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Interpolation and Extrapolation of IRFs

.. currentmodule:: pyirf.interpolation

This module contains functions to inter- or extrapolate from a set of IRFs for different conditions to a new IRF. Implementations of interpolation and extrapolation algorithms exist as interpolator and extrapolator classes and are applied by top-level estimator classes to IRF components. Direct usage of the inter- and extrapolator classes is discouraged, as only the estimator classes check the data for consistency.

Most methods support an arbitrary number of interpolation dimensions although it is strongly advised to limit those for reasonable results. The herein provided functionalities can e.g. be used to interpolate the IRF for a zenith angle of 30° from available IRFs at 20° and 40°.

IRF Component Estimator Classes

.. autosummary::
   :nosignatures:

   EffectiveAreaEstimator       Estimate AEFF tables.
   RadMaxEstimator              Estimate RadMax tables.
   EnergyDispersionEstimator    Estimate 2D EDISPs.
   PSFTableEstimator            Estimate PSF tables.

Inter- and Extrapolation Classes

This module provides inter- and extrapolation classes that can be plugged into the estimator classes. Not all of these classes support arbitrary grid-dimensions where the grid in this context is the grid of e.g. observation parameters like zenith angle and magnetic field inclination (this would be a 2D grid) on which template IRFs exist and are meant to be inter- or extrapolated.

For parametrized components (Effective Areas and Rad-Max tables) these classes are:

Name Type Grid-Dim Note
:any:`GridDataInterpolator` Interpolation Arbitrary See also :any:`scipy.interpolate.griddata`.
:any:`ParametrizedNearestSimplexExtrapolator` Extrapolation 1D or 2D Linear (1D) or baryzentric (2D) extension outside the grid's convex hull from the nearest simplex.
:any:`ParametrizedVisibleEdgesExtrapolator` Extrapolation 1D or 2D Like :any:`ParametrizedNearestSimplexExtrapolator` but blends over all visible simplices [Alf84] and is thus smooth outside the convex hull.
:any:`ParametrizedNearestNeighborSearcher` Nearest Neighbor Arbitrary Nearest neighbor finder usable instead of inter- and/or extrapolation.

For components represented by discretized PDFs (PSF and EDISP tables) these classes are:

Name Type Grid-Dim Note
:any:`QuantileInterpolator` Interpolation Arbitrary Adaption of [Hol+13] and [Rea99] to discretized PDFs.
:any:`MomentMorphInterpolator` Interpolation 1D or 2D Adaption of [Baa+15] to discretized PDFs.
:any:`MomentMorphNearestSimplexExtrapolator` Extrapolation 1D or 2D Extension of [Baa+15] beyond the grid's convex hull from the nearest simplex.
:any:`DiscretePDFNearestNeighborSearcher` Nearest Neighbor Arbitrary Nearest neighbor finder usable instead of inter- and/or extrapolation.
[Alf84]P. Alfred (1984). Triangular Extrapolation. Technical summary rept., Univ. of Wisconsin-Madison. https://apps.dtic.mil/sti/pdfs/ADA144660.pdf
[Hol+13]B. E. Hollister and A. T. Pang (2013). Interpolation of Non-Gaussian Probability Distributions for Ensemble Visualization. https://engineering.ucsc.edu/sites/default/files/technical-reports/UCSC-SOE-13-13.pdf
[Rea99]A. L. Read (1999). Linear Interpolation of Histograms. Nucl. Instrum. Methods Phys. Res. A 425, 357-360. https://doi.org/10.1016/S0168-9002(98)01347-3
[Baa+15](1, 2) M. Baak, S. Gadatsch, R. Harrington and W. Verkerke (2015). Interpolation between multi-dimensional histograms using a new non-linear moment morphing method Nucl. Instrum. Methods Phys. Res. A 771, 39-48. https://doi.org/10.1016/j.nima.2014.10.033

Using Estimator Classes

Usage of the estimator classes is simple. As an example, consider CTA's Prod5 IRFs [CTA+21], they can be downloaded manually or by executing download_irfs.py in pyirf's root directory, which downloads them to .../pyirf/irfs/. The estimator classes can simply be used by first creating an instance of the respective class with all relevant information and then using the object's __call__ interface the obtain results for a specific target point. As the energy dispersion represents one of the discretized PDF IRF components, one can use the MomentMorphInterpolator for interpolation and the DiscretePDFNearestNeighborSearcher for extrapolation.

import numpy as np

from gammapy.irf import load_irf_dict_from_file
from glob import glob
from pyirf.interpolation import (
   EnergyDispersionEstimator,
   MomentMorphInterpolator,
   DiscretePDFNearestNeighborSearcher
)

# Load IRF data, replace path with actual path
PROD5_IRF_PATH = "pyirf/irfs/*.fits.gz"

irfs = [load_irf_dict_from_file(path) for path in sorted(glob(PROD5_IRF_PATH))]

edisps = np.array([irf["edisp"].quantity for irf in irfs])
bin_edges = irfs[0]["edisp"].axes["migra"].edges
# IRFs are for zenith distances of 20, 40 and 60 deg
zen_pnt = np.array([[20], [40], [60]])

# Create estimator instance
edisp_estimator = EnergyDispersionEstimator(
     grid_points=zen_pnt,
     migra_bins=bin_edges,
     energy_dispersion=edisps,
     interpolator_cls=MomentMorphInterpolator,
     interpolator_kwargs=None,
     extrapolator_cls=DiscretePDFNearestNeighborSearcher,
     extrapolator_kwargs=None,
 )

 # Estimate energy dispersions
 interpolated_edisp = edisp_estimator(np.array([[30]]))
 extrapolated_edisp = edisp_estimator(np.array([[10]]))
[CTA+21]Cherenkov Telescope Array Observatory & Cherenkov Telescope Array Consortium. (2021). CTAO Instrument Response Functions - prod5 version v0.1 (v0.1) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.5499840

Creating new Estimator Classes

To create a estimator class for an IRF component not yet implemented, one can simply inherit from respective base class. There are two, tailored to either parametrized or discrete PDF components.

.. autosummary::
   :nosignatures:

   ParametrizedComponentEstimator   Parametrized components
   DiscretePDFComponentEstimator    Discrete PDF components

Consider an example, where one is interested in an estimator for simple Gaussians. As this is already the scope of the DiscretePDFComponentEstimator base class and for the sake of this demonstration, let the Gaussians come with some units attached that need handling:

import astropy.units as u
from pyirf.interpolation import (DiscretePDFComponentEstimator,
                                 MomentMorphInterpolator)

class GaussianEstimatior(DiscretePDFComponentEstimator):
   @u.quantity_input(gaussians=u.m)
   def __init__(
      self,
      grid_points,
      bin_edges,
      gaussians,
      interpolator_cls=MomentMorphInterpolator,
      interpolator_kwargs=None,
      extrapolator_cls=None,
      extrapolator_kwargs=None,
   ):
      if interpolator_kwargs is None:
         interpolator_kwargs = {}

      if extrapolator_kwargs is None:
         extrapolator_kwargs = {}

      self.unit = gaussians.unit

      super().__init__(
         grid_points=grid_points,
         bin_edges=bin_edges,
         binned_pdf=gaussians.to_value(u.m),
         interpolator_cls=interpolator_cls,
         interpolator_kwargs=interpolator_kwargs,
         extrapolator_cls=extrapolator_cls,
         extrapolator_kwargs=extrapolator_kwargs,
      )

   def __call__(self, target_point):
      res = super().__call__(target_point)

      # Return result with correct unit
      return u.Quantity(res, u.m, copy=False).to(self.unit)

This new estimator class can now be used just like any other estimator class already implemented in pyirf.interpolation. While the extrapolator_cls argument can be empty when creating an instance of GaussianEstimator, effectively disabling extrapolation and raising an error in case it would be needed regardless, assume the desired extrapolation method to be MomentMorphNearestSimplexExtrapolator:

import numpy as np
from pyirf.interpolation import MomentMorphNearestSimplexExtrapolator
from scipy.stats import norm

bins = np.linspace(-10, 10, 51)
grid = np.array([[1], [2], [3]])

gaussians = np.array([np.diff(norm(loc=x, scale=1/x).cdf(bins))/np.diff(bins) for x in grid])

estimator = GaussianEstimatior(
   grid_points = grid,
   bin_edges = bins,
   gaussians = gaussians * u.m,
   interpolator_cls = MomentMorphInterpolator,
   extrapolator_cls = MomentMorphNearestSimplexExtrapolator
)

This estimator object can now easily be used to estimate Gaussians at arbitrary target points:

targets = np.array([[0.9], [1.5]])

results = u.Quantity([estimator(target).squeeze() for target in targets])

Helper Classes

.. autosummary::
   :nosignatures:

   PDFNormalization

Base Classes

.. autosummary::
   :nosignatures:

   BaseComponentEstimator
   ParametrizedComponentEstimator
   DiscretePDFComponentEstimator
   BaseInterpolator
   ParametrizedInterpolator
   DiscretePDFInterpolator
   BaseExtrapolator
   ParametrizedExtrapolator
   DiscretePDFExtrapolator
   BaseNearestNeighborSearcher

Full API

.. automodapi:: pyirf.interpolation
   :no-heading:
   :no-main-docstr:
   :inherited-members:
   :no-inheritance-diagram: