Multivariate Discriminant Analysis applied to classification of ne CC events in MINOS

1
Multivariate Discriminant Analysisapplied to
classification of ne CC events in MINOS

• Alex Sousa
• Tufts University
• MINOS Collaboration Meeting
• Fermilab
• 03/28/2004

2
Multivariate Discriminant Analysis

• Introduction
• Developed by Pearson, Fisher and Mahalanobis,
  among others.
• Applied to data that can be described by several
  variables, possibly correlated
• Aims to find combinations of those variables that
  maximize separation between two or more classes
  present in the data.
• Main uses include signal/background distinction,
  taxonomic classification, medical diagnosis, etc.

Karl Pearson (1857 1936)
R.A. Fisher (1890 1962)
P.C. Mahalanobis (1893 1972)
Simple Example
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MDA Procedure

• Define a set of variables that
  appropriately describes the data sample.
• Calculate the covariance matrix for each class
  
• Determine the Mahalanobis distance to each class
  for each event
• Compute the probabilities for an event to belong
  to each class (scores).

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Implementation

• Use the SAS package
• Very complete Statistical Analysis software
  bundle. Available at the amber cluster at Tufts.
• Built-in procedures such as DISCRIM and STEPDISC
  perform most of the tasks necessary in MDA for an
  arbitrary number of variables and classes.

• Do your own code
• Validated and debugged with Soudan2 data and SAS
  for 2 classes and arbitrary number of variables.
• Higher flexibility will be useful for future
  sensitivity studies.

5
Samples and Variables

• Sample contents
• Constructed from combination of two old Far MC
  files.
• Variables
• Total of 60 variables assembled from samples.

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Tufts Variables
y

• Proposed by J. Schneps, based on 3D Hits obtained
  from matching U and V views.

m
nm
q
z
x
E_Hit_long
ThetaL_Hit_Max
ne
ne
NC
NC
Ratio_Hit_Energ
ThetaE_Hit_Max
ne
ne
NC
NC
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Variable Selection

• Occams Razor Frustra fit per plura quod potest
  fieri per pauciora
• (it is vain to
  do with more what can be done with fewer).

• MDA provides more reliable results when highly
  redundant (correlated) variables with similar
  discriminating power are eliminated. A lower set
  of variables also allows a more manageable
  application of the method.
• SAS built-in Stepwise procedure
• measures the discriminating power of each
  variable and adds the best one at each step.
• Ideally the selected variables would be normally
  distributed, If not so, continuous variables can
  still be gaussianized. However, this needs to
  be looked at carefully.

8
Probability Distributions
Training Sample
Training Sample(E Cut)

• We can make an educated guess on an appropriate
  threshold by calculating the Figure of Merit for
  each threshold value.

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Efficiency and Purity vs Energy
Training Sample
Training Sample (E Cut)
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Results
Training Sample
Training Sample (E Cut)
Test Sample (E Cut)
Test Sample
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Results (contd)
Test Sample (E Cut)
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Results(contd)

• Comparison with Numi-714

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Conclusions

• Preliminary Results of application of a
  Multivariate Discriminant Analysis method to ne
  CC/NC/nm CC separation were presented. Still much
  work to be done
• Define a variable selection method. Make sure
  variables are close to gaussian behavior as to
  improve separation efficiency.
• Need to test on several samples for each training
  to assess possible biases.
• Look at evolution of results for the new MC ne
  files. Will start processing Roberts new files
  as soon as possible.
• Work with Harvard group on developing a common
  framework to generate working samples.
• Will count with the help of C. Gomez-Abajo.
• W.A. Mann has volunteered his experience to do a
  limited blind manual scanning of a sample
  containing ne CC,NC,nm CC events.