Example Plot for Separation Power

distributions for kaons and pions 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).

separation power expressed in sigma

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.

-- KlausFoehl - 26 Jun 2006

From Sigma to Efficiencies

  • curves:
    curves half size
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:
    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:
    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.

-- KlausFoehl - 19 Junl 2006

Topic attachments
I Attachment ActionSorted ascending Size Date Who Comment
betasigma30port.epseps betasigma30port.eps manage 31.9 K 26 Jun 2006 - 14:51 KlausFoehl separation power expressed in sigma
betasigma30port.gifgif betasigma30port.gif manage 5.9 K 26 Jun 2006 - 14:55 KlausFoehl separation power expressed in sigma
tagplot1.gifgif tagplot1.gif manage 6.9 K 19 Jul 2006 - 10:24 KlausFoehl curves
tagplot1a.gifgif tagplot1a.gif manage 3.1 K 19 Jul 2006 - 10:21 KlausFoehl curves with red dot
tagplot1h.gifgif tagplot1h.gif manage 3.1 K 19 Jul 2006 - 10:35 KlausFoehl curves half size
visualisation1.gifgif visualisation1.gif manage 2.6 K 26 Jun 2006 - 15:20 KlausFoehl distributions for kaons and pions
visualisation2.gifgif visualisation2.gif manage 2.8 K 19 Jul 2006 - 10:20 KlausFoehl 2gauss with red line
Topic revision: r2 - 19 Jul 2006, KlausFoehl
 
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