little additional information may be recovered by the diagonalisation procedure. In these cases, non-linear methods must be applied. */

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 standard constructor

 MethodLikelihood options:
 format and syntax of option string: "Spline2:0:25:D"

 where:
  SplineI [I=0,12,3,5] - which spline is used for smoothing the pdfs
                    0  - how often the input histos are smoothed
                    25 - average num of events per PDF bin to trigger warning
                    D  - use square-root-matrix to decorrelate variable space 
 

 construct likelihood references from file

 default initialisation called by all constructors

 define the options (their key words) that can be set in the option string 
 know options:
 Spline            <int>    spline used to interpolate reference histograms
    available values are:        0, 1, 2 <default>, 3, 5

 NSmooth           <int>    how often the input histos are smoothed
 NAvEvtPerBin      <int>    minimum average number of events per PDF bin (less trigger warning)
 TransformOutput   <bool>   transform (often strongly peaked) likelihood output through sigmoid inversion

 process user options

 destructor  

 create reference distributions (PDFs) from signal and background events:
 fill histograms and smooth them; if decorrelation is required, compute 
 corresponding square-root matrices

 returns the likelihood estimator for signal

 write reference PDFs to file

 read reference PDFs from file   if (istr.eof());

 write histograms and PDFs to file for monitoring purposes

 ranking of input variables

 overload test event reading