Welcome to ctaplot’s documentation!

Contents

ctaplot

ctaplot is a collection of functions to make instrument response functions (IRF) and reconstruction quality-checks plots for Imaging Atmospheric Cherenkov Telescopes such as CTA

Given a list of reconstructed and simulated quantities, compute and plot the Instrument Response Functions such as:

  • angular resolution
  • energy resolution
  • effective surface
  • impact point resolution

You may find examples in the documentation.

The CTA instrument response functions data used in ctaplot come from the CTA Consortium and Observatory and may be found on the cta-observatory website .

In cases for which the CTA instrument response functions are used in a research project, we ask to add the following acknowledgement in any resulting publication:

“This research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see http://www.cta-observatory.org/science/cta-performance/ (version prod3b-v2) for more details.”

Documentation Status https://travis-ci.org/vuillaut/ctaplot.svg?branch=master License: MIT

Install

Requirements packages:

  • python > 3.6
  • numpy
  • scipy>=0.19
  • matplotlib>=2.0
  • astropy

We recommend the use of anaconda

The package is available through pip:

pip install ctaplot

export GAMMABOARD_DATA=path_to_the_data_directory

We recommend that you add this line to your bash source file ($HOME/.bashrc or $HOME/.bash_profile)

GammaBoard

A dashboard to show them all.

GammaBoard is a simple jupyter dashboard thought to display metrics assessing the reconstructions performances of Imaging Atmospheric Cherenkov Telescopes (IACTs). Deep learning is a lot about bookkeeping and trials and errors. GammaBoard ease this bookkeeping and allows quick comparison of the reconstruction performances of your machine learning experiments.

It is a working prototype used in CTA, especially by the [GammaLearn](https://gitlab.lapp.in2p3.fr/GammaLearn/) project.

Run GammaBoard

To launch the dashboard, you can simply try the command:

gammaboard

This will run a temporary copy of the dashboard (a jupyter notebook). Local changes that you make in the dashboard will be discarded afterwards.

GammaBoard is using data in a specific directory storing all your experiments files. This directory is known under $GAMMABOARD_DATA by default. However, you can change the path access at any time in the dashboard itself.

Demo

Here is a simple demo of GammaBoard:

  • On top the plots (metrics) such as angular resolution and energy resolution.
  • Below, the list of experiments in the user folder.

When an experiment is selected in the list, the data is automatically loaded, the metrics computed and displayed. A list of information provided during the training phase is also displayed. As many experiments results can be overlaid. When an experiment is deselected, it simply is removed from the plots.

gammaboard_demo

License

MIT License

Copyright (c) 2018 ctaplot

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Modules

plots.py

Functions to make IRF and other reconstruction quality-check plots

ctaplot.plots.plot_resolution(bins, res, log=False, ax=None, **kwargs)

Plot the passed resolution.

Parameters:
  • bins (1D numpy.ndarray) –
  • res (2D numpy.ndarray - output from ctpalot.ana.resolution) – res[:,0]: resolution res[:,1]: lower confidence limit res[:,2]: upper confidence limit
  • log (bool) – if true, x is logscaled
  • ax (matplotlib.pyplot.axes) –
  • kwargs (kwargs for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_resolution_difference(bins, reference_resolution, new_resolution, ax=None, **kwargs)

Plot the algebric difference between a new resolution and reference resolution.

Parameters:
  • bins (numpy.ndarray) –
  • reference_resolution (numpy.ndarray) – output from ctaplot.ana.resolution
  • new_resolution (numpy.ndarray) – output from ctaplot.ana.resolution
  • ax (matplotlib.pyplot.axis) –
  • kwargs (args for ctaplot.plots.plot_resolution) –
Returns:

ax
Return type:

matplotlib.pyplot.axis
ctaplot.plots.plot_energy_resolution(simu_energy, reco_energy, percentile=68.27, confidence_level=0.95, bias_correction=False, ax=None, **kwargs)

Plot the enregy resolution as a function of the energy

Parameters:
  • simu_energy (numpy.ndarray) –
  • reco_energy (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • bias_correction (bool) –
  • kwargs (args for matplotlib.pyplot.plot) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_binned_bias(simu, reco, x, relative_scaling_method=None, ax=None, bins=10, log=False, **kwargs)

Plot the bias between simu and reco as a function of bins of x

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
  • x (numpy.ndarray) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
  • ax (matplotlib.pyplot.axis) –
  • bins (bins for numpy.histogram) –
  • log (bool) – if True, logscale is applied to the x axis
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axis
ctaplot.plots.plot_energy_bias(simu_energy, reco_energy, ax=None, **kwargs)

Plot the energy bias

Parameters:
  • simu_energy (numpy.ndarray) –
  • reco_energy (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.plot) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_impact_parameter_error_per_bin(x, reco_x, reco_y, simu_x, simu_y, bins=10, ax=None, **kwargs)

Plot the impact parameter error per bin

Parameters:
  • x (numpy.ndarray) –
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • bins (arg for np.histogram) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for plot_resolution) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_layout_map(tel_x, tel_y, tel_type=None, ax=None, **kwargs)

Plot the layout map of telescopes positions

Parameters:
  • tel_x (numpy.ndarray) –
  • tel_y (numpy.ndarray) –
  • TelId (numpy.ndarray) –
  • tel_type (numpy.ndarray) –
  • LayoutId (numpy.ndarray) –
  • Outfile (string) –
ctaplot.plots.plot_multiplicity_per_telescope_type(multiplicity, telescope_type, ax=None, outfile=None, quartils=False, **kwargs)

Plot the multiplicity for each telescope type

Parameters:
  • multiplicity (numpy.ndarray) –
  • telescope_type (numpy.ndarray) – same shape as multiplicity
  • ax (matplotlib.pyplot.axes) –
  • outfile (path) –
  • quartils (bool - True to plot 50% and 90% quartil mark) –
  • kwargs (args for matplotlib.pyplot.hist) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_multiplicity_hist(multiplicity, ax=None, outfile=None, quartils=False, **kwargs)

Histogram of the telescopes multiplicity

Parameters:
  • multiplicity (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • outfile (string) –
  • **kwargs (args for matplotlib.pyplot.bar) –
ctaplot.plots.plot_angles_distribution(reco_alt, reco_az, source_alt, source_az, outfile=None)

Plot the distribution of reconstructed angles in two axes. Save figure to outfile in png format.

Parameters:
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • source_alt (float) –
  • source_az (float) –
  • outfile (string) –
Returns:
Return type:

matplotlib.pyplot.figure
ctaplot.plots.plot_angles_map_distri(reco_alt, reco_az, source_alt, source_az, energies, outfile=None)

Plot the angles map distribution

Parameters:
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • source_alt (float) –
  • source_az (float) –
  • energies (numpy.ndarray) –
  • outfile (str) –
Returns:

fig
Return type:

matplotlib.pyplot.figure
ctaplot.plots.plot_angular_resolution_cta_performance(cta_site, ax=None, **kwargs)

Plot the official CTA performances (June 2018) for the angular resolution

Parameters:
  • cta_site (string, see ana.cta_performance) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.plot) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_angular_resolution_cta_requirement(cta_site, ax=None, **kwargs)

Plot the CTA requirement for the angular resolution :param cta_site: see ctaplot.ana.cta_requirement :type cta_site: string :param ax: :type ax: matplotlib.pyplot.axes :param kwargs: :type kwargs: args for matplotlib.pyplot.plot

Returns:ax
Return type:matplotlib.pyplot.axes
ctaplot.plots.plot_angular_resolution_per_energy(reco_alt, reco_az, mc_alt, mc_az, energy, percentile=68.27, confidence_level=0.95, bias_correction=False, ax=None, **kwargs)

Plot the angular resolution as a function of the energy

Parameters:
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • mc_alt (numpy.ndarray) –
  • mc_az (numpy.ndarray) –
  • energy (numpy.ndarray) – energies in TeV
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_angular_resolution_per_off_pointing_angle(simu_alt, simu_az, reco_alt, reco_az, alt_pointing, az_pointing, res_degree=False, bins=10, ax=None, **kwargs)

Plot the angular resolution as a function of the angular separation between events true position and the pointing direction. Angles must be given in radians.

Parameters:
  • simu_alt (numpy.ndarray) –
  • simu_az (numpy.ndarray) –
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • alt_pointing (numpy.ndarray) –
  • az_pointing (numpy.ndarray) –
  • res_degree (bool) – if True, the angular resolution is computed in degrees.
  • bins (int or numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (kwargs for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_bias_per_energy(simu, reco, energy, relative_scaling_method=None, ax=None, **kwargs)

Plot the bias per bins of energy

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
  • energy (numpy.ndarray) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
  • ax (matplotlib.pyplot.axis) –
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axis
ctaplot.plots.plot_binned_stat(x, y, statistic='mean', bins=20, errorbar=False, percentile=68.27, ax=None, **kwargs)

Plot statistics on the quantity y binned following the quantity x. The statistic can be given by a string (￿’mean￿’, ￿’sum’, ￿’max￿’…) or a function. See scipy.stats.binned_statistic. Errorbars may be added and represents the dispersion (given by the percentile option) of the y distribution around the measured value in a bin. These error bars might not make sense for some statistics, it is left to the user to use the function responsibly.

Parameters:
  • x (numpy.ndarray) –
  • y (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • errorbar (bool) –
  • statistic (string or callable - see scipy.stats.binned_statistic) –
  • bins (bins for scipy.stats.binned_statistic) –
  • kwargs (if errorbar: kwargs for matplotlib.pyplot.hlines else: kwargs for matplotlib.pyplot.plot) –
Returns:
Return type:

matplotlib.pyplot.axes

Examples

>>> from ctaplot.plots import plot_binned_stat
>>> import numpy as np
>>> x = np.random.rand(1000)
>>> y = x**2
>>> plot_binned_stat(x, y, statistic='median', bins=40, percentile=95, marker='o', linestyle='')
ctaplot.plots.plot_dispersion(simu_x, reco_x, x_log=False, ax=None, **kwargs)

Plot the dispersion around an expected value X_true

Parameters:
  • simu_x (numpy.ndarray) –
  • reco_x (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.hist2d) –
Returns:
Return type:

maptlotlib.pyplot.axes
ctaplot.plots.plot_effective_area_cta_performance(cta_site, ax=None, **kwargs)

Plot the CTA performances for the effective area

Parameters:
  • cta_site (string) – see ctaplot.ana.cta_requirement
  • ax (matplotlib.pyplot.axes) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_effective_area_cta_requirement(cta_site, ax=None, **kwargs)

Plot the CTA requirement for the effective area

Parameters:
  • cta_site (string) – see ctaplot.ana.cta_requirement
  • ax (matplotlib.pyplot.axes) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_effective_area_per_energy(simu_energy, reco_energy, simulated_area, ax=None, **kwargs)

Plot the effective area as a function of the energy

Parameters:
  • simu_energy (numpy.ndarray - all simulated event energies) –
  • reco_energy (numpy.ndarray - all reconstructed event energies) –
  • simulated_area (float) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (options for maplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes

Example

>>> import numpy as np
>>> import ctaplot
>>> irf = ctaplot.ana.irf_cta()
>>> simu_e = 10**(-2 + 4*np.random.rand(1000))
>>> reco_e = 10**(-2 + 4*np.random.rand(100))
>>> ax = ctaplot.plots.plot_effective_area_per_energy(simu_e, reco_e, irf.LaPalmaArea_prod3)
ctaplot.plots.plot_effective_area_per_energy_power_law(emin, emax, total_number_events, spectral_index, reco_energy, simu_area, ax=None, **kwargs)

Plot the effective area as a function of the energy. The effective area is computed using the ctaplot.ana.effective_area_per_energy_power_law.

Parameters:
  • emin (float) – min simulated energy
  • emax (float) – max simulated energy
  • total_number_events (int) – total number of simulated events
  • spectral_index (float) – spectral index of the simulated power-law
  • reco_energy (numpy.ndarray) – reconstructed energies
  • simu_area (float) – simulated core area
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_energy_distribution(mc_energy, reco_energy, ax=None, outfile=None, mask_mc_detected=True)

Plot the energy distribution of the simulated particles, detected particles and reconstructed particles The plot might be saved automatically if outfile is provided.

Parameters:
  • mc_energy (Numpy 1d array of simulated energies) –
  • reco_energy (Numpy 1d array of reconstructed energies) –
  • ax (matplotlib.pyplot.axes) –
  • outfile (string - output file path) –
  • mask_mc_detected (Numpy 1d array - mask of detected particles for the SimuE array) –
ctaplot.plots.plot_energy_resolution_cta_performance(cta_site, ax=None, **kwargs)

Plot the cta performances (June 2018) for the energy resolution

Parameters:
  • cta_site (string) – see ctaplot.ana.cta_performance
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.plot) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_energy_resolution_cta_requirement(cta_site, ax=None, **kwargs)

Plot the cta requirement for the energy resolution

Parameters:
  • cta_site (string) – see ana.cta_requirement
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.plot) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_feature_importance(feature_keys, feature_importances, ax=None)

Plot features importance after model training (typically from scikit-learn)

Parameters:
  • feature_keys (list of string) –
  • feature_importances (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
Returns:
Return type:

ax
ctaplot.plots.plot_field_of_view_map(reco_alt, reco_az, source_alt, source_az, energies=None, ax=None, outfile=None)

Plot a map in angles [in degrees] of the photons seen by the telescope (after reconstruction)

Parameters:
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • source_alt (float, source Altitude) –
  • source_az (float, source Azimuth) –
  • energies (numpy.ndarray - if given, set the colorbar) –
  • ax (matplotlib.pyplot.axes) –
  • outfile (string - if None, the plot is not saved) –
Returns:

ax
Return type:

matplitlib.pyplot.axes
ctaplot.plots.plot_impact_map(impact_x, impact_y, tel_x, tel_y, tel_types=None, ax=None, Outfile='ImpactMap.png', hist_kwargs={}, scatter_kwargs={})

Map of the site with telescopes positions and impact points heatmap

Parameters:
  • impact_x (numpy.ndarray) –
  • impact_y (numpy.ndarray) –
  • tel_x (numpy.ndarray) –
  • tel_y (numpy.ndarray) –
  • tel_types (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • hist_kwargs (kwargs for matplotlib.pyplot.hist) –
  • scatter_kwargs (kwargs for matplotlib.pyplot.scatter) –
  • Outfile (string - name of the output file) –
ctaplot.plots.plot_impact_parameter_error_per_energy(reco_x, reco_y, simu_x, simu_y, energy, ax=None, **kwargs)

Deprecated since version 18/08/2019: plot_impact_parameter_error_per_energy will be removed in a future release.Use plot_impact_parameter_resolution_per_energy instead

plot the impact parameter error distance as a function of energy and save the plot as Outfile :param reco_x: :type reco_x: numpy.ndarray :param reco_y: :type reco_y: numpy.ndarray :param simu_x: :type simu_x: numpy.ndarray :param simu_y: :type simu_y: numpy.ndarray :param energy: :type energy: numpy.ndarray :param ax: :type ax: matplotlib.pyplot.axes :param kwargs: :type kwargs: args for matplotlib.pyplot.errorbar

Returns:energy, err_mean
Return type:numpy arrays
ctaplot.plots.plot_impact_parameter_error_per_multiplicity(reco_x, reco_y, simu_x, simu_y, multiplicity, max_mult=None, ax=None, **kwargs)

Plot the impact parameter error as a function of multiplicity TODO: refactor and clean code

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • multiplicity (numpy.ndarray) –
  • max_mult (optional, max multiplicity - float) –
  • ax (matplotlib.pyplot.axes) –
ctaplot.plots.plot_impact_parameter_error_site_center(reco_x, reco_y, simu_x, simu_y, ax=None, **kwargs)

Plot the impact parameter error as a function of the distance to the site center.

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (kwargs for matplotlib.pyplot.hist2d) –
Returns:
Return type:

ax
ctaplot.plots.plot_impact_parameter_resolution_per_energy(reco_x, reco_y, simu_x, simu_y, energy, ax=None, **kwargs)
Parameters:
  • reco_x –
  • reco_y –
  • simu_x –
  • simu_y –
  • energy –
  • ax –
  • kwargs –
ctaplot.plots.plot_impact_point_heatmap(reco_x, reco_y, ax=None, outfile=None)

Plot the heatmap of the impact points on the site ground and save it under Outfile

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • outfile (string) –
ctaplot.plots.plot_impact_point_map_distri(reco_x, reco_y, tel_x, tel_y, fit=False, outfile=None, **kwargs)

Map and distributions of the reconstructed impact points.

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • tel_x (numpy.ndarray) – X positions of the telescopes
  • tel_y (numpy.ndarray) – Y positions of the telescopes
  • kde (bool - if True, makes a gaussian fit of the point density) –
  • outfile ('str' - save a png image of the plot under 'string.png') –
Returns:

fig
Return type:

matplotlib.pyplot.figure
ctaplot.plots.plot_impact_resolution_per_energy(reco_x, reco_y, simu_x, simu_y, simu_energy, percentile=68.27, confidence_level=0.95, bias_correction=False, ax=None, **kwargs)

Plot the angular resolution as a function of the energy

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (float) –
  • simu_y (float) –
  • simu_energy (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_migration_matrix(x, y, ax=None, colorbar=False, xy_line=False, hist2d_args={}, line_args={})

Make a simple plot of a migration matrix

Parameters:
  • x (list or numpy.ndarray) –
  • y (list or numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • colorbar (matplotlib.colorbar) –
  • hist2d_args (dict, args for matplotlib.pyplot.hist2d) –
  • line_args (dict, args for matplotlib.pyplot.plot) –
Returns:
Return type:

matplotlib.pyplot.axes

Examples

>>> from ctaplot.plots import plot_migration_matrix
>>> import matplotlib
>>> x = np.random.rand(10000)
>>> y = x**2
>>> plot_migration_matrix(x, y, colorbar=True, hist2d_args=dict(norm=matplotlib.colors.LogNorm()))
In this example, the colorbar will be log normed
ctaplot.plots.plot_multiplicity_per_energy(multiplicity, energies, ax=None, outfile=None)

Plot the telescope multiplicity as a function of the energy The plot might be saved automatically if outfile is provided.

Parameters:
  • multiplicity (numpy.ndarray) –
  • energies (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • outfile (string) –
ctaplot.plots.plot_resolution_per_energy(reco, simu, energy, ax=None, **kwargs)

Plot a variable resolution as a function of the energy

Parameters:
  • reco (numpy.ndarray) –
  • simu (numpy.ndarray) –
  • energy (numpy.ndarray) –
  • ax (matplotlib.pyplot.axes) –
  • kwargs (args for matplotlib.pyplot.errorbar) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_sensitivity_cta_performance(cta_site, ax=None, **kwargs)

Plot the CTA performances for the sensitivity

Parameters:
  • cta_site (string - see ctaplot.ana.cta_requirement) –
  • ax (matplotlib.pyplot.axes, optional) –
Returns:

ax
Return type:

matplotlib.pyplot.axes
ctaplot.plots.plot_sensitivity_cta_requirement(cta_site, ax=None, **kwargs)

Plot the CTA requirement for the sensitivity :param cta_site: :type cta_site: string - see ctaplot.ana.cta_requirement :param ax: :type ax: matplotlib.pyplot.axes, optional

Returns:ax
Return type:matplotlib.pyplot.axes
ctaplot.plots.plot_theta2(reco_alt, reco_az, mc_alt, mc_az, ax=None, **kwargs)

Plot the theta2 distribution and display the corresponding angular resolution in degrees. The input must be given in radians.

Parameters:
  • reco_alt (numpy.ndarray - reconstructed altitude angle in radians) –
  • reco_az (numpy.ndarray - reconstructed azimuth angle in radians) –
  • mc_alt (numpy.ndarray - true altitude angle in radians) –
  • mc_az (numpy.ndarray - true azimuth angle in radians) –
  • ax (matplotlib.pyplot.axes) –
  • **kwargs (options for matplotlib.pyplot.hist) –
Returns:

ax
Return type:

matplotlib.pyplot.axes

ana.py

Contain mathematical functions to make results analysis (compute angular resolution, effective surface, energy resolution… )

class ctaplot.ana.irf_cta

Bases: object

Class to handle Instrument Response Function data

set_E_bin(E_bin)
class ctaplot.ana.cta_performance(site)

Bases: object

get_angular_resolution()
get_effective_area(observation_time=50)

Return the effective area at the given observation time in hours. NB: Only 50h supported Returns the energy array and the effective area array :param observation_time: :type observation_time: optional

Returns:
Return type:numpy.ndarray, numpy.ndarray
get_energy_resolution()
get_sensitivity(observation_time=50)
class ctaplot.ana.cta_requirement(site)

Bases: object

get_angular_resolution()
get_effective_area(observation_time=50)

Return the effective area at the given observation time in hours. NB: Only 0.5h supported Returns the energy array and the effective area array :param observation_time: :type observation_time: optional

Returns:
Return type:numpy.ndarray, numpy.ndarray
get_energy_resolution()
get_sensitivity(observation_time=50)
ctaplot.ana.stat_per_energy(energy, y, statistic='mean')

Return statistic for the given quantity per energy bins. The binning is given by irf_cta

Parameters:
  • energy (numpy.ndarray (1d)) – event energies
  • y (numpy.ndarray (1d)) –
  • statistic (string) – see scipy.stat.binned_statistic
Returns:

bin_stat, bin_edges, binnumber
Return type:

numpy.ndarray, numpy.ndarray, numpy.ndarray
ctaplot.ana.bias(simu, reco)

Compute the bias of a reconstructed variable as median(reco-simu)

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
Returns:
Return type:

float
ctaplot.ana.relative_bias(simu, reco, relative_scaling_method='s1')

Compute the relative bias of a reconstructed variable as median(reco-simu)/relative_scaling(simu, reco)

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
ctaplot.ana.relative_scaling(simu, reco, method='s0')

Define the relative scaling for the relative error calculation. There are different ways to calculate this scaling factor. The easiest and most spread one is simply np.abs(simu). However this is possible only when simu != 0. Possible methods:

  • None or ‘s0’: scale = 1
  • ‘s1’: scale = np.abs(simu)
  • ‘s2’: scale = np.abs(reco)
  • ‘s3’: scale = (np.abs(simu) + np.abs(reco))/2.
  • ‘s4’: scale = np.max([np.abs(reco), np.abs(simu)], axis=0)

This method is not exposed but kept for tests and future reference. The s1 method is used in all ctaplot functions.

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
Returns:
Return type:

numpy.ndarray
ctaplot.ana.angular_resolution(reco_alt, reco_az, simu_alt, simu_az, percentile=68.27, confidence_level=0.95, bias_correction=False)

Compute the angular resolution as the Qth (standard being 68) containment radius of theta2 with lower and upper limits on this value corresponding to the confidence value required (1.645 for 95% confidence)

Parameters:
  • reco_alt (numpy.ndarray - reconstructed altitude angle in radians) –
  • reco_az (numpy.ndarray - reconstructed azimuth angle in radians) –
  • simu_alt (numpy.ndarray - true altitude angle in radians) –
  • simu_az (numpy.ndarray - true azimuth angle in radians) –
  • percentile (float - percentile, 68 corresponds to one sigma) –
  • confidence_level (float) –
Returns:
Return type:

numpy.array [angular_resolution, lower limit, upper limit]
ctaplot.ana.angular_separation_altaz(alt1, az1, alt2, az2, unit='rad')

Compute the angular separation in radians or degrees between two pointing direction given with alt-az

Parameters:
  • alt1 (1d numpy.ndarray, altitude of the first pointing direction) –
  • az1 (1d numpy.ndarray azimuth of the first pointing direction) –
  • alt2 (1d numpy.ndarray, altitude of the second pointing direction) –
  • az2 (1d numpy.ndarray, azimuth of the second pointing direction) –
  • unit ('deg' or 'rad') –
Returns:
Return type:

1d numpy.ndarray or float, angular separation
ctaplot.ana.angular_resolution_per_bin(simu_alt, simu_az, reco_alt, reco_az, x, percentile=68.27, confidence_level=0.95, bias_correction=False, bins=10)

Compute the angular resolution per binning of x

Parameters:
  • simu_alt (numpy.ndarray) –
  • simu_az (numpy.ndarray) –
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • x (numpy.ndarray) –
  • percentile (float) – 0 < percentile < 100
  • confidence_level (float) – 0 < confidence_level < 1
  • bias_correction (bool) –
  • bins (int or numpy.ndarray) –
Returns:

bins: 1D numpy.ndarray ang_res: 2D numpy.ndarray
Return type:

bins, ang_res
ctaplot.ana.angular_resolution_per_energy(reco_alt, reco_az, simu_alt, simu_az, energy, percentile=68.27, confidence_level=0.95, bias_correction=False)

Plot the angular resolution as a function of the event simulated energy

Parameters:
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • simu_alt (numpy.ndarray) –
  • simu_az (numpy.ndarray) –
  • energy (numpy.ndarray) –
  • **kwargs (args for angular_resolution) –
Returns:

(E, RES)
Return type:

(1d numpy array, 1d numpy array) = Energies, Resolution
ctaplot.ana.angular_resolution_per_off_pointing_angle(simu_alt, simu_az, reco_alt, reco_az, alt_pointing, az_pointing, bins=10)

Compute the angular resolution as a function of separation angle for the pointing direction

Parameters:
  • simu_alt (numpy.ndarray) –
  • simu_az (numpy.ndarray) –
  • reco_alt (numpy.ndarray) –
  • reco_az (numpy.ndarray) –
  • alt_pointing (numpy.ndarray) –
  • az_pointing (numpy.ndarray) –
  • bins (float or numpy.ndarray) –
Returns:

bins: 1D numpy.ndarray res: 2D numpy.ndarray - resolutions with confidence intervals (output from ctaplot.ana.resolution)
Return type:

(bins, res)
ctaplot.ana.energy_resolution(true_energy, reco_energy, percentile=68.27, confidence_level=0.95, bias_correction=False)

Compute the energy resolution of reco_energy as the percentile (68 as standard) containment radius of `true_energy-reco_energy)/simu_energy with the lower and upper confidence limits defined by the given confidence level

Parameters:
  • true_energy (1d numpy array of simulated energies) –
  • reco_energy (1d numpy array of reconstructed energies) –
  • percentile (float) – <= 100
Returns:
Return type:

numpy.array - [energy_resolution, lower_confidence_limit, upper_confidence_limit]
ctaplot.ana.energy_bias(simu_energy, reco_energy)

Compute the energy relative bias per energy bin.

Parameters:
  • simu_energy (1d numpy array of simulated energies) –
  • reco_energy (1d numpy array of reconstructed energies) –
Returns:

(energy_bins, bias)
Return type:

tuple of 1d numpy arrays - energy, energy bias
ctaplot.ana.energy_resolution_per_energy(simu_energy, reco_energy, percentile=68.27, confidence_level=0.95, bias_correction=False)
Parameters:
  • simu_energy (1d numpy array of simulated energies) –
  • reco_energy (1d numpy array of reconstructed energies) –
Returns:

(e, e_res)
Return type:

tuple of 1d numpy arrays - energy, resolution in energy
ctaplot.ana.bias_per_energy(simu, reco, energy, relative_scaling_method=None)

Bias between simu and reco per bins of energy

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
  • energy (: numpy.ndarray) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
Returns:

bins, bias
Return type:

numpy.ndarray, numpy.ndarray
ctaplot.ana.resolution_per_bin(x, y_true, y_reco, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method=None, bins=10)

Resolution of y as a function of binned x.

Parameters:
  • x (numpy.ndarray) –
  • y_true (numpy.ndarray) –
  • y_reco (numpy.ndarray) –
  • percentile (float) –
  • confidence_level (float) –
  • bias_correction (bool) –
  • relative_scaling_method (see ctaplot.ana.relative_scaling) –
  • bins (int or numpy.ndarray (see numpy.histogram)) –
Returns:

(x_bins, res) – x_bins: bins for x res: resolutions with confidence level intervals for each bin
Return type:

(numpy.ndarray, numpy.ndarray)
ctaplot.ana.resolution(simu, reco, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method='s1')

Compute the resolution of reco as the Qth (68.27 as standard = 1 sigma) containment radius of (simu-reco)/relative_scaling with the lower and upper confidence limits defined the values inside

the error_percentile
Parameters:
  • simu (numpy.ndarray (1d)) – simulated quantity
  • reco (numpy.ndarray (1d)) – reconstructed quantity
  • percentile (float) – percentile for the resolution containment radius
  • error_percentile (float) – percentile for the confidence limits
  • bias_correction (bool) – if True, the resolution is corrected with the bias computed on simu and reco
  • relative_scaling (str) – see ctaplot.ana.relative_scaling
Returns:
Return type:

numpy.ndarray - [resolution, lower_confidence_limit, upper_confidence_limit]
ctaplot.ana.resolution_per_energy(simu, reco, simu_energy, percentile=68.27, confidence_level=0.95, bias_correction=False)
Parameters:
  • simu (1d numpy.ndarray of simulated energies) –
  • reco (1d numpy.ndarray of reconstructed energies) –
Returns:

energy_bins - 1D numpy.ndarray resolution: - 3D numpy.ndarray see ctaplot.ana.resolution
Return type:

(energy_bins, resolution)
ctaplot.ana.impact_resolution_per_energy(reco_x, reco_y, simu_x, simu_y, energy, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method=None)

Plot the angular resolution as a function of the event simulated energy

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • energy (numpy.ndarray) –
  • percentile (float) – see ctaplot.ana.resolution
  • confidence_level (float) – see ctaplot.ana.resolution
  • bias_correction (bool) – see ctaplot.ana.resolution
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
Returns:

(energy, resolution)
Return type:

(1d numpy array, 1d numpy array)
ctaplot.ana.impact_parameter_error(reco_x, reco_y, simu_x, simu_y)

compute the error distance between simulated and reconstructed impact parameters :param reco_x: :type reco_x: 1d numpy array :param reco_y: :type reco_y: 1d numpy array :param simu_x: :type simu_x: 1d numpy array :param simu_y: :type simu_y: 1d numpy array

Returns:1d numpy array
Return type:distances
ctaplot.ana.impact_resolution(reco_x, reco_y, simu_x, simu_y, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method=None)

Compute the shower impact parameter resolution as the Qth (68 as standard) containment radius of the square distance to the simulated one with the lower and upper limits corresponding to the required confidence level

Parameters:
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • percentile (float) – see ctaplot.ana.resolution
  • confidence_level (float) – see ctaplot.ana.resolution
  • bias_correction (bool) – see ctaplot.ana.resolution
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
Returns:

(impact_resolution, lower_confidence_level, upper_confidence_level)
Return type:

(numpy.array, numpy.array, numpy.array)
ctaplot.ana.distance2d_resolution(reco_x, reco_y, simu_x, simu_y, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method=None)

Compute the 2D distance resolution as the Qth (standard being 68) containment radius of the relative distance with lower and upper limits on this value corresponding to the confidence value required (1.645 for 95% confidence)

Parameters:
  • reco_x (numpy.ndarray - reconstructed x position) –
  • reco_y (numpy.ndarray - reconstructed y position) –
  • simu_x (numpy.ndarray - true x position) –
  • simu_y (numpy.ndarray - true y position) –
  • percentile (float - percentile, 68.27 corresponds to one sigma) –
  • confidence_level (float) –
  • bias_correction (bool) –
  • relative_scaling_method (str) –
    • see ctaplot.ana.relative_scaling
Returns:
Return type:

numpy.array [angular_resolution, lower limit, upper limit]
ctaplot.ana.distance2d_resolution_per_bin(x, reco_x, reco_y, simu_x, simu_y, bins=10, percentile=68.27, confidence_level=0.95, bias_correction=False, relative_scaling_method=None)

Compute the 2D distance per bin of x

Parameters:
  • x (numpy.ndarray) –
  • reco_x (numpy.ndarray) –
  • reco_y (numpy.ndarray) –
  • simu_x (numpy.ndarray) –
  • simu_y (numpy.ndarray) –
  • bins (bins args of np.histogram) –
  • percentile (float - percentile, 68.27 corresponds to one sigma) –
  • confidence_level (float) –
  • bias_correction (bool) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
Returns:
Return type:

x_bins, distance_res
ctaplot.ana.power_law_integrated_distribution(xmin, xmax, total_number_events, spectral_index, bins)

For each bin, return the expected number of events for a power-law distribution. bins: numpy.ndarray, e.g. np.logspace(np.log10(emin), np.logspace(xmax))

Parameters:
  • xmin (float, min of the simulated power-law) –
  • xmax (float, max of the simulated power-law) –
  • total_number_events (int) –
  • spectral_index (float) –
  • bins (numpy.ndarray) –
Returns:

y
Return type:

numpy.ndarray, len(y) = len(bins) - 1
ctaplot.ana.effective_area(simu_energy, reco_energy, simu_area)

Compute the effective area from a list of simulated energies and reconstructed energies :param simu_energy: :type simu_energy: 1d numpy array :param reco_energy: :type reco_energy: 1d numpy array :param simu_area: :type simu_area: float - area on which events are simulated

Returns:
Return type:float = effective area
ctaplot.ana.effective_area_per_energy(simu_energy, reco_energy, simu_area)

Compute the effective area per energy bins from a list of simulated energies and reconstructed energies

Parameters:
  • simu_energy (1d numpy array) –
  • reco_energy (1d numpy array) –
  • simu_area (float - area on which events are simulated) –
Returns:

(E, Seff)
Return type:

(1d numpy array, 1d numpy array)
ctaplot.ana.effective_area_per_energy_power_law(emin, emax, total_number_events, spectral_index, reco_energy, simu_area)

Compute the effective area per energy bins from a list of simulated energies and reconstructed energies

Parameters:
  • emin (float) –
  • emax (float) –
  • total_number_events (int) –
  • spectral_index (float) –
  • reco_energy (1d numpy array) –
  • simu_area (float - area on which events are simulated) –
Returns:

(energy, effective_area)
Return type:

(1d numpy array, 1d numpy array)
ctaplot.ana.bias_per_bin(simu, reco, x, relative_scaling_method=None, bins=10)

Bias between simu and reco per bin of x.

Parameters:
  • simu (numpy.ndarray) –
  • reco (numpy.ndarray) –
  • x (: numpy.ndarray) –
  • relative_scaling_method (str) – see ctaplot.ana.relative_scaling
  • bins (bins for numpy.histogram) –
Returns:

bins, bias
Return type:

numpy.ndarray, numpy.ndarray
ctaplot.ana.percentile_confidence_interval(x, percentile=68, confidence_level=0.95)

Return the confidence interval for the qth percentile of x for a given confidence level

REF: http://people.stat.sfu.ca/~cschwarz/Stat-650/Notes/PDF/ChapterPercentiles.pdf S. Chakraborti and J. Li, Confidence Interval Estimation of a Normal Percentile, doi:10.1198/000313007X244457

Parameters:
  • x (numpy.ndarray) –
  • percentile (float) – 0 < percentile < 100
  • confidence_level (float) – 0 < confidence level (by default 95%) < 1
ctaplot.ana.logbin_mean(x_bin)

Function that gives back the mean of each bin in logscale

Parameters:x_bin (numpy.ndarray) –
Returns:
Return type:numpy.ndarray
ctaplot.ana.binned_statistic(x, values, statistic='mean', bins=10, range=None)

Compute a binned statistic for one or more sets of data.

This is a generalization of a histogram function. A histogram divides the space into bins, and returns the count of the number of points in each bin. This function allows the computation of the sum, mean, median, or other statistic of the values (or set of values) within each bin.

Parameters:
  • x ((N,) array_like) – A sequence of values to be binned.
  • values ((N,) array_like or list of (N,) array_like) – The data on which the statistic will be computed. This must be the same shape as x, or a set of sequences - each the same shape as x. If values is a set of sequences, the statistic will be computed on each independently.
  • statistic (string or callable, optional) –

    The statistic to compute (default is ‘mean’). The following statistics are available:

    • ’mean’ : compute the mean of values for points within each bin. Empty bins will be represented by NaN.
    • ’std’ : compute the standard deviation within each bin. This is implicitly calculated with ddof=0.
    • ’median’ : compute the median of values for points within each bin. Empty bins will be represented by NaN.
    • ’count’ : compute the count of points within each bin. This is identical to an unweighted histogram. values array is not referenced.
    • ’sum’ : compute the sum of values for points within each bin. This is identical to a weighted histogram.
    • ’min’ : compute the minimum of values for points within each bin. Empty bins will be represented by NaN.
    • ’max’ : compute the maximum of values for point within each bin. Empty bins will be represented by NaN.
    • function : a user-defined function which takes a 1D array of values, and outputs a single numerical statistic. This function will be called on the values in each bin. Empty bins will be represented by function([]), or NaN if this returns an error.
  • bins (int or sequence of scalars, optional) – If bins is an int, it defines the number of equal-width bins in the given range (10 by default). If bins is a sequence, it defines the bin edges, including the rightmost edge, allowing for non-uniform bin widths. Values in x that are smaller than lowest bin edge are assigned to bin number 0, values beyond the highest bin are assigned to bins[-1]. If the bin edges are specified, the number of bins will be, (nx = len(bins)-1).
  • range ((float, float) or [(float, float)], optional) – The lower and upper range of the bins. If not provided, range is simply (x.min(), x.max()). Values outside the range are ignored.
Returns:

  • statistic (array) – The values of the selected statistic in each bin.
  • bin_edges (array of dtype float) – Return the bin edges (length(statistic)+1).
  • binnumber (1-D ndarray of ints) – Indices of the bins (corresponding to bin_edges) in which each value of x belongs. Same length as values. A binnumber of i means the corresponding value is between (bin_edges[i-1], bin_edges[i]).

See also

numpy.digitize(), numpy.histogram(), binned_statistic_2d(), binned_statistic_dd()

Notes

All but the last (righthand-most) bin is half-open. In other words, if bins is [1, 2, 3, 4], then the first bin is [1, 2) (including 1, but excluding 2) and the second [2, 3). The last bin, however, is [3, 4], which includes 4.

New in version 0.11.0.

Examples

>>> from scipy import stats
>>> import matplotlib.pyplot as plt

First some basic examples:

Create two evenly spaced bins in the range of the given sample, and sum the corresponding values in each of those bins:

>>> values = [1.0, 1.0, 2.0, 1.5, 3.0]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
BinnedStatisticResult(statistic=array([4. , 4.5]), bin_edges=array([1., 4., 7.]), binnumber=array([1, 1, 1, 2, 2]))

Multiple arrays of values can also be passed. The statistic is calculated on each set independently:

>>> values = [[1.0, 1.0, 2.0, 1.5, 3.0], [2.0, 2.0, 4.0, 3.0, 6.0]]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
BinnedStatisticResult(statistic=array([[4. , 4.5],
       [8. , 9. ]]), bin_edges=array([1., 4., 7.]), binnumber=array([1, 1, 1, 2, 2]))

>>> stats.binned_statistic([1, 2, 1, 2, 4], np.arange(5), statistic='mean',
...                        bins=3)
BinnedStatisticResult(statistic=array([1., 2., 4.]), bin_edges=array([1., 2., 3., 4.]), binnumber=array([1, 2, 1, 2, 3]))

As a second example, we now generate some random data of sailing boat speed as a function of wind speed, and then determine how fast our boat is for certain wind speeds:

>>> windspeed = 8 * np.random.rand(500)
>>> boatspeed = .3 * windspeed**.5 + .2 * np.random.rand(500)
>>> bin_means, bin_edges, binnumber = stats.binned_statistic(windspeed,
...                 boatspeed, statistic='median', bins=[1,2,3,4,5,6,7])
>>> plt.figure()
>>> plt.plot(windspeed, boatspeed, 'b.', label='raw data')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=5,
...            label='binned statistic of data')
>>> plt.legend()

Now we can use binnumber to select all datapoints with a windspeed below 1:

>>> low_boatspeed = boatspeed[binnumber == 0]

As a final example, we will use bin_edges and binnumber to make a plot of a distribution that shows the mean and distribution around that mean per bin, on top of a regular histogram and the probability distribution function:

>>> x = np.linspace(0, 5, num=500)
>>> x_pdf = stats.maxwell.pdf(x)
>>> samples = stats.maxwell.rvs(size=10000)

>>> bin_means, bin_edges, binnumber = stats.binned_statistic(x, x_pdf,
...         statistic='mean', bins=25)
>>> bin_width = (bin_edges[1] - bin_edges[0])
>>> bin_centers = bin_edges[1:] - bin_width/2

>>> plt.figure()
>>> plt.hist(samples, bins=50, density=True, histtype='stepfilled',
...          alpha=0.2, label='histogram of data')
>>> plt.plot(x, x_pdf, 'r-', label='analytical pdf')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=2,
...            label='binned statistic of data')
>>> plt.plot((binnumber - 0.5) * bin_width, x_pdf, 'g.', alpha=0.5)
>>> plt.legend(fontsize=10)
>>> plt.show()

Examples

How to easily plot CTA IRF requirements and performances

CTA performances are up-to-date and public and can be found on the cta-observatory website

[1]:

import ctaplot
from ctaplot.dataset import get
import numpy as np
import matplotlib.pyplot as plt
import matplotlib

%matplotlib inline
font = {'size'   : 20}
matplotlib.rc('font', **font)
Angular resolution
[3]:

fig, ax = plt.subplots(figsize=(12,8))
ax = ctaplot.plot_angular_res_requirement('north', ax=ax, linewidth=3)
ax = ctaplot.plot_angular_res_cta_performance('north', ax=ax, marker='o')
ax = ctaplot.plot_angular_res_requirement('south', ax=ax,  linewidth=3)
ax = ctaplot.plot_angular_res_cta_performance('south', ax=ax, marker='o')
ax.grid()
plt.legend(prop = font);
_images/notebooks_plot_cta_irf_4_0.png
Energy resolution
[5]:

fig, ax = plt.subplots(figsize=(12,8))
ax = ctaplot.plot_energy_resolution_requirements('north', ax=ax, linewidth=3)
ax = ctaplot.plot_energy_resolution_cta_performances('north', ax=ax, marker='o')
ax = ctaplot.plot_energy_resolution_requirements('south', ax=ax,  linewidth=3)
ax = ctaplot.plot_energy_resolution_cta_performances('south', ax=ax, marker='o')
ax.grid()
plt.legend(prop = font);
_images/notebooks_plot_cta_irf_6_0.png
Effective Area
[7]:

fig, ax = plt.subplots(figsize=(12,8))
ax = ctaplot.plot_effective_area_requirement('north', ax=ax, linewidth=3)
ax = ctaplot.plot_effective_area_performances('north', ax=ax, marker='o')
ax = ctaplot.plot_effective_area_requirement('south', ax=ax,  linewidth=3)
ax = ctaplot.plot_effective_area_performances('south', ax=ax, marker='o')
ax.grid()
plt.legend(prop = font);
_images/notebooks_plot_cta_irf_8_0.png
Sensitivity
[8]:

fig, ax = plt.subplots(figsize=(12,8))
ax = ctaplot.plot_sensitivity_requirement('north', ax=ax, linewidth=3)
ax = ctaplot.plot_sensitivity_performances('north', ax=ax, marker='o')
ax = ctaplot.plot_sensitivity_requirement('south', ax=ax,  linewidth=3)
ax = ctaplot.plot_sensitivity_performances('south', ax=ax, marker='o')
ax.set_ylabel(r'Flux Sensitivity $[erg.cm^{-2}.s^{-1}]$')
ax.grid()
plt.legend(prop = font);
_images/notebooks_plot_cta_irf_10_0.png 
[ ]:
Sub-arrays
[10]:

lst_north_angres_requirements = np.loadtxt(get('cta_requirements_North-50h-LST-AngRes.dat'))

[11]:

fig, ax = plt.subplots(figsize=(12,8))
ax = ctaplot.plot_angular_res_requirement('north', ax=ax, linewidth=3)
ax = ctaplot.plot_angular_res_cta_performance('north', ax=ax, marker='o')
ax.scatter(lst_north_angres_requirements[:,0], lst_north_angres_requirements[:,1],
           label="LST North requirements",
           color='red')

ax.grid()
plt.legend(prop = font);
_images/notebooks_plot_cta_irf_14_0.png 
[ ]:

How is resolution computed

Index
[1]:

import numpy as np
import matplotlib.pyplot as plt
from ctaplot.ana import resolution, relative_scaling
Normal distribution
For a nomal distribution, \sigma corresponds to the 68 percentile of the distribution
See the 68–95–99.7 rule
[2]:

loc = 10
scale = 3

X = np.random.normal(size=1000000, scale=scale, loc=loc)
plt.hist(np.abs(X - loc), bins=80, density=True)

sig_68 = np.percentile(np.abs(X - loc), 68.27)
sig_95 = np.percentile(np.abs(X - loc), 95.45)
sig_99 = np.percentile(np.abs(X - loc), 99.73)

plt.vlines(sig_68, 0, 0.3, label='68%', color='red')
plt.vlines(sig_95, 0, 0.3, label='95%', color='green')
plt.vlines(sig_99, 0, 0.3, label='99%', color='yellow')
plt.ylim(0,0.3)
plt.legend()


print("68th percentile = {:.4f}".format(sig_68))
print("95th percentile = {:.4f}".format(sig_95))
print("99th percentile = {:.4f}".format(sig_99))

assert np.isclose(sig_68, scale, rtol=1e-2)
assert np.isclose(sig_95, 2 * scale, rtol=1e-2)
assert np.isclose(sig_99, 3 * scale, rtol=1e-2)

68th percentile = 3.0016
95th percentile = 5.9941
99th percentile = 8.9812
_images/notebooks_resolution_definition_4_1.png
Resolution
The resolution is defined as the 68th percentile of the relative error err = (reco - simu)/scaling.
(see the relative scaling section for more info on it).
Hence, if the relative error follows a normal distribution, the resolution is equal to the sigma of the distribution
[3]:

simu = loc * np.ones(X.shape[0])
reco = np.random.normal(loc=loc, scale=scale, size=X.shape[0])

err = np.abs(simu - reco)

fig, axes = plt.subplots(1, 2, figsize=(20,7))
axes[0].hist(reco, bins=80, label='reco')
axes[0].axvline(loc, 0, 1, color='red', label='simu')
axes[0].legend()
axes[1].hist(err, bins=80)
axes[1].set_title('err')

[3]:

Text(0.5, 1.0, 'err')
_images/notebooks_resolution_definition_6_1.png
Let’s define reco in order to have a relative error equals to err.
Its resolution is equals to the sigma of the distribution
[4]:

res = resolution(simu, reco, relative_scaling_method=None)
print(res)
assert np.isclose(res[0], scale, rtol=1e-2)

[3.00210261 2.99740973 3.0068767 ]
Relative scaling

The resolution can be measured on the absolute error or on the relative one.

err = (reco - simu)/scaling

There is no absolute definition of the relative scaling and several are proposed in ctaplot.
The choice can be passed through the relative_scaling_method:
Methods:
- s0 : no scaling (scaling = 1)
- s1 : $ scaling = |simu| $
- s2 : $ scaling = |reco| $
- s3 : $ scaling = (|simu| + |reco|)/2 $
- s4 : $ scaling = max(|reco|, |simu|) $

The default one for the resolution is s1.

Note that methods involving reco are more subject to deviation from the expected result:

[5]:

for method in ['s1', 's2', 's3', 's4']:
    res = resolution(simu, reco, relative_scaling_method=method)
    print("Method {} gives a resolution = {:.3f} to be compared with the expected value = {}".format(method, res[0], scale/loc))

Method s1 gives a resolution = 0.300 to be compared with the expected value = 0.3
Method s2 gives a resolution = 0.288 to be compared with the expected value = 0.3
Method s3 gives a resolution = 0.297 to be compared with the expected value = 0.3
Method s4 gives a resolution = 0.258 to be compared with the expected value = 0.3
NB:
- the angular resolution uses no scaling
- the energy resolution uses scaling s1
- the impact resolution uses no scaling by default
Error bars
Errors bars in resolution plots are given by the confidence interval (by default at 95%, equivalent to 2 sigmas for a normal distribution).
This means that we can be confident at 95% that the resolution values are within the range given by the error bars.

The implementation for percentile confidence interval follows: - http://people.stat.sfu.ca/~cschwarz/Stat-650/Notes/PDF/ChapterPercentiles.pdf

Example with a normal distribution:

[6]:

nbins, bins, patches = plt.hist(X, bins=100, density=True)
ymax = 1.1 * nbins.max()
plt.ylim(0, ymax)
plt.vlines(np.percentile(X, 68.27), 0, ymax, color='red')
plt.show()
print("The true 68th percentile of this distribution is: {:.4f}".format(np.percentile(X, 68.27)))
_images/notebooks_resolution_definition_16_0.png 
The true 68th percentile of this distribution is: 11.4245
We can consider this as our underlying (infinite) distribution.
If we take a random sample in this ditribution, the measurement of a percentile will come with a measurement error. We can assess this error by taking multiple random samples and taking the distribution of measured percentile values.
[7]:

n = 1000
p = 0.6827
all_68 = []
for i in range(int(len(X)/n)):
    all_68.append(np.percentile(X[i*n:(i+1)*n], p*100))

all_68 = np.array(all_68)
nbins, bins, patches = plt.hist(all_68, density=True);
ymax = 1.1 * nbins.max()
plt.vlines(all_68.mean(), 0, ymax)
plt.vlines(all_68.mean() + all_68.std(), 0, ymax, color='red')
plt.vlines(all_68.mean() - all_68.std(), 0, ymax, color='red')
plt.ylim(0, ymax)
plt.title("Distribution of the measured 68th percentile")

print("Standard deviation = {:.5f}".format(all_68.std()))

Standard deviation = 0.12287
_images/notebooks_resolution_definition_18_1.png
To evaluate directly the confidence interval from a sub-sample of the distribution, one can use the following formulae:

R_{low} = n*p − z * \sqrt{n * p * (1 − p)}

R_{up} = n*p + z * \sqrt{n * p * (1 − p)}

with p the percentile and z the confidence level desired.

And the confidence interval given by: (X[R_{low}], X[R_{up}])

The confidence level is given by the cumulative distribution function (scipy.stats.norm.ppf).
Some useful values:
  • z = 0.47 for a confidence level of 68%
  • z = 1.645 for a confidence level of 95%
  • z = 2.33 for a confidence level of 99%
[8]:

# confidence level:
z = 2.33

# sub-sample:
x = X[:n]

rl = int(n * p - z * np.sqrt(n * p * (1-p)))
ru = int(n * p + z * np.sqrt(n * p * (1-p)))
print("Measured percentile: {:.4f}".format(np.percentile(x, p*100)))
print("Confidence interval: ({:.4f}, {:.4f})".format(np.sort(x)[rl], np.sort(x)[ru]))
print("To be compared with: ({:.4f}, {:.4f})".format(all_68.mean()-all_68.std()*3, all_68.mean()+all_68.std()*3))
print("which is the corresponding confidence interval given by the normal distribution of the measured percentiles")

Measured percentile: 11.4301
Confidence interval: (11.1279, 11.7104)
To be compared with: (11.0539, 11.7911)
which is the corresponding confidence interval given by the normal distribution of the measured percentiles

In ctaplot, this is computed by the function percentile_confidence interval:

[9]:

from ctaplot.ana import percentile_confidence_interval

p = 68.27
confidence_level = 0.99
pci = percentile_confidence_interval(x, percentile=p, confidence_level=0.99)
print("68th percentile: {:.3f}".format(np.percentile(x, p)))
print("Interval with a confidence level of {}%: ({:.3f}, {:.3f})".format(confidence_level*100, pci[0], pci[1]))


plt.figure(figsize=(12,7))
nbins, bins, patches = plt.hist(x, bins=100, density=True);
ymax = 1.1 * nbins.max()

plt.vlines(np.percentile(x, 50), 0, ymax, color='black')
plt.vlines(np.percentile(x, p), 0, ymax, color='red')
plt.vlines(pci[0], 0, ymax, linestyles='--', color='red')
plt.vlines(pci[1], 0, ymax, linestyles='--', color='red',)
plt.ylim(0, ymax);

68th percentile: 11.430
Interval with a confidence level of 99.0%: (11.128, 11.710)
_images/notebooks_resolution_definition_22_1.png
Note: The same method could be applied around the median.
In this case, the confidence interval is also given by -\sigma/\sqrt{n} for a normal distribution.
[10]:

pci = percentile_confidence_interval(X, percentile=50, confidence_level=0.99)
print("Median: {:.5f}".format(np.median(X)))
print("Confidence interval at 99%: {}".format(pci))
print("To be compared with: ({}, {})".format(np.median(X)-3*scale/np.sqrt(len(X)), np.median(X)+3*scale/np.sqrt(len(X))))

Median: 10.00197
Confidence interval at 99%: (9.992948666358563, 10.01069362511663)
To be compared with: (9.992972062483608, 10.010972062483608)