<jupyter_start><jupyter_text>Low level API Prerequisites- Understanding the gammapy data workflow, in particular what are DL3 events and instrument response functions (IRF).- Understanding of the data reduction and modeling fitting process as shown in the [analysis with the high level interface tutorial](analysis_1.ipynb) ContextThis notebook is an introduction to gammapy analysis this time using the lower level classes and functionsthe library.This allows to understand what happens during two main gammapy analysis steps, data reduction and modeling/fitting. **Objective: Create a 3D dataset of the Crab using the H.E.S.S. DL3 data release 1 and perform a simple model fitting of the Crab nebula using the lower level gammapy API.** Proposed approachHere, we have to interact with the data archive (with the `~gammapy.data.DataStore`) to retrieve a list of selected observations (`~gammapy.data.Observations`). Then, we define the geometry of the `~gammapy.datasets.MapDataset` object we want to produce and the maker object that reduce an observationto a dataset. We can then proceed with data reduction with a loop over all selected observations to produce datasets in the relevant geometry and stack them together (i.e. sum them all).In practice, we have to:- Create a `~gammapy.data.DataStore` poiting to the relevant data - Apply an observation selection to produce a list of observations, a `~gammapy.data.Observations` object.- Define a geometry of the Map we want to produce, with a sky projection and an energy range.    - Create a `~gammapy.maps.MapAxis` for the energy    - Create a `~gammapy.maps.WcsGeom` for the geometry- Create the necessary makers :     - the map dataset maker : `~gammapy.makers.MapDatasetMaker`    - the background normalization maker, here a `~gammapy.makers.FoVBackgroundMaker`    - and usually the safe range maker : `~gammapy.makers.SafeRangeMaker`- Perform the data reduction loop. And for every observation:    - Apply the makers sequentially to produce the current `~gammapy.datasets.MapDataset`    - Stack it on the target one.- Define the `~gammapy.modeling.models.SkyModel` to apply to the dataset.- Create a `~gammapy.modeling.Fit` object and run it to fit the model parameters- Apply a `~gammapy.estimators.FluxPointsEstimator` to compute flux points for the spectral part of the fit. SetupFirst, we setup the analysis by performing required imports.<jupyter_code>%matplotlib inline
import matplotlib.pyplot as plt
from pathlib import Path
from astropy import units as u
from astropy.coordinates import SkyCoord
from regions import CircleSkyRegion
from gammapy.data import DataStore
from gammapy.datasets import MapDataset
from gammapy.maps import WcsGeom, MapAxis
from gammapy.makers import MapDatasetMaker, SafeMaskMaker, FoVBackgroundMaker
from gammapy.modeling.models import (
    SkyModel,
    PowerLawSpectralModel,
    PointSpatialModel,
    FoVBackgroundModel,
)
from gammapy.modeling import Fit
from gammapy.estimators import FluxPointsEstimator<jupyter_output><empty_output><jupyter_text>Defining the datastore and selecting observationsWe first use the `~gammapy.data.DataStore` object to access the observations we want to analyse. Here the H.E.S.S. DL3 DR1.<jupyter_code>data_store = DataStore.from_dir("$GAMMAPY_DATA/hess-dl3-dr1")<jupyter_output><empty_output><jupyter_text>We can now define an observation filter to select only the relevant observations. Here we use a cone search which we define with a python dict.We then filter the `ObservationTable` with `~gammapy.data.ObservationTable.select_observations()`.<jupyter_code>selection = dict(
    type="sky_circle",
    frame="icrs",
    lon="83.633 deg",
    lat="22.014 deg",
    radius="5 deg",
)
selected_obs_table = data_store.obs_table.select_observations(selection)<jupyter_output><empty_output><jupyter_text>We can now retrieve the relevant observations by passing their `obs_id` to the`~gammapy.data.DataStore.get_observations()` method.<jupyter_code>observations = data_store.get_observations(selected_obs_table["OBS_ID"])<jupyter_output><empty_output><jupyter_text>Preparing reduced datasets geometryNow we define a reference geometry for our analysis, We choose a WCS based geometry with a binsize of 0.02 deg and also define an energy axis:<jupyter_code>energy_axis = MapAxis.from_energy_bounds(1.0, 10.0, 4, unit="TeV")

geom = WcsGeom.create(
    skydir=(83.633, 22.014),
    binsz=0.02,
    width=(2, 2),
    frame="icrs",
    proj="CAR",
    axes=[energy_axis],
)

# Reduced IRFs are defined in true energy (i.e. not measured energy).
energy_axis_true = MapAxis.from_energy_bounds(
    0.5, 20, 10, unit="TeV", name="energy_true"
)<jupyter_output><empty_output><jupyter_text>Now we can define the target dataset with this geometry.<jupyter_code>stacked = MapDataset.create(
    geom=geom, energy_axis_true=energy_axis_true, name="crab-stacked"
)<jupyter_output><empty_output><jupyter_text>Data reduction Create the maker classes to be usedThe `~gammapy.datasets.MapDatasetMaker` object is initialized as well as the `~gammapy.makers.SafeMaskMaker` that carries here a maximum offset selection.<jupyter_code>offset_max = 2.5 * u.deg
maker = MapDatasetMaker()
maker_safe_mask = SafeMaskMaker(
    methods=["offset-max", "aeff-max"], offset_max=offset_max
)
circle = CircleSkyRegion(
    center=SkyCoord("83.63 deg", "22.14 deg"), radius=0.2 * u.deg
)
exclusion_mask = ~geom.region_mask(regions=[circle])
maker_fov = FoVBackgroundMaker(method="fit", exclusion_mask=exclusion_mask)<jupyter_output><empty_output><jupyter_text>Perform the data reduction loop<jupyter_code>%%time

for obs in observations:
    # First a cutout of the target map is produced
    cutout = stacked.cutout(
        obs.pointing_radec, width=2 * offset_max, name=f"obs-{obs.obs_id}"
    )
    # A MapDataset is filled in this cutout geometry
    dataset = maker.run(cutout, obs)
    # The data quality cut is applied
    dataset = maker_safe_mask.run(dataset, obs)
    # fit background model
    dataset = maker_fov.run(dataset)
    print(
        f"Background norm obs {obs.obs_id}: {dataset.background_model.spectral_model.norm.value:.2f}"
    )
    # The resulting dataset cutout is stacked onto the final one
    stacked.stack(dataset)
print(stacked)<jupyter_output><empty_output><jupyter_text>Inspect the reduced dataset<jupyter_code>stacked.counts.sum_over_axes().smooth(0.05 * u.deg).plot(
    stretch="sqrt", add_cbar=True
);<jupyter_output><empty_output><jupyter_text>Save dataset to diskIt is common to run the preparation step independent of the likelihood fit, because often the preparation of maps, PSF and energy dispersion is slow if you have a lot of data. We first create a folder:<jupyter_code>path = Path("analysis_2")
path.mkdir(exist_ok=True)<jupyter_output><empty_output><jupyter_text>And then write the maps and IRFs to disk by calling the dedicated `~gammapy.datasets.MapDataset.write()` method:<jupyter_code>filename = path / "crab-stacked-dataset.fits.gz"
stacked.write(filename, overwrite=True)<jupyter_output><empty_output><jupyter_text>Define the modelWe first define the model, a `SkyModel`, as the combination of a point source `SpatialModel` with a powerlaw `SpectralModel`:<jupyter_code>target_position = SkyCoord(ra=83.63308, dec=22.01450, unit="deg")
spatial_model = PointSpatialModel(
    lon_0=target_position.ra, lat_0=target_position.dec, frame="icrs"
)

spectral_model = PowerLawSpectralModel(
    index=2.702,
    amplitude=4.712e-11 * u.Unit("1 / (cm2 s TeV)"),
    reference=1 * u.TeV,
)

sky_model = SkyModel(
    spatial_model=spatial_model, spectral_model=spectral_model, name="crab"
)

bkg_model = FoVBackgroundModel(dataset_name="crab-stacked")<jupyter_output><empty_output><jupyter_text>Now we assign this model to our reduced dataset:<jupyter_code>stacked.models = [sky_model, bkg_model]<jupyter_output><empty_output><jupyter_text>Fit the modelThe `~gammapy.modeling.Fit` class is orchestrating the fit, connecting the `stats` method of the dataset to the minimizer. By default, it uses `iminuit`.Its constructor takes a list of dataset as argument.<jupyter_code>%%time
fit = Fit(optimize_opts={"print_level": 1})
result = fit.run([stacked])<jupyter_output><empty_output><jupyter_text>The `FitResult` contains information about the optimization and parameter error calculation.<jupyter_code>print(result)<jupyter_output><empty_output><jupyter_text>The fitted parameters are visible from the `~astropy.modeling.models.Models` object.<jupyter_code>stacked.models.to_parameters_table()<jupyter_output><empty_output><jupyter_text>Inspecting residualsFor any fit it is useful to inspect the residual images. We have a few options on the dataset object to handle this. First we can use `.plot_residuals_spatial()` to plot a residual image, summed over all energies:<jupyter_code>stacked.plot_residuals_spatial(method="diff/sqrt(model)", vmin=-0.5, vmax=0.5);<jupyter_output><empty_output><jupyter_text>In addition, we can also specify a region in the map to show the spectral residuals:<jupyter_code>region = CircleSkyRegion(
    center=SkyCoord("83.63 deg", "22.14 deg"), radius=0.5 * u.deg
)
stacked.plot_residuals(
    kwargs_spatial=dict(method="diff/sqrt(model)", vmin=-0.5, vmax=0.5),
    kwargs_spectral=dict(region=region),
);<jupyter_output><empty_output><jupyter_text>We can also directly access the `.residuals()` to get a map, that we can plot interactively:<jupyter_code>residuals = stacked.residuals(method="diff")
residuals.smooth("0.08 deg").plot_interactive(
    cmap="coolwarm", vmin=-0.2, vmax=0.2, stretch="linear", add_cbar=True
);<jupyter_output><empty_output><jupyter_text>Plot the fitted spectrum  Making a butterfly plot The `SpectralModel` component can be used to produce a, so-called, butterfly plot showing the envelope of the model taking into account parameter uncertainties:<jupyter_code>spec = sky_model.spectral_model<jupyter_output><empty_output><jupyter_text>Now we can actually do the plot using the `plot_error` method:<jupyter_code>energy_bounds = [1, 10] * u.TeV
spec.plot(energy_bounds=energy_bounds, energy_power=2)
ax = spec.plot_error(energy_bounds=energy_bounds, energy_power=2)<jupyter_output><empty_output><jupyter_text>Computing flux pointsWe can now compute some flux points using the `~gammapy.estimators.FluxPointsEstimator`. Besides the list of datasets to use, we must provide it the energy intervals on which to compute flux points as well as the model component name.<jupyter_code>energy_edges = [1, 2, 4, 10] * u.TeV
fpe = FluxPointsEstimator(energy_edges=energy_edges, source="crab")
%%time
flux_points = fpe.run(datasets=[stacked])
ax = spec.plot_error(energy_bounds=energy_bounds, energy_power=2)
flux_points.plot(ax=ax, energy_power=2)<jupyter_output><empty_output>
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