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Modeling and fitting

Prerequisites

Proposed approach

This is a hands-on tutorial to gammapy.modeling, showing how the model, dataset and fit classes work together. As an example we are going to work with HESS data of the Crab Nebula and show in particular how to : - perform a spectral analysis - use different fitting backends - acces covariance matrix informations and parameter errors - compute likelihood profile - compute confidence contours

See also: Models gallery tutorial and docs/modeling/index.rst.

The setup

[1]:
import numpy as np
from astropy import units as u
import matplotlib.pyplot as plt
from gammapy.modeling import Fit
from gammapy.datasets import Datasets, SpectrumDatasetOnOff
from gammapy.modeling.models import LogParabolaSpectralModel, SkyModel
from gammapy.visualization.utils import plot_contour_line
from itertools import combinations

Model and dataset

First we define the source model, here we need only a spectral model for which we choose a log-parabola

[2]:
crab_spectrum = LogParabolaSpectralModel(
    amplitude=1e-11 / u.cm ** 2 / u.s / u.TeV,
    reference=1 * u.TeV,
    alpha=2.3,
    beta=0.2,
)

crab_spectrum.alpha.max = 3
crab_spectrum.alpha.min = 1
crab_model = SkyModel(spectral_model=crab_spectrum, name="crab")

The data and background are read from pre-computed ON/OFF datasets of HESS observations, for simplicity we stack them together. Then we set the model and fit range to the resulting dataset.

[3]:
datasets = []
for obs_id in [23523, 23526]:
    dataset = SpectrumDatasetOnOff.from_ogip_files(
        f"$GAMMAPY_DATA/joint-crab/spectra/hess/pha_obs{obs_id}.fits"
    )
    datasets.append(dataset)

dataset_hess = Datasets(datasets).stack_reduce(name="HESS")

# Set model and fit range
dataset_hess.models = crab_model
e_min = 0.66 * u.TeV
e_max = 30 * u.TeV
dataset_hess.mask_fit = dataset_hess.counts.geom.energy_mask(e_min, e_max)
No background model defined for dataset 23523
No background model defined for dataset 23526
No background model defined for dataset HESS
No background model defined for dataset HESS

Fitting options

First let’s create a Fit instance:

[4]:
fit = Fit([dataset_hess])

By default the fit is performed using MINUIT, you can select alternative optimizers and set their option using the optimize_opts argument of the Fit.run() method.

Note that, for now, covaraince matrix and errors are computed only for the fitting with MINUIT. However depending on the problem other optimizers can better perform, so somethimes it can be usefull to run a pre-fit with alternative optimization methods.

For the “scipy” backend the available options are desribed in detail here:
[5]:
%%time
scipy_opts = {"method": "L-BFGS-B", "options": {"ftol": 1e-4, "gtol": 1e-05}}
result_scipy = fit.run(backend="scipy", optimize_opts=scipy_opts)
No covariance estimate - not supported by this backend.
CPU times: user 532 ms, sys: 3.77 ms, total: 536 ms
Wall time: 530 ms
For the “sherpa” backend you can choose the optimization algorithm between method = {“simplex”, “levmar”, “moncar”, “gridsearch”}.
Those methods are described and compared in detail on http://cxc.cfa.harvard.edu/sherpa/methods/index.html.
The available options of the optimization methods are described on the following page https://cxc.cfa.harvard.edu/sherpa/methods/opt_methods.html
[6]:
%%time
sherpa_opts = {"method": "simplex", "ftol": 1e-3, "maxfev": int(1e4)}
results_simplex = fit.run(backend="sherpa", optimize_opts=sherpa_opts)
No covariance estimate - not supported by this backend.
CPU times: user 602 ms, sys: 3.93 ms, total: 606 ms
Wall time: 604 ms

For the “minuit” backend see https://iminuit.readthedocs.io/en/latest/reference.html for a detailed description of the available options. If there is an entry ‘migrad_opts’, those options will be passed to iminuit.Minuit.migrad. Additionnaly you can set the fit tolerance using the tol option. The minimization will stop when the estimated distance to the minimum is less than 0.001*tol (by default tol=0.1). The strategy option change the speed and accuracy of the optimizer: 0 fast, 1 default, 2 slow but accurate. If you want more reliable error estimates, you should run the final fit with strategy 2.

[7]:
%%time
minuit_opts = {"tol": 0.001, "strategy": 1}
result_minuit = fit.run(backend="minuit", optimize_opts=minuit_opts)
CPU times: user 185 ms, sys: 0 ns, total: 185 ms
Wall time: 184 ms

Fit quality assessment

There are various ways to check the convergence and quality of a fit. Among them:

  • Refer to the automatically-generated results dictionary

[8]:
print(result_scipy)
OptimizeResult

        backend    : scipy
        method     : L-BFGS-B
        success    : True
        message    : b'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
        nfev       : 148
        total stat : 75.60

[9]:
print(results_simplex)
OptimizeResult

        backend    : sherpa
        method     : simplex
        success    : True
        message    : Optimization terminated successfully
        nfev       : 170
        total stat : 30.35

[10]:
print(result_minuit)
OptimizeResult

        backend    : minuit
        method     : minuit
        success    : True
        message    : Optimization terminated successfully.
        nfev       : 52
        total stat : 30.35

  • Check that the fitted values and errors for all parameters are reasonable, and no fitted parameter value is “too close” - or even outside - its allowed min-max range

[11]:
result_minuit.parameters.to_table()
[11]:
Table length=4
namevalueunitminmaxfrozenerror
str9float64str14float64float64boolfloat64
amplitude3.814e-11cm-2 s-1 TeV-1nannanFalse3.140e-12
reference1.000e+00TeVnannanTrue0.000e+00
alpha2.196e+001.000e+003.000e+00False1.122e-01
beta2.265e-01nannanFalse6.962e-02
[12]:
total_stat = result_minuit.total_stat

for par in dataset_hess.models.parameters:
    if par.frozen is False:
        profile = fit.stat_profile(parameter=par)
        plt.plot(profile["values"], profile["stat"] - total_stat)
        plt.xlabel(f"{par.unit}")
        plt.ylabel("Delta TS")
        plt.title(f"{par.name}: {par.value} +- {par.error}")
        plt.show()
        plt.close()
../_images/tutorials_modeling_22_0.png
../_images/tutorials_modeling_22_1.png
../_images/tutorials_modeling_22_2.png
  • Inspect model residuals. Those can always be accessed using ~Dataset.residuals(), that will return an array in case a the fitted Dataset is a SpectrumDataset and a full cube in case of a MapDataset. For more details, we refer here to the dedicated fitting tutorials: analysis_3d.ipynb (for MapDataset fitting) and spectrum_analysis.ipynb (for SpectrumDataset fitting).

Covariance and parameters errors

After the fit the covariance matrix is attached to the model. You can get the error on a specific parameter by accessing the .error attribute:

[13]:
crab_model.spectral_model.alpha.error
[13]:
0.11220813559921301

As an example, this step is needed to produce a butterfly plot showing the envelope of the model taking into account parameter uncertainties.

[14]:
energy_range = [1, 10] * u.TeV
crab_spectrum.plot(energy_range=energy_range, energy_power=2)
ax = crab_spectrum.plot_error(energy_range=energy_range, energy_power=2)
/home/runner/work/gammapy-docs/gammapy-docs/gammapy/gammapy/modeling/models/spectral.py:323: MatplotlibDeprecationWarning: The 'nonposx' parameter of __init__() has been renamed 'nonpositive' since Matplotlib 3.3; support for the old name will be dropped two minor releases later.
  ax.set_xscale("log", nonposx="clip")
/home/runner/work/gammapy-docs/gammapy-docs/gammapy/gammapy/modeling/models/spectral.py:324: MatplotlibDeprecationWarning: The 'nonposy' parameter of __init__() has been renamed 'nonpositive' since Matplotlib 3.3; support for the old name will be dropped two minor releases later.
  ax.set_yscale("log", nonposy="clip")
../_images/tutorials_modeling_27_1.png

Confidence contours

In most studies, one wishes to estimate parameters distribution using observed sample data. A confidence interval gives an estimated range of values which is likely to include an unknown parameter. The selection of a confidence level for an interval determines the probability that the confidence interval produced will contain the true parameter value. A confidence contour is a 2D generalization of a confidence interval, often represented as an ellipsoid around the best-fit value.

After the fit, MINUIT offers the possibility to compute the confidence confours. gammapy provides an interface to this functionnality throught the Fit object using the minos_contour method. Here we defined a function to automatize the contour production for the differents parameterer and confidence levels (expressed in term of sigma):

[15]:
def make_contours(fit, result, npoints, sigmas):
    cts_sigma = []
    for sigma in sigmas:
        contours = dict()
        for par_1, par_2 in combinations(["alpha", "beta", "amplitude"], r=2):
            contour = fit.minos_contour(
                result.parameters[par_1],
                result.parameters[par_2],
                numpoints=npoints,
                sigma=sigma,
            )
            contours[f"contour_{par_1}_{par_2}"] = {
                par_1: contour["x"].tolist(),
                par_2: contour["y"].tolist(),
            }
        cts_sigma.append(contours)
    return cts_sigma

Now we can compute few contours.

[16]:
%%time
sigma = [1, 2]
cts_sigma = make_contours(fit, result_minuit, 10, sigma)
CPU times: user 15.9 s, sys: 2.3 ms, total: 15.9 s
Wall time: 15.9 s

Then we prepare some aliases and annotations in order to make the plotting nicer.

[17]:
pars = {
    "phi": r"$\phi_0 \,/\,(10^{-11}\,{\rm TeV}^{-1} \, {\rm cm}^{-2} {\rm s}^{-1})$",
    "alpha": r"$\alpha$",
    "beta": r"$\beta$",
}

panels = [
    {
        "x": "alpha",
        "y": "phi",
        "cx": (lambda ct: ct["contour_alpha_amplitude"]["alpha"]),
        "cy": (
            lambda ct: np.array(1e11)
            * ct["contour_alpha_amplitude"]["amplitude"]
        ),
    },
    {
        "x": "beta",
        "y": "phi",
        "cx": (lambda ct: ct["contour_beta_amplitude"]["beta"]),
        "cy": (
            lambda ct: np.array(1e11)
            * ct["contour_beta_amplitude"]["amplitude"]
        ),
    },
    {
        "x": "alpha",
        "y": "beta",
        "cx": (lambda ct: ct["contour_alpha_beta"]["alpha"]),
        "cy": (lambda ct: ct["contour_alpha_beta"]["beta"]),
    },
]

Finally we produce the confidence contours figures.

[18]:
fig, axes = plt.subplots(1, 3, figsize=(16, 5))
colors = ["m", "b", "c"]
for p, ax in zip(panels, axes):
    xlabel = pars[p["x"]]
    ylabel = pars[p["y"]]
    for ks in range(len(cts_sigma)):
        plot_contour_line(
            ax,
            p["cx"](cts_sigma[ks]),
            p["cy"](cts_sigma[ks]),
            lw=2.5,
            color=colors[ks],
            label=f"{sigma[ks]}" + r"$\sigma$",
        )
    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
plt.legend()
plt.tight_layout()
../_images/tutorials_modeling_35_0.png
[19]: