Profile histograms

CALL HBPROF (ID,CHTITL,NCX,XLOW,XUP,YMIN,YMAX,CHOPT)
   

Action: Create a profile histogram. Profile histograms are used to display the mean value of Y and its RMS for each bin in X. Profile histograms are in many cases an elegant replacement of two-dimensional histograms : the inter-relation of two measured quantities X and Y can always be visualized by a two-dimensional histogram or scatter-plot; its representation on the line-printer is not particularly satisfactory, except for sparse data. If Y is an unknown (but single-valued) approximate function of X, this function is displayed by a profile histogram with much better precision than by a scatter-plot.

The following formulae show the cumulated contents (capital letters) and the values displayed by the printing or plotting routines (small letters) of the elements for bin J.

$H(J)$	$\sum \; Y$	$E(J)$	$\sum \; Y2$
$l(J)$	$\sum \; l$	$L(J)$	$\sum \; l$
$h(J)$	$H(J)/L(J)$	$s(J)$	$E(J)/L(J)-h(J)**2$
$e(J)$	$s(J)\; /L(J)$

The first five parameters are similar to HBOOK1. Only the values of Y between YMIN and YMAX will be considered. To fill a profile histogram, one must use CALL \Rind{HFILL (ID,X,Y,1.)}.

$H(J)$ is printed as the channel contents. The errors displayed are $s(J)$ if CHOPT='S' (spread option), or $e(J)$ if CHOPT=' ' (error on mean).

Profile histograms can be filled with weights.

The computation of the errors and the text below is based on a proposal by Stephane Coutu.

If a bin has N data points all with the same value Y (especially possible when dealing with integers), the spread in Y for that bin is zero, and the uncertainty assigned is also zero, and the bin is ignored in making subsequent fits. This is a problem. If $Y$ was the correct error in the case above, then $Y/N$

would be the correct error here. In fact, any bin with non-zero number of entries N but with zero spread should have an uncertainty $Y/N$ .

Yet, is $Y/N$

really the correct uncertainty? Probably it is only true in the case where the variable Y is some sort of counting statistics, following a Poisson distribution. This should probably correspond to the default case. However, Y can be any variable from an original Nntuple, not necessarily distributed according to a Poisson distribution. Therefore several settings for the option variable CHOPT are possible to determine how errors should be calculated (S stands for the spread in the formulae below):

' '

(Default)

'S'

Errors are	$S$	for $S\ne 0.$ ,
&quad;""	$Y$	for $S=0,N>0$ ,
&quad;""	$0.$	for $N=0$ .

'I'

Errors are	$S/N$	for $S\ne 0.$ ,
&quad;""	$1./12.N$	for $S=0,N>0$ ,
&quad;""	$0.$	for $N=0$ .

The third case above corresponds to integer Y values for which the uncertainty is $\pm 0.5$ , with the assumption that the probability that Y takes any value between $Y-0.5$ and $Y+0.5$

is uniform (the same argument goes for Y uniformly distributed between $Y$ and $Y+1$ ); this could be useful, for instance, for the case where $Y$ are ADC measurements.