#include "TSPlot.h"
#include "TVirtualFitter.h"
#include "TH1.h"
#include "TTreePlayer.h"
#include "TTreeFormula.h"
#include "TTreeFormulaManager.h"
#include "TSelectorDraw.h"
#include "TBrowser.h"
#include "TClass.h"
#include "TMath.h"
extern void Yields(Int_t &, Double_t *, Double_t &f, Double_t *x, Int_t iflag);
ClassImp(TSPlot)
/* -->
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<p>
<b><font size="+2">Overview</font></b>
</p><p>
A common method used in High Energy Physics to perform measurements is
the maximum Likelihood method, exploiting discriminating variables to
disentangle signal from background. The crucial point for such an
analysis to be reliable is to use an exhaustive list of sources of
events combined with an accurate description of all the Probability
Density Functions (PDF).
</p><p>To assess the validity of the fit, a convincing quality check
is to explore further the data sample by examining the distributions of
control variables. A control variable can be obtained for instance by
removing one of the discriminating variables before performing again
the maximum Likelihood fit: this removed variable is a control
variable. The expected distribution of this control variable, for
signal, is to be compared to the one extracted, for signal, from the
data sample. In order to be able to do so, one must be able to unfold
from the distribution of the whole data sample.
</p><p>The TSPlot method allows to reconstruct the distributions for
the control variable, independently for each of the various sources of
events, without making use of any <em>a priori</em> knowledge on <u>this</u>
variable. The aim is thus to use the knowledge available for the
discriminating variables to infer the behaviour of the individual
sources of events with respect to the control variable.
</p><p>
TSPlot is optimal if the control variable is uncorrelated with the discriminating variables.
</p><p>
A detail description of the formalism itself, called <!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48">, is given in [<a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/node1.html#bib:sNIM">1</a>].
</p><p>
<b><font size="+2">The method</font></b>
</p><p>
The <!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48"> technique is developped in the above context of a maximum Likelihood method making use of discriminating variables.
</p><p>One considers a data sample in which are merged several species
of events. These species represent various signal components and
background components which all together account for the data sample.
The different terms of the log-Likelihood are:
</p><ul>
<li><img src="gif/sPlot_img6.png" alt="$N$" align="bottom" border="0" height="17" width="22">: the total number of events in the data sample,
</li>
<li><!-- MATH
${\rm N}_{\rm s}$
-->
<img src="gif/sPlot_img7.png" alt="${\rm N}_{\rm s}$" align="middle" border="0" height="34" width="25">: the number of species of events populating the data sample,
</li>
<li><img src="gif/sPlot_img8.png" alt="$N_i$" align="middle" border="0" height="34" width="25">: the number of events expected on the average for the <img src="gif/sPlot_img9.png" alt="$i^{\rm th}$" align="bottom" border="0" height="20" width="25"> species,
</li>
<li><!-- MATH
${\rm f}_i(y_e)$
-->
<img src="gif/sPlot_img10.png" alt="${\rm f}_i(y_e)$" align="middle" border="0" height="37" width="47">: the value of the PDFs of the discriminating variables <img src="gif/sPlot_img11.png" alt="$y$" align="middle" border="0" height="33" width="15"> for the <img src="gif/sPlot_img12.png" alt="$i^{th}$" align="bottom" border="0" height="20" width="25"> species and for event <img src="gif/sPlot_img13.png" alt="$e$" align="bottom" border="0" height="17" width="13">,
</li>
<li><img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">: the set of control variables which, by definition, do not appear in the expression of the Likelihood function <img src="gif/sPlot_img15.png" alt="${\cal L}$" align="bottom" border="0" height="18" width="18">.
</li>
</ul>
The extended log-Likelihood reads:
<br>
<div align="right">
<!-- MATH
\begin{equation}
{\cal L}=\sum_{e=1}^{N}\ln \Big\{ \sum_{i=1}^{{\rm N}_{\rm s}}N_i{\rm f}_i(y_e) \Big\} -\sum_{i=1}^{{\rm N}_{\rm s}}N_i ~.
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:eLik"></a><img src="gif/sPlot_img16.png" alt="\begin{displaymath}
{\cal L}=\sum_{e=1}^{N}\ln \Big\{ \sum_{i=1}^{{\rm N}_{\rm s}}N_i{\rm f}_i(y_e) \Big\} -\sum_{i=1}^{{\rm N}_{\rm s}}N_i ~.
\end{displaymath}" border="0" height="59" width="276"></td>
<td align="right" width="10">
(1)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
From this expression, after maximization of <img src="gif/sPlot_img15.png" alt="${\cal L}$" align="bottom" border="0" height="18" width="18"> with respect to the <img src="gif/sPlot_img8.png" alt="$N_i$" align="middle" border="0" height="34" width="25"> parameters, a weight can be computed for every event and each species, in order to obtain later the true distribution <!-- MATH
${\hbox{\bf {M}}}_i(x)$
-->
<img src="gif/sPlot_img17.png" alt="${\hbox{\bf {M}}}_i(x)$" align="middle" border="0" height="37" width="56"> of variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">. If <img src="gif/sPlot_img18.png" alt="${\rm n}$" align="bottom" border="0" height="17" width="15"> is one of the <!-- MATH
${\rm N}_{\rm s}$
-->
<img src="gif/sPlot_img7.png" alt="${\rm N}_{\rm s}$" align="middle" border="0" height="34" width="25"> species present in the data sample, the weight for this species is defined by:
<br>
<div align="right">
<!-- MATH
\begin{equation}
\begin{Large}\fbox{$
{_s{\cal P}}_{\rm n}(y_e)={\sum_{j=1}^{{\rm N}_{\rm s}} \hbox{\bf V}_{{\rm n}j}{\rm f}_j(y_e)\over\sum_{k=1}^{{\rm N}_{\rm s}}N_k{\rm f}_k(y_e) } $}\end{Large} ~,
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:weightxnotiny"></a><img src="gif/sPlot_img19.png" alt="\begin{displaymath}
\begin{Large}
\fbox{$
{_s{\cal P}}_{\rm n}(y_e)={\sum_{j=1}^...
...um_{k=1}^{{\rm N}_{\rm s}}N_k{\rm f}_k(y_e) } $}\end{Large} ~,
\end{displaymath}" border="0" height="76" width="279"></td>
<td align="right" width="10">
(2)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
where <!-- MATH
$\hbox{\bf V}_{{\rm n}j}$
-->
<img src="gif/sPlot_img20.png" alt="$\hbox{\bf V}_{{\rm n}j}$" align="middle" border="0" height="34" width="35">
is the covariance matrix resulting from the Likelihood maximization.
This matrix can be used directly from the fit, but this is numerically
less accurate than the direct computation:
<br>
<div align="right">
<!-- MATH
\begin{equation}
\hbox{\bf V}^{-1}_{{\rm n}j}~=~
{\partial^2(-{\cal L})\over\partial N_{\rm n}\partial N_j}~=~
\sum_{e=1}^N {{\rm f}_{\rm n}(y_e){\rm f}_j(y_e)\over(\sum_{k=1}^{{\rm N}_{\rm s}}N_k{\rm f}_k(y_e))^2} ~.
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:VarianceMatrixDirect"></a><img src="gif/sPlot_img21.png" alt="\begin{displaymath}
\hbox{\bf V}^{-1}_{{\rm n}j}~=~
{\partial^2(-{\cal L})\over\...
...y_e)\over(\sum_{k=1}^{{\rm N}_{\rm s}}N_k{\rm f}_k(y_e))^2} ~.
\end{displaymath}" border="0" height="58" width="360"></td>
<td align="right" width="10">
(3)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
The distribution of the control variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15"> obtained by histogramming the weighted events reproduces, on average, the true distribution <!-- MATH
${\hbox{\bf {M}}}_{\rm n}(x)$
-->
<img src="gif/sPlot_img22.png" alt="${\hbox{\bf {M}}}_{\rm n}(x)$" align="middle" border="0" height="37" width="59">.
<p>
The class TSPlot allows to reconstruct the true distribution <!-- MATH
${\hbox{\bf {M}}}_{\rm n}(x)$
-->
<img src="gif/sPlot_img22.png" alt="${\hbox{\bf {M}}}_{\rm n}(x)$" align="middle" border="0" height="37" width="59"> of a control variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15"> for each of the <!-- MATH
${\rm N}_{\rm s}$
-->
<img src="gif/sPlot_img7.png" alt="${\rm N}_{\rm s}$" align="middle" border="0" height="34" width="25"> species from the sole knowledge of the PDFs of the discriminating variables <img src="gif/sPlot_img23.png" alt="${\rm f}_i(y)$" align="middle" border="0" height="37" width="40">. The plots obtained thanks to the TSPlot class are called <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57">.
</p><p>
<b><font size="+2">Some properties and checks</font></b>
</p><p>
Beside reproducing the true distribution, <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> bear remarkable properties:
</p><ul>
<li>
Each <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">-distribution is properly normalized:
<br>
<div align="right">
<!-- MATH
\begin{equation}
\sum_{e=1}^{N} {_s{\cal P}}_{\rm n}(y_e)~=~N_{\rm n}~.
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:NormalizationOK"></a><img src="gif/sPlot_img24.png" alt="\begin{displaymath}
\sum_{e=1}^{N} {_s{\cal P}}_{\rm n}(y_e)~=~N_{\rm n}~.
\end{displaymath}" border="0" height="58" width="158"></td>
<td align="right" width="10">
(4)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
</li>
<li>
For any event:
<br>
<div align="right">
<!-- MATH
\begin{equation}
\sum_{l=1}^{{\rm N}_{\rm s}} {_s{\cal P}}_l(y_e) ~=~1 ~.
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:numberconservation"></a><img src="gif/sPlot_img25.png" alt="\begin{displaymath}
\sum_{l=1}^{{\rm N}_{\rm s}} {_s{\cal P}}_l(y_e) ~=~1 ~.
\end{displaymath}" border="0" height="59" width="140"></td>
<td align="right" width="10">
(5)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
That is to say that, summing up the <!-- MATH
${\rm N}_{\rm s}$
-->
<img src="gif/sPlot_img7.png" alt="${\rm N}_{\rm s}$" align="middle" border="0" height="34" width="25"> <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57">, one recovers the data sample distribution in <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">, and summing up the number of events entering in a <!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48"> for a given species, one recovers the yield of the species, as provided by the fit. The property <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#eq:NormalizationOK">4</a> is implemented in the TSPlot class as a check.
</li>
<li>the sum of the statistical uncertainties per bin
<br>
<div align="right">
<!-- MATH
\begin{equation}
\sigma[N_{\rm n}\ _s\tilde{\rm M}_{\rm n}(x) {\delta x}]~=~\sqrt{\sum_{e \subset {\delta x}} ({_s{\cal P}}_{\rm n})^2} ~.
\end{equation}
-->
<table align="center" width="100%">
<tbody><tr valign="middle"><td align="center" nowrap="nowrap"><a name="eq:ErrorPerBin"></a><img src="gif/sPlot_img26.png" alt="\begin{displaymath}
\sigma[N_{\rm n}\ _s\tilde{\rm M}_{\rm n}(x) {\delta x}]~=~\sqrt{\sum_{e \subset {\delta x}} ({_s{\cal P}}_{\rm n})^2} ~.
\end{displaymath}" border="0" height="55" width="276"></td>
<td align="right" width="10">
(6)</td></tr>
</tbody></table>
<br clear="all"></div><p></p>
reproduces the statistical uncertainty on the yield <img src="gif/sPlot_img27.png" alt="$N_{\rm n}$" align="middle" border="0" height="34" width="28">, as provided by the fit: <!-- MATH
$\sigma[N_{\rm n}]\equiv\sqrt{\hbox{\bf V}_{{\rm n}{\rm n}}}$
-->
<img src="gif/sPlot_img28.png" alt="$\sigma[N_{\rm n}]\equiv\sqrt{\hbox{\bf V}_{{\rm n}{\rm n}}}$" align="middle" border="0" height="40" width="123">.
Because of that and since the determination of the yields is optimal
when obtained using a Likelihood fit, one can conclude that the<!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48"> technique is itself an optimal method to reconstruct distributions of control variables.
</li>
</ul>
<p>
<b><font size="+2">Different steps followed by TSPlot</font></b>
</p><p>
</p><ol>
<li>A maximum Likelihood fit is performed to obtain the yields <img src="gif/sPlot_img8.png" alt="$N_i$" align="middle" border="0" height="34" width="25"> of the various species.
The fit relies on discriminating variables <img src="gif/sPlot_img11.png" alt="$y$" align="middle" border="0" height="33" width="15"> uncorrelated with a control variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">:
the later is therefore totally absent from the fit.
</li>
<li>The weights <img src="gif/sPlot_img29.png" alt="${_s{\cal P}}$" align="middle" border="0" height="34" width="27"> are calculated using Eq. (<a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#eq:weightxnotiny">2</a>) where the covariance matrix is taken from Minuit.
</li>
<li>Histograms of <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15"> are filled by weighting the events with <img src="gif/sPlot_img29.png" alt="${_s{\cal P}}$" align="middle" border="0" height="34" width="27">.
</li>
<li>Error bars per bin are given by Eq. (<a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#eq:ErrorPerBin">6</a>).
</li>
</ol>
The <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> reproduce the true distributions of the species in the control variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">, within the above defined statistical uncertainties.
<p>
<b><font size="+2">Illustrations</font></b>
</p><p>
To illustrate the technique, one considers an example derived from the analysis where <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57">
have been first used (charmless B decays). One is dealing with a data
sample in which two species are present: the first is termed signal and
the second background. A maximum Likelihood fit is performed to obtain
the two yields <img src="gif/sPlot_img30.png" alt="$N_1$" align="middle" border="0" height="34" width="27"> and <img src="gif/sPlot_img31.png" alt="$N_2$" align="middle" border="0" height="34" width="27">. The fit relies on two discriminating variables collectively denoted <img src="gif/sPlot_img11.png" alt="$y$" align="middle" border="0" height="33" width="15"> which are chosen within three possible variables denoted <img src="gif/sPlot_img1.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39">, <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35"> and <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20">.
The variable which is not incorporated in <img src="gif/sPlot_img11.png" alt="$y$" align="middle" border="0" height="33" width="15"> is used as the control variable <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15">. The six distributions of the three variables are assumed to be the ones depicted in Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:pdfs">1</a>.
</p><p>
</p><div align="center"><a name="fig:pdfs"></a><a name="106"></a>
<table>
<caption align="bottom"><strong>Figure 1:</strong>
Distributions of the three discriminating variables available to perform the Likelihood fit:
<img src="gif/sPlot_img32.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39">, <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35">, <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20">.
Among the three variables, two are used to perform the fit while one is
kept out of the fit to serve the purpose of a control variable. The
three distributions on the top (resp. bottom) of the figure correspond
to the signal (resp. background). The unit of the vertical axis is
chosen such that it indicates the number of entries per bin, if one
slices the histograms in 25 bins.</caption>
<tbody><tr><td><img src="gif/sPlot_img33.png" alt="\begin{figure}\begin{center}
\mbox{{\psfig{file=pdfmesNIM.eps,width=0.33\linewi...
...th}}
{\psfig{file=pdffiNIM.eps,width=0.33\linewidth}}}
\end{center}\end{figure}" border="0" height="162" width="544"></td></tr>
</tbody></table>
</div>
<p>
A data sample being built through a Monte Carlo simulation based on the distributions shown in Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:pdfs">1</a>, one obtains the three distributions of Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:pdfstot">2</a>. Whereas the distribution of <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35"> clearly indicates the presence of the signal, the distribution of <img src="gif/sPlot_img1.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39"> and <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20"> are less obviously populated by signal.
</p><p>
</p><div align="center"><a name="fig:pdfstot"></a><a name="169"></a>
<table>
<caption align="bottom"><strong>Figure 2:</strong>
Distributions of the three discriminating variables for signal plus
background. The three distributions are the ones obtained from a data
sample obtained through a Monte Carlo simulation based on the
distributions shown in Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:pdfs">1</a>. The data sample consists of 500 signal events and 5000 background events.</caption>
<tbody><tr><td><img src="gif/sPlot_img34.png" alt="\begin{figure}\begin{center}
\mbox{{\psfig{file=genmesTOTNIM.eps,width=0.33\lin...
...}
{\psfig{file=genfiTOTNIM.eps,width=0.33\linewidth}}}
\end{center}\end{figure}" border="0" height="160" width="545"></td></tr>
</tbody></table>
</div>
<p>
Chosing <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35"> and <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20"> as discriminating variables to determine <img src="gif/sPlot_img30.png" alt="$N_1$" align="middle" border="0" height="34" width="27"> and <img src="gif/sPlot_img31.png" alt="$N_2$" align="middle" border="0" height="34" width="27"> through a maximum Likelihood fit, one builds, for the control variable <img src="gif/sPlot_img1.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39"> which is unknown to the fit, the two <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> for signal and background shown in Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:messPlots">3</a>. One observes that the <!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48">
for signal reproduces correctly the PDF even where the latter vanishes,
although the error bars remain sizeable. This results from the almost
complete cancellation between positive and negative weights: the sum of
weights is close to zero while the sum of weights squared is not. The
occurence of negative weights occurs through the appearance of the
covariance matrix, and its negative components, in the definition of
Eq. (<a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#eq:weightxnotiny">2</a>).
</p><p>
A word of caution is in order with respect to the error bars. Whereas
their sum in quadrature is identical to the statistical uncertainties
of the yields determined by the fit, and if, in addition, they are
asymptotically correct, the error bars should be handled with care for
low statistics and/or for too fine binning. This is because the error
bars do not incorporate two known properties of the PDFs: PDFs are
positive definite and can be non-zero in a given x-bin, even if in the
particular data sample at hand, no event is observed in this bin. The
latter limitation is not specific to<!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57">,
rather it is always present when one is willing to infer the PDF at the
origin of an histogram, when, for some bins, the number of entries does
not guaranty the applicability of the Gaussian regime. In such
situations, a satisfactory practice is to attach allowed ranges to the
histogram to indicate the upper and lower limits of the PDF value which
are consistent with the actual observation, at a given confidence
level.
</p><p>
</p><div align="center"><a name="fig:messPlots"></a><a name="127"></a>
<table>
<caption align="bottom"><strong>Figure 3:</strong>
The <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> (signal on the left, background on the right) obtained for <img src="gif/sPlot_img32.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39"> are represented as dots with error bars. They are obtained from a fit using only information from <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35"> and <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20">.</caption>
<tbody><tr><td><img src="gif/sPlot_img35.png" alt="\begin{figure}\begin{center}
\mbox{\psfig{file=mass-sig-sPlot.eps,width=0.48\li...
... \psfig{file=mass-bkg-sPlot.eps,width=0.48\linewidth}}
\end{center}\end{figure}" border="0" height="181" width="539"></td></tr>
</tbody></table>
</div>
<p>
Chosing <img src="gif/sPlot_img1.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39"> and <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35"> as discriminating variables to determine <img src="gif/sPlot_img30.png" alt="$N_1$" align="middle" border="0" height="34" width="27"> and <img src="gif/sPlot_img31.png" alt="$N_2$" align="middle" border="0" height="34" width="27"> through a maximum Likelihood fit, one builds, for the control variable <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20"> which is unknown to the fit, the two <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> for signal and background shown in Fig. <a href="http://www.slac.stanford.edu/%7Epivk/sPlot/sPlot_ROOT/sPlot_ROOT.html#fig:FisPlots">4</a>. In the <!-- MATH
$\hbox{$_s$}{\cal P}lot$
-->
<img src="gif/sPlot_img5.png" alt="$\hbox{$_s$}{\cal P}lot$" align="middle" border="0" height="34" width="48"> for signal one observes that error bars are the largest in the <img src="gif/sPlot_img14.png" alt="$x$" align="bottom" border="0" height="17" width="15"> regions where the background is the largest.
</p><p>
</p><div align="center"><a name="fig:FisPlots"></a><a name="136"></a>
<table>
<caption align="bottom"><strong>Figure 4:</strong>
The <!-- MATH
$\hbox{$_s$}{\cal P}lots$
-->
<img src="gif/sPlot_img4.png" alt="$\hbox {$_s$}{\cal P}lots$" align="middle" border="0" height="34" width="57"> (signal on the left, background on the right) obtained for <img src="gif/sPlot_img3.png" alt="${\cal F}$" align="bottom" border="0" height="18" width="20"> are represented as dots with error bars. They are obtained from a fit using only information from <img src="gif/sPlot_img32.png" alt="${m_{\rm ES}}$" align="middle" border="0" height="33" width="39"> and <img src="gif/sPlot_img2.png" alt="$\Delta E$" align="bottom" border="0" height="17" width="35">.</caption>
<tbody><tr><td><img src="gif/sPlot_img36.png" alt="\begin{figure}\begin{center}
\mbox{\psfig{file=fisher-sig-sPlot.eps,width=0.48\...
...psfig{file=fisher-bkg-sPlot.eps,width=0.48\linewidth}}
\end{center}\end{figure}" border="0" height="180" width="539"></td></tr>
</tbody></table>
</div>
<p>
The results above can be obtained by running the tutorial TestSPlot.C
</p>
<!--*/
//-->End_Html
TSPlot::TSPlot() :
fTree(0),
fTreename(0),
fVarexp(0),
fSelection(0)
{
fNx = 0;
fNy=0;
fNevents = 0;
fNSpecies=0;
fNumbersOfEvents=0;
}
TSPlot::TSPlot(Int_t nx, Int_t ny, Int_t ne, Int_t ns, TTree *tree) :
fTreename(0),
fVarexp(0),
fSelection(0)
{
fNx = nx;
fNy=ny;
fNevents = ne;
fNSpecies=ns;
fXvar.ResizeTo(fNevents, fNx);
fYvar.ResizeTo(fNevents, fNy);
fYpdf.ResizeTo(fNevents, fNSpecies*fNy);
fSWeights.ResizeTo(fNevents, fNSpecies*(fNy+1));
fTree = tree;
fNumbersOfEvents = 0;
}
TSPlot::~TSPlot()
{
if (fNumbersOfEvents)
delete [] fNumbersOfEvents;
if (!fXvarHists.IsEmpty())
fXvarHists.Delete();
if (!fYvarHists.IsEmpty())
fYvarHists.Delete();
if (!fYpdfHists.IsEmpty())
fYpdfHists.Delete();
}
void TSPlot::Browse(TBrowser *b)
{
if (!fSWeightsHists.IsEmpty()) {
TIter next(&fSWeightsHists);
TH1D* h = 0;
while ((h = (TH1D*)next()))
b->Add(h,h->GetName());
}
if (!fYpdfHists.IsEmpty()) {
TIter next(&fYpdfHists);
TH1D* h = 0;
while ((h = (TH1D*)next()))
b->Add(h,h->GetName());
}
if (!fYvarHists.IsEmpty()) {
TIter next(&fYvarHists);
TH1D* h = 0;
while ((h = (TH1D*)next()))
b->Add(h,h->GetName());
}
if (!fXvarHists.IsEmpty()) {
TIter next(&fXvarHists);
TH1D* h = 0;
while ((h = (TH1D*)next()))
b->Add(h,h->GetName());
}
b->Add(&fSWeights, "sWeights");
}
void TSPlot::SetInitialNumbersOfSpecies(Int_t *numbers)
{
if (!fNumbersOfEvents)
fNumbersOfEvents = new Double_t[fNSpecies];
for (Int_t i=0; i<fNSpecies; i++)
fNumbersOfEvents[i]=numbers[i];
}
void TSPlot::MakeSPlot(Option_t *option)
{
if (!fNumbersOfEvents){
Error("MakeSPlot","Initial numbers of events in species have not been set");
return;
}
Int_t i, j, ispecies;
TString opt = option;
opt.ToUpper();
opt.ReplaceAll("VV", "W");
char s[]="TFitter";
if (TVirtualFitter::GetFitter()){
Int_t strdiff=strcmp(TVirtualFitter::GetFitter()->IsA()->GetName(), s);
if (strdiff!=0)
delete TVirtualFitter::GetFitter();
}
TVirtualFitter *minuit = TVirtualFitter::Fitter(0, 2);
fPdfTot.ResizeTo(fNevents, fNSpecies);
for (Int_t iplot=-1; iplot<fNy; iplot++){
for (i=0; i<fNevents; i++){
for (ispecies=0; ispecies<fNSpecies; ispecies++){
fPdfTot(i, ispecies)=1;
for (j=0; j<fNy; j++){
if (j!=iplot)
fPdfTot(i, ispecies)*=fYpdf(i, ispecies*fNy+j);
}
}
}
minuit->Clear();
minuit->SetFCN(Yields);
Double_t arglist[10];
if (opt.Contains("Q")||opt.Contains("V")){
arglist[0]=-1;
}
if (opt.Contains("W"))
arglist[0]=0;
minuit->ExecuteCommand("SET PRINT", arglist, 1);
minuit->SetObjectFit(&fPdfTot);
for (ispecies=0; ispecies<fNSpecies; ispecies++)
minuit->SetParameter(ispecies, "", fNumbersOfEvents[ispecies], 1, 0, 0);
minuit->ExecuteCommand("MIGRAD", arglist, 0);
for (ispecies=0; ispecies<fNSpecies; ispecies++){
fNumbersOfEvents[ispecies]=minuit->GetParameter(ispecies);
if (!opt.Contains("Q"))
printf("estimated #of events in species %d = %f\n", ispecies, fNumbersOfEvents[ispecies]);
}
if (!opt.Contains("Q"))
printf("\n");
Double_t *covmat = minuit->GetCovarianceMatrix();
SPlots(covmat, iplot);
if (opt.Contains("W")){
Double_t *sumweight = new Double_t[fNSpecies];
for (i=0; i<fNSpecies; i++){
sumweight[i]=0;
for (j=0; j<fNevents; j++)
sumweight[i]+=fSWeights(j, (iplot+1)*fNSpecies + i);
printf("checking sum of weights[%d]=%f\n", i, sumweight[i]);
}
printf("\n");
delete [] sumweight;
}
}
}
void TSPlot::SPlots(Double_t *covmat, Int_t i_excl)
{
Int_t i, ispecies, k;
Double_t numerator, denominator;
for (i=0; i<fNevents; i++){
denominator=0;
for (ispecies=0; ispecies<fNSpecies; ispecies++)
denominator+=fNumbersOfEvents[ispecies]*fPdfTot(i, ispecies);
for (ispecies=0; ispecies<fNSpecies; ispecies++){
numerator=0;
for (k=0; k<fNSpecies; k++)
numerator+=covmat[ispecies*fNSpecies+k]*fPdfTot(i, k);
fSWeights(i, (i_excl+1)*fNSpecies + ispecies)=numerator/denominator;
}
}
}
void TSPlot::GetSWeights(TMatrixD &weights)
{
if (weights.GetNcols()!=fNSpecies*(fNy+1) || weights.GetNrows()!=fNevents)
weights.ResizeTo(fNevents, fNSpecies*(fNy+1));
weights = fSWeights;
}
void TSPlot::GetSWeights(Double_t *weights)
{
for (Int_t i=0; i<fNevents; i++){
for (Int_t j=0; j<fNSpecies; j++){
weights[i*fNSpecies+j]=fSWeights(i, j);
}
}
}
void TSPlot::FillXvarHists(Int_t nbins)
{
Int_t i, j;
if (!fXvarHists.IsEmpty()){
if (((TH1D*)fXvarHists.First())->GetNbinsX()!=nbins)
fXvarHists.Delete();
else
return;
}
char name[10];
for (i=0; i<fNx; i++){
snprintf(name,10, "x%d", i);
TH1D *h = new TH1D(name, name, nbins, fMinmax(0, i), fMinmax(1, i));
for (j=0; j<fNevents; j++)
h->Fill(fXvar(j, i));
fXvarHists.Add(h);
}
}
TObjArray* TSPlot::GetXvarHists()
{
Int_t nbins = 100;
if (fXvarHists.IsEmpty())
FillXvarHists(nbins);
else if (((TH1D*)fXvarHists.First())->GetNbinsX()!=nbins)
FillXvarHists(nbins);
return &fXvarHists;
}
TH1D *TSPlot::GetXvarHist(Int_t ixvar)
{
Int_t nbins = 100;
if (fXvarHists.IsEmpty())
FillXvarHists(nbins);
else if (((TH1D*)fXvarHists.First())->GetNbinsX()!=nbins)
FillXvarHists(nbins);
return (TH1D*)fXvarHists.UncheckedAt(ixvar);
}
void TSPlot::FillYvarHists(Int_t nbins)
{
Int_t i, j;
if (!fYvarHists.IsEmpty()){
if (((TH1D*)fYvarHists.First())->GetNbinsX()!=nbins)
fYvarHists.Delete();
else
return;
}
char name[10];
for (i=0; i<fNy; i++){
snprintf(name,10, "y%d", i);
TH1D *h=new TH1D(name, name, nbins, fMinmax(0, fNx+i), fMinmax(1, fNx+i));
for (j=0; j<fNevents; j++)
h->Fill(fYvar(j, i));
fYvarHists.Add(h);
}
}
TObjArray* TSPlot::GetYvarHists()
{
Int_t nbins = 100;
if (fYvarHists.IsEmpty())
FillYvarHists(nbins);
else if (((TH1D*)fYvarHists.First())->GetNbinsX()!=nbins)
FillYvarHists(nbins);
return &fYvarHists;
}
TH1D *TSPlot::GetYvarHist(Int_t iyvar)
{
Int_t nbins = 100;
if (fYvarHists.IsEmpty())
FillYvarHists(nbins);
else if (((TH1D*)fYvarHists.First())->GetNbinsX()!=nbins)
FillYvarHists(nbins);
return (TH1D*)fYvarHists.UncheckedAt(iyvar);
}
void TSPlot::FillYpdfHists(Int_t nbins)
{
Int_t i, j, ispecies;
if (!fYpdfHists.IsEmpty()){
if (((TH1D*)fYpdfHists.First())->GetNbinsX()!=nbins)
fYpdfHists.Delete();
else
return;
}
char name[30];
for (ispecies=0; ispecies<fNSpecies; ispecies++){
for (i=0; i<fNy; i++){
snprintf(name,30, "pdf_species%d_y%d", ispecies, i);
TH1D *h = new TH1D(name, name, nbins, fMinmax(0, fNx+fNy+ispecies*fNy+i), fMinmax(1, fNx+fNy+ispecies*fNy+i));
for (j=0; j<fNevents; j++)
h->Fill(fYpdf(j, ispecies*fNy+i));
fYpdfHists.Add(h);
}
}
}
TObjArray* TSPlot::GetYpdfHists()
{
Int_t nbins = 100;
if (fYpdfHists.IsEmpty())
FillYpdfHists(nbins);
return &fYpdfHists;
}
TH1D *TSPlot::GetYpdfHist(Int_t iyvar, Int_t ispecies)
{
Int_t nbins = 100;
if (fYpdfHists.IsEmpty())
FillYpdfHists(nbins);
return (TH1D*)fYpdfHists.UncheckedAt(fNy*ispecies+iyvar);
}
void TSPlot::FillSWeightsHists(Int_t nbins)
{
if (fSWeights.GetNoElements()==0){
Error("GetSWeightsHists", "SWeights were not computed");
return;
}
if (!fSWeightsHists.IsEmpty()){
if (((TH1D*)fSWeightsHists.First())->GetNbinsX()!=nbins)
fSWeightsHists.Delete();
else
return;
}
char name[30];
for (Int_t ivar=0; ivar<fNx; ivar++){
for (Int_t ispecies=0; ispecies<fNSpecies; ispecies++){
snprintf(name,30, "x%d_species%d", ivar, ispecies);
TH1D *h = new TH1D(name, name, nbins, fMinmax(0, ivar), fMinmax(1, ivar));
h->Sumw2();
for (Int_t ievent=0; ievent<fNevents; ievent++)
h->Fill(fXvar(ievent, ivar), fSWeights(ievent, ispecies));
fSWeightsHists.AddLast(h);
}
}
for (Int_t iexcl=0; iexcl<fNy; iexcl++){
for(Int_t ispecies=0; ispecies<fNSpecies; ispecies++){
snprintf(name,30, "y%d_species%d", iexcl, ispecies);
TH1D *h = new TH1D(name, name, nbins, fMinmax(0, fNx+iexcl), fMinmax(1, fNx+iexcl));
h->Sumw2();
for (Int_t ievent=0; ievent<fNevents; ievent++)
h->Fill(fYvar(ievent, iexcl), fSWeights(ievent, fNSpecies*(iexcl+1)+ispecies));
fSWeightsHists.AddLast(h);
}
}
}
TObjArray *TSPlot::GetSWeightsHists()
{
Int_t nbins = 50;
if (fSWeightsHists.IsEmpty())
FillSWeightsHists(nbins);
return &fSWeightsHists;
}
void TSPlot::RefillHist(Int_t type, Int_t nvar, Int_t nbins, Double_t min, Double_t max, Int_t nspecies)
{
if (type<1 || type>5){
Error("RefillHist", "type must lie between 1 and 5");
return;
}
char name[20];
Int_t j;
TH1D *hremove;
if (type==1){
hremove = (TH1D*)fXvarHists.RemoveAt(nvar);
delete hremove;
snprintf(name,20,"x%d",nvar);
TH1D *h = new TH1D(name, name, nbins, min, max);
for (j=0; j<fNevents;j++)
h->Fill(fXvar(j, nvar));
fXvarHists.AddAt(h, nvar);
}
if (type==2){
hremove = (TH1D*)fYvarHists.RemoveAt(nvar);
delete hremove;
snprintf(name,20, "y%d", nvar);
TH1D *h = new TH1D(name, name, nbins, min, max);
for (j=0; j<fNevents;j++)
h->Fill(fYvar(j, nvar));
fXvarHists.AddAt(h, nvar);
}
if (type==3){
hremove = (TH1D*)fYpdfHists.RemoveAt(nspecies*fNy+nvar);
delete hremove;
snprintf(name,20, "pdf_species%d_y%d", nspecies, nvar);
TH1D *h=new TH1D(name, name, nbins, min, max);
for (j=0; j<fNevents; j++)
h->Fill(fYpdf(j, nspecies*fNy+nvar));
fYpdfHists.AddAt(h, nspecies*fNy+nvar);
}
if (type==4){
hremove = (TH1D*)fSWeightsHists.RemoveAt(fNSpecies*nvar+nspecies);
delete hremove;
snprintf(name,20, "x%d_species%d", nvar, nspecies);
TH1D *h = new TH1D(name, name, nbins, min, max);
h->Sumw2();
for (Int_t ievent=0; ievent<fNevents; ievent++)
h->Fill(fXvar(ievent, nvar), fSWeights(ievent, nspecies));
fSWeightsHists.AddAt(h, fNSpecies*nvar+nspecies);
}
if (type==5){
hremove = (TH1D*)fSWeightsHists.RemoveAt(fNx*fNSpecies + fNSpecies*nvar+nspecies);
delete hremove;
snprintf(name,20, "y%d_species%d", nvar, nspecies);
TH1D *h = new TH1D(name, name, nbins, min, max);
h->Sumw2();
for (Int_t ievent=0; ievent<fNevents; ievent++)
h->Fill(fYvar(ievent, nvar), fSWeights(ievent, nspecies));
fSWeightsHists.AddAt(h, fNx*fNSpecies + fNSpecies*nvar+nspecies);
}
}
TH1D *TSPlot::GetSWeightsHist(Int_t ixvar, Int_t ispecies,Int_t iyexcl)
{
Int_t nbins = 50;
if (fSWeightsHists.IsEmpty())
FillSWeightsHists(nbins);
if (ixvar==-1)
return (TH1D*)fSWeightsHists.UncheckedAt(fNx*fNSpecies + fNSpecies*iyexcl+ispecies);
else
return (TH1D*)fSWeightsHists.UncheckedAt(fNSpecies*ixvar + ispecies);
}
void TSPlot::SetTree(TTree *tree)
{
fTree = tree;
}
void TSPlot::SetTreeSelection(const char* varexp, const char *selection, Long64_t firstentry)
{
TTreeFormula **var;
std::vector<TString> cnames;
TList *formulaList = new TList();
TSelectorDraw *selector = (TSelectorDraw*)(((TTreePlayer*)fTree->GetPlayer())->GetSelector());
Long64_t entry, entryNumber;
Int_t i,nch;
Int_t ncols;
TObjArray *leaves = fTree->GetListOfLeaves();
fTreename= new TString(fTree->GetName());
if (varexp)
fVarexp = new TString(varexp);
if (selection)
fSelection= new TString(selection);
nch = varexp ? strlen(varexp) : 0;
TTreeFormula *select = 0;
if (selection && strlen(selection)) {
select = new TTreeFormula("Selection",selection,fTree);
if (!select) return;
if (!select->GetNdim()) { delete select; return; }
formulaList->Add(select);
}
if (nch == 0) {
ncols = fNx + fNy + fNy*fNSpecies;
for (i=0;i<ncols;i++) {
cnames.push_back( leaves->At(i)->GetName() );
}
} else {
ncols = selector->SplitNames(varexp,cnames);
}
var = new TTreeFormula* [ncols];
Double_t *xvars = new Double_t[ncols];
fMinmax.ResizeTo(2, ncols);
for (i=0; i<ncols; i++){
fMinmax(0, i)=1e30;
fMinmax(1, i)=-1e30;
}
for (i=0;i<ncols;i++) {
var[i] = new TTreeFormula("Var1",cnames[i].Data(),fTree);
formulaList->Add(var[i]);
}
TTreeFormulaManager *manager=0;
if (formulaList->LastIndex()>=0) {
manager = new TTreeFormulaManager;
for(i=0;i<=formulaList->LastIndex();i++) {
manager->Add((TTreeFormula*)formulaList->At(i));
}
manager->Sync();
}
Int_t tnumber = -1;
Long64_t selectedrows=0;
for (entry=firstentry;entry<firstentry+fNevents;entry++) {
entryNumber = fTree->GetEntryNumber(entry);
if (entryNumber < 0) break;
Long64_t localEntry = fTree->LoadTree(entryNumber);
if (localEntry < 0) break;
if (tnumber != fTree->GetTreeNumber()) {
tnumber = fTree->GetTreeNumber();
if (manager) manager->UpdateFormulaLeaves();
}
Int_t ndata = 1;
if (manager && manager->GetMultiplicity()) {
ndata = manager->GetNdata();
}
for(Int_t inst=0;inst<ndata;inst++) {
Bool_t loaded = kFALSE;
if (select) {
if (select->EvalInstance(inst) == 0) {
continue;
}
}
if (inst==0) loaded = kTRUE;
else if (!loaded) {
for (i=0;i<ncols;i++) {
var[i]->EvalInstance(0);
}
loaded = kTRUE;
}
for (i=0;i<ncols;i++) {
xvars[i] = var[i]->EvalInstance(inst);
}
for (i=0; i<fNx; i++){
fXvar(selectedrows, i) = xvars[i];
if (fXvar(selectedrows, i) < fMinmax(0, i))
fMinmax(0, i)=fXvar(selectedrows, i);
if (fXvar(selectedrows, i) > fMinmax(1, i))
fMinmax(1, i)=fXvar(selectedrows, i);
}
for (i=0; i<fNy; i++){
fYvar(selectedrows, i) = xvars[i+fNx];
if (fYvar(selectedrows, i) < fMinmax(0, i+fNx))
fMinmax(0, i+fNx) = fYvar(selectedrows, i);
if (fYvar(selectedrows, i) > fMinmax(1, i+fNx))
fMinmax(1, i+fNx) = fYvar(selectedrows, i);
for (Int_t j=0; j<fNSpecies; j++){
fYpdf(selectedrows, j*fNy + i)=xvars[j*fNy + i+fNx+fNy];
if (fYpdf(selectedrows, j*fNy+i) < fMinmax(0, j*fNy+i+fNx+fNy))
fMinmax(0, j*fNy+i+fNx+fNy) = fYpdf(selectedrows, j*fNy+i);
if (fYpdf(selectedrows, j*fNy+i) > fMinmax(1, j*fNy+i+fNx+fNy))
fMinmax(1, j*fNy+i+fNx+fNy) = fYpdf(selectedrows, j*fNy+i);
}
}
selectedrows++;
}
}
fNevents=selectedrows;
delete [] xvars;
delete [] var;
}
void Yields(Int_t &, Double_t *, Double_t &f, Double_t *x, Int_t )
{
Double_t lik;
Int_t i, ispecies;
TVirtualFitter *fitter = TVirtualFitter::GetFitter();
TMatrixD *pdftot = (TMatrixD*)fitter->GetObjectFit();
Int_t nev = pdftot->GetNrows();
Int_t nes = pdftot->GetNcols();
f=0;
for (i=0; i<nev; i++){
lik=0;
for (ispecies=0; ispecies<nes; ispecies++)
lik+=x[ispecies]*(*pdftot)(i, ispecies);
if (lik<0) lik=1;
f+=TMath::Log(lik);
}
Double_t ntot=0;
for (i=0; i<nes; i++)
ntot += x[i];
f = -2*(f-ntot);
}