## Example Plot for Separation Power

This histogram visualises how distributions in beta look for kaons and pions, the example being a Time-of-Propagation calculation with perspex radiator, 1000 particle in each peak. One can determine the centroid and second moment (peak width in sigma) of these peaks, the ratio distance over sigma then gives a measure for the separation power. Plotting beta is preferred over plotting the reconstructed mass because it does not introduce a singularity at beta=1, the analysis happily giving values beta>1 sometimes.

- This quantity makes sense as long as the peaks are Gaussian in shape.
- This quantity is independent of the acceptance interval choice for pions and/or kaons.
- Such quantities from several subdetectors can be combined.

For each particle pair (here pion-kaon) this

**sigma separation power** is a function of particle angle (at target vertex) and particle momentum (or energy).

Visualising the full function is a challenge due to the three independent variables. Phi symmetry may reduce this to two, theta and momentum, and (colour coded or contour) 2D plots are possible. The 1D plot above is giving the values for fixed momentum.

Approximate PID momentum limits (3.5 sigma or 4 sigma or some other value) can be derived from the date used for the curve(s) above as the sigma value scales with p^(-2), the assumption being that in the beta histogram (top of this page) only the peak position changes but not the peak width.

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KlausFoehl - 26 Jun 2006

## From Sigma to Efficiencies

- curves:

The separation of two distributions does not yet specify the acceptance and rejection efficiencies. The graph (horizontal axis background particle 0, vertical axis particle 1) above shows probability lines for hypothesis (particle identified as) 0 and 1 (0.3 i.e. 70% 0 and 30% 1) for Gaussian distributions separated by 0,1,2,3,4 sigma (black lines). For non-gaussian behaviour see coloured lines examples.

- 2gauss with red line:

It is only the next step that the choice of a separation criterion (red line)
determines the efficiencies.

- curves with red dot:

This corresponds to selecting a point (red blob, happens to be on the 2sigma line) in parameter space. Here only 60% of particles 1 are correctly identified, as only 4% of particles 0 are false positives.

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KlausFoehl - 19 Junl 2006