\subsection{H-Matrix discriminant}

\label{sec:hmatrix}

The origins of the H-Matrix\index{H-Matrix} approach dates back to works

of Fisher and Mahalanobis in the context of Gaussian

classifiers~\cite{Fisher,Mahalanobis}.

It discriminates one class (signal) of a feature vector from another

(background). The correlated elements of the vector are assumed to be

Gaussian distributed, and the inverse of the covariance matrix is

the {\em H-Matrix}. A multivariate $\chi^2$ estimator\index{Chi-squared estimator}

is built that exploits differences in the mean values of the vector elements

between the two classes for the purpose of discrimination.

The H-Matrix classifier as it is implemented in TMVA is equal or less performing

than the Fisher discriminant (see Sec.~\ref{sec:fisher}), and has been only

included for completeness.

\subsubsection{Booking options}

The H-Matrix discriminant is booked via the command:

\begin{codeexample}

\begin{tmvacode}

factory->BookMethod( Types::kHMatrix, "HMatrix", "<options>" );

\end{tmvacode}

\caption[.]{\codeexampleCaptionSize Booking of the H-Matrix classifier: the first argument is

a predefined enumerator, the second argument is a user-defined

string identifier, and the third argument is the configuration options string.

Individual options are separated by a ':'.

See Sec.~\ref{sec:usingtmva:booking} for more information on the booking.}

\end{codeexample}

No specific options are defined for this method beyond those shared with all the other

methods (\cf Option Table~\ref{opt:mva::methodbase} on page~\pageref{opt:mva::methodbase}).

\subsubsection{Description and implementation}

For an event $i$, each one $\chi^2$ estimator ($\chi^2_{S(B)}$) is computed for

signal ($S$) and background ($B$), using estimates for the sample means

($\overline x_{S(B),k}$) and covariance matrices ($C_{S(B)}$) obtained

from the training data

\beq

\chi^2_U(i)=\sum_{k,\ell=1}^{\Nvar}

\left(x_k(i) - \overline x_{U,k}\right)

C^{-1}_{U,k\ell}

\left(x_\ell(i) - \overline x_{U,\ell}\right)\,,

\eeq

where $U=S,B$. From this, the discriminant

\beq

\HMATRIX(i) = \frac{\chi^2_B(i)-\chi^2_S(i)}{\chi^2_B(i)+\chi^2_S(i)}\,,

\eeq

is computed to discriminate between the signal and background classes.

\subsubsection{Variable ranking}

The present implementation of the H-Matrix discriminant does not provide a ranking

of the input variables.

\subsubsection{Performance}

The TMVA implementation of the H-Matrix classifier has been shown to underperform

in comparison with the corresponding Fisher discriminant (\cf\ Sec.~\ref{sec:fisher}),

when using similar assumptions and complexity. It might therefore be considered to be depreciated.