Gulliver-Bootstrapping

A tool to apply the bootstrapping technique to Gulliver-based fits. This is useful e.g. to extract an estimation of the angular uncertainty of track reconstructions.

• Release Notes

This project hosts two Gulliver-based Icetray modules:

• BootstrappingLikelihoodService

  Takes an existing likelihood service and feeds it with resampled pulses.

• BootstrappingSeedService

  Uses the original best fit (before resampling) as the seed for each iteration of the resampling.

A third Icetray module is:

• BootstrapSeedTweak

  Calculation of a given containment radius, in which a certain fraction of the bootstrapped fits falls. The result can serve as an estimator of angular resolution.

See also

Gulliver, Lilliput, Paraboloid.

Background

Bootstrapping is a statistical method aimed at providing estimates of distributions of statistics in cases where the underlying theoretical distribution is unknown. It does so by employing random resampling with replacement to the given dataset.

The method was first described by B. Efron in Bootstrap Methods: Another Look at the Jackknife.

Given a sample \(X_0 = (x_1, \dots, x_n)\) defining the empirical distribution function \(\hat{F}\): to the theoretical distribution \(F\), the approach generates the bootstrap samples \(X_b = (x_1^*, \dots, x_n^*), b=1 \dots B\) by resampling with replacement \(n\) times from the given distribution \(X_0\).

The distribution of the statistical measure of interest \(T(F)\) is then approximated by \(T_b=T(X_b)\).

In the case of IceCube, this approach can be used to obtain an estimate for the angular error of a track reconstruction. In the language used before, \(T\) corresponds to the direction of the track and \(x\) can be understood as the pulses seen in the detector from a given track. Modelling the detector behaviour, it is possible to choose between resampling the pulses from a Poisson or Multinomial distribution. From the empirical distribution of fits after resampling it is finally possible to obtain e.g. the desired containment radius as a measure of the angular uncertainty.

Parameters

• BootstrappingLikelihoodService

  • Pulses: original set of pulses, that was also used for the initial track fit.

  • Bootstrapping: The type of resampling to use. Options are Multinomial (default) and Poisson.

  • Iterations: The number of resamplings to perform. Set to 4 by default.

    Increasing to 8 yields narrower pull distributions at the expense of longer runtime.

  • RandomService: The random service to use for the resampling.

  • WrappedLikelihood: The likelihood service used in the initial track fit.

• BootstrappingSeedService

  • WrappedSeed: A seed service pointing to the result of the initial track fit.

  • BootstrappingLikelihood: The name of the aforementioned likelihood service.

• BootstrapSeedTweak

  • BootstrappedRecos: The name of the output of the fitter.

  • ContainmentLevel: The size of the desired containment radius. Default: 50%.

  • AngularError: Where to store the angular size corresponding to the containment radius.

Application

The general sequence of applying this method is as follows.

1. Do the initial track fit (like SPE, MPE, SplineMPE, etc.).

2. Create a BootstrappingLikelihoodService and a BootstrappingSeedService pointing to the pulses and and likelihood configuration used in step 1.

3. Use an I3SimpleFitter to fit each set of resampled pulses provided by the classes from step 2. Set the output to AllFitsAndFitParams so that the result from each iteration is accessible afterwards.

4. Use the BootstrapSeedTweak to calculate the desired containment radius given the output of step 3.

Example

The following code shows just the application of the bootstrapping itself. It assumes that you have done a track fit (firstguess) before:

mininame = lilliput.segments.add_minuit_simplex_minimizer_service(tray)
paraname = lilliput.segments.add_simple_track_parametrization_service(tray)

pulses = "CleanedMuonPulses"        # <- The pulses used in the initial fit
firstguess = "OnlineL2_SplineMPE"   # <- The name of the initial fit
firstllh = "SplineMPEllh"           # <- The name of the llh of the initial fit

tray.AddService("I3BasicSeedServiceFactory", "MySplineMPESeed",
        FirstGuesses =                 [ firstguess ],
        ChargeFraction =               0.9,                      # Default
        FixedEnergy =                  float( "nan" ),           # Default
        MaxMeanTimeResidual =          1000.0*I3Units.ns,        # Default
        NChEnergyGuessPolynomial =     [],                       # Default
        SpeedPolice =                  True,                     # Default
        AddAlternatives =              "None",                   # Default
        OnlyAlternatives =             False                     # Default
        )

if 'I3RandomService' not in tray.tray_info.factories_in_order:
    tray.AddService("I3GSLRandomServiceFactory", "I3RandomService")

tray.AddService("BootstrappingLikelihoodServiceFactory", "MyBootstrapLLH",
        Pulses            = pulses,
        Bootstrapping     = gulliver_bootstrap.BootstrapOption.Multinomial,
        Iterations        = 8,
        WrappedLikelihood = firstllh,
        RandomService     = "I3RandomService"
        )

tray.AddService("BootstrappingSeedServiceFactory", "MyBootstrapSeed",
        WrappedSeed             = "MySplineMPESeed",
        BootstrappingLikelihood = "MyBootstrapLLH"
        )

tray.AddModule("I3SimpleFitter", "SplineMPE_Bootstrap",
        SeedService       = "MyBootstrapSeed",
        Parametrization   = paraname,
        LogLikelihood     = "MyBootstrapLLH",
        Minimizer         = mininame,
        StoragePolicy     = "AllFitsAndFitParams",
        If = If
        )

tray.AddModule("BootstrapSeedTweak", "MyBootstrapTweakSeeds",
        BootstrappedRecos = "SplineMPE_BootstrapVect",
        ContainmentLevel  = 0.5,
        AngularError      = "SplineMPE_Bootstrap_Angular",
        If = If
        )

For a more thorough example, take a look at the example script provided along with the project.