# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Functions to compute TS images."""
import contextlib
import functools
import logging
import warnings
from multiprocessing import Pool
import numpy as np
import scipy.optimize
from astropy import units as u
from astropy.coordinates import Angle
from astropy.utils import lazyproperty
from gammapy.datasets import Datasets
from gammapy.datasets.map import MapEvaluator
from gammapy.maps import Map, WcsGeom
from gammapy.modeling.models import PointSpatialModel, PowerLawSpectralModel, SkyModel
from gammapy.stats import cash_sum_cython, f_cash_root_cython, norm_bounds_cython
from gammapy.utils.array import shape_2N, symmetric_crop_pad_width
from .core import Estimator
from .utils import estimate_exposure_reco_energy
__all__ = ["TSMapEstimator"]
log = logging.getLogger(__name__)
def round_up_to_odd(f):
return int(np.ceil(f) // 2 * 2 + 1)
def _extract_array(array, shape, position):
"""Helper function to extract parts of a larger array.
Simple implementation of an array extract function , because
`~astropy.ndata.utils.extract_array` introduces too much overhead.`
Parameters
----------
array : `~numpy.ndarray`
The array from which to extract.
shape : tuple or int
The shape of the extracted array.
position : tuple of numbers or number
The position of the small array's center with respect to the
large array.
"""
x_width = shape[2] // 2
y_width = shape[1] // 2
y_lo = position[0] - y_width
y_hi = position[0] + y_width + 1
x_lo = position[1] - x_width
x_hi = position[1] + x_width + 1
return array[:, y_lo:y_hi, x_lo:x_hi]
[docs]class TSMapEstimator(Estimator):
r"""Compute TS map from a MapDataset using different optimization methods.
The map is computed fitting by a single parameter norm fit. The fit is
simplified by finding roots of the the derivative of the fit statistics using
various root finding algorithms. The approach is described in Appendix A
in Stewart (2009).
Parameters
----------
model : `~gammapy.modeling.model.SkyModel`
Source model kernel. If set to None,
assume spatail model: point source model, PointSpatialModel.
spectral model: PowerLawSpectral Model of index 2
kernel_width : `~astropy.coordinates.Angle`
Width of the kernel to use: the kernel will be truncated at this size
n_sigma : int
Number of sigma for flux error. Default is 1.
n_sigma_ul : int
Number of sigma for flux upper limits. Default is 2.
downsampling_factor : int
Sample down the input maps to speed up the computation. Only integer
values that are a multiple of 2 are allowed. Note that the kernel is
not sampled down, but must be provided with the downsampled bin size.
threshold : float (None)
If the TS value corresponding to the initial flux estimate is not above
this threshold, the optimizing step is omitted to save computing time.
rtol : float (0.001)
Relative precision of the flux estimate. Used as a stopping criterion for
the norm fit.
selection_optional : list of str or 'all'
Which maps to compute besides TS, sqrt(TS), flux and symmetric error on flux.
Available options are:
* "errn-errp": estimate assymmetric error on flux.
* "ul": estimate upper limits on flux.
By default all steps are executed.
energy_edges : `~astropy.units.Quantity`
Energy edges of the maps bins.
sum_over_energy_groups : bool
Whether to sum over the energy groups or fit the norm on the full energy
cube.
n_jobs : int
Number of processes used in parallel for the computation.
Notes
-----
Negative :math:`TS` values are defined as following:
.. math::
TS = \left \{
\begin{array}{ll}
-TS \text{ if } F < 0 \\
TS \text{ else}
\end{array}
\right.
Where :math:`F` is the fitted flux norm.
References
----------
[Stewart2009]_
"""
tag = "TSMapEstimator"
_available_selection_optional = ["errn-errp", "ul"]
def __init__(
self,
model=None,
kernel_width="0.2 deg",
downsampling_factor=None,
n_sigma=1,
n_sigma_ul=2,
threshold=None,
rtol=0.01,
selection_optional="all",
energy_edges=None,
sum_over_energy_groups=True,
n_jobs=None,
):
self.kernel_width = Angle(kernel_width)
if model is None:
model = SkyModel(
spectral_model=PowerLawSpectralModel(),
spatial_model=PointSpatialModel(),
name="ts-kernel",
)
self.model = model
self.downsampling_factor = downsampling_factor
self.n_sigma = n_sigma
self.n_sigma_ul = n_sigma_ul
self.threshold = threshold
self.rtol = rtol
self.n_jobs = n_jobs
self.sum_over_energy_groups = sum_over_energy_groups
self.selection_optional = selection_optional
self.energy_edges = energy_edges
self._flux_estimator = BrentqFluxEstimator(
rtol=self.rtol,
n_sigma=self.n_sigma,
n_sigma_ul=self.n_sigma_ul,
selection_optional=selection_optional,
ts_threshold=threshold,
)
@property
def selection_all(self):
"""Which quantities are computed"""
selection = ["ts", "flux", "niter", "flux_err"]
if "errn-errp" in self.selection_optional:
selection += ["flux_errp", "flux_errn"]
if "ul" in self.selection_optional:
selection += ["flux_ul"]
return selection
[docs] def estimate_kernel(self, dataset):
"""Get the convolution kernel for the input dataset.
Convolves the model with the PSFKernel at the center of the dataset.
Parameters
----------
dataset : `~gammapy.datasets.MapDataset`
Input dataset.
Returns
-------
kernel : `Map`
Kernel map
"""
# TODO: further simplify the code below
geom = dataset.counts.geom
model = self.model.copy()
model.spatial_model.position = geom.center_skydir
binsz = np.mean(geom.pixel_scales)
width_pix = self.kernel_width / binsz
npix = round_up_to_odd(width_pix.to_value(""))
axis = dataset.exposure.geom.axes["energy_true"]
geom_kernel = WcsGeom.create(
skydir=model.position, proj="TAN", npix=npix, axes=[axis], binsz=binsz
)
exposure = Map.from_geom(geom_kernel, unit="cm2 s1")
exposure.data += 1.0
# We use global evaluation mode to not modify the geometry
evaluator = MapEvaluator(model, evaluation_mode="global")
evaluator.update(exposure, dataset.psf, dataset.edisp, dataset.counts.geom)
kernel = evaluator.compute_npred()
kernel.data /= kernel.data.sum()
if (self.kernel_width + binsz >= geom.width).any():
raise ValueError(
"Kernel shape larger than map shape, please adjust"
" size of the kernel"
)
return kernel
[docs] def estimate_flux_default(self, dataset, kernel, exposure=None):
"""Estimate default flux map using a given kernel.
Parameters
----------
dataset : `~gammapy.datasets.MapDataset`
Input dataset.
kernel : `~numpy.ndarray`
Source model kernel.
Returns
-------
flux : `~gammapy.maps.WcsNDMap`
Approximate flux map.
"""
if exposure is None:
exposure = estimate_exposure_reco_energy(dataset, self.model.spectral_model)
kernel = kernel / np.sum(kernel ** 2)
flux = (dataset.counts - dataset.npred()) / exposure
flux.quantity = flux.quantity.to("1 / (cm2 s)")
flux = flux.convolve(kernel)
return flux.sum_over_axes()
[docs] @staticmethod
def estimate_mask_default(dataset, kernel):
"""Compute default mask where to estimate TS values.
Parameters
----------
dataset : `~gammapy.datasets.MapDataset`
Input dataset.
kernel : `~numpy.ndarray`
Source model kernel.
Returns
-------
mask : `gammapy.maps.WcsNDMap`
Mask map.
"""
geom = dataset.counts.geom.to_image()
# mask boundary
mask = np.zeros(geom.data_shape, dtype=bool)
slice_x = slice(kernel.shape[2] // 2, -kernel.shape[2] // 2 + 1)
slice_y = slice(kernel.shape[1] // 2, -kernel.shape[1] // 2 + 1)
mask[slice_y, slice_x] = 1
if dataset.mask is not None:
mask &= dataset.mask.reduce_over_axes(func=np.logical_or, keepdims=False)
# in some image there are pixels, which have exposure, but zero
# background, which doesn't make sense and causes the TS computation
# to fail, this is a temporary fix
background = dataset.npred().sum_over_axes(keepdims=False)
mask[background.data == 0] = False
return Map.from_geom(data=mask, geom=geom)
[docs] def estimate_sqrt_ts(self, map_ts, norm):
r"""Compute sqrt(TS) map.
Compute sqrt(TS) as defined by:
.. math::
\sqrt{TS} = \left \{
\begin{array}{ll}
-\sqrt{-TS} & : \text{if} \ TS < 0 \\
\sqrt{TS} & : \text{else}
\end{array}
\right.
Parameters
----------
map_ts : `gammapy.maps.WcsNDMap`
Input TS map.
Returns
-------
sqrt_ts : `gammapy.maps.WcsNDMap`
Sqrt(TS) map.
"""
sqrt_ts = self.get_sqrt_ts(map_ts.data, norm.data)
return map_ts.copy(data=sqrt_ts)
[docs] def estimate_flux_map(self, dataset):
"""Estimate flux and ts maps for single dataset
Parameters
----------
dataset : `MapDataset`
Map dataset
"""
# First create 2D map arrays
counts = dataset.counts
background = dataset.npred()
exposure = estimate_exposure_reco_energy(dataset, self.model.spectral_model)
kernel = self.estimate_kernel(dataset)
mask = self.estimate_mask_default(dataset, kernel.data)
flux = self.estimate_flux_default(dataset, kernel.data, exposure=exposure)
energy_axis = counts.geom.axes["energy"]
flux_ref = self.model.spectral_model.integral(
energy_axis.edges[0], energy_axis.edges[-1]
)
exposure_npred = (exposure * flux_ref).quantity.to_value("")
wrap = functools.partial(
_ts_value,
counts=counts.data.astype(float),
exposure=exposure_npred.astype(float),
background=background.data.astype(float),
kernel=kernel.data,
norm=(flux.quantity / flux_ref).to_value(""),
flux_estimator=self._flux_estimator,
)
x, y = np.where(np.squeeze(mask.data))
positions = list(zip(x, y))
if self.n_jobs is None:
results = list(map(wrap, positions))
else:
with contextlib.closing(Pool(processes=self.n_jobs)) as pool:
log.info("Using {} jobs to compute TS map.".format(self.n_jobs))
results = pool.map(wrap, positions)
pool.join()
result = {}
j, i = zip(*positions)
geom = counts.geom.squash(axis_name="energy")
for name in self.selection_all:
unit = 1 / exposure.unit if "flux" in name else ""
m = Map.from_geom(geom=geom, data=np.nan, unit=unit)
m.data[0, j, i] = [_[name.replace("flux", "norm")] for _ in results]
if "flux" in name:
m.data *= flux_ref.to_value(m.unit)
m.quantity = m.quantity.to("1 / (cm2 s)")
result[name] = m
return result
[docs] def run(self, dataset):
"""
Run TS map estimation.
Requires a MapDataset with counts, exposure and background_model
properly set to run.
Parameters
----------
dataset : `~gammapy.datasets.MapDataset`
Input MapDataset.
Returns
-------
maps : dict
Dictionary containing result maps. Keys are:
* ts : delta TS map
* sqrt_ts : sqrt(delta TS), or significance map
* flux : flux map
* flux_err : symmetric error map
* flux_ul : upper limit map
"""
if self.downsampling_factor:
shape = dataset.counts.geom.to_image().data_shape
pad_width = symmetric_crop_pad_width(shape, shape_2N(shape))[0]
dataset = dataset.pad(pad_width).downsample(self.downsampling_factor)
# TODO: add support for joint likelihood fitting to TSMapEstimator
datasets = Datasets(dataset)
if self.energy_edges is None:
energy_axis = dataset.counts.geom.axes["energy"]
energy_edges = u.Quantity([energy_axis.edges[0], energy_axis.edges[-1]])
else:
energy_edges = self.energy_edges
results = []
for energy_min, energy_max in zip(energy_edges[:-1], energy_edges[1:]):
dataset = datasets.slice_by_energy(energy_min, energy_max)[0]
if self.sum_over_energy_groups:
dataset = dataset.to_image()
result = self.estimate_flux_map(dataset)
results.append(result)
result_all = {}
for name in self.selection_all:
map_all = Map.from_images(images=[_[name] for _ in results])
if self.downsampling_factor:
order = 0 if name == "niter" else 1
map_all = map_all.upsample(
factor=self.downsampling_factor, preserve_counts=False, order=order
)
map_all = map_all.crop(crop_width=pad_width)
result_all[name] = map_all
result_all["sqrt_ts"] = self.estimate_sqrt_ts(
result_all["ts"], result_all["flux"]
)
return result_all
# TODO: merge with MapDataset?
class SimpleMapDataset:
"""Simple map dataset
Parameters
----------
counts : `~numpy.ndarray`
Counts array
background : `~numpy.ndarray`
Background array
model : `~numpy.ndarray`
Kernel array
"""
def __init__(self, model, counts, background, norm_guess):
self.model = model
self.counts = counts
self.background = background
self.norm_guess = norm_guess
@lazyproperty
def norm_bounds(self):
"""Bounds for x"""
return norm_bounds_cython(
self.counts.ravel(), self.background.ravel(), self.model.ravel()
)
def npred(self, norm):
"""Predicted number of counts"""
return self.background + norm * self.model
def stat_sum(self, norm):
"""Stat sum"""
return cash_sum_cython(self.counts.ravel(), self.npred(norm).ravel())
def stat_derivative(self, norm):
"""Stat derivative"""
return f_cash_root_cython(
norm, self.counts.ravel(), self.background.ravel(), self.model.ravel()
)
def stat_2nd_derivative(self, norm):
"""Stat 2nd derivative"""
with np.errstate(invalid="ignore", divide="ignore"):
return (
self.model ** 2
* self.counts
/ (self.background + norm * self.model) ** 2
).sum()
@classmethod
def from_arrays(cls, counts, background, exposure, norm, position, kernel):
""""""
counts_cutout = _extract_array(counts, kernel.shape, position)
background_cutout = _extract_array(background, kernel.shape, position)
exposure_cutout = _extract_array(exposure, kernel.shape, position)
norm_guess = norm[0, position[0], position[1]]
return cls(
counts=counts_cutout,
background=background_cutout,
model=kernel * exposure_cutout,
norm_guess=norm_guess,
)
# TODO: merge with `FluxEstimator`?
class BrentqFluxEstimator(Estimator):
"""Single parameter flux estimator"""
_available_selection_optional = ["errn-errp", "ul"]
tag = "BrentqFluxEstimator"
def __init__(
self,
rtol,
n_sigma,
n_sigma_ul,
selection_optional=None,
max_niter=20,
ts_threshold=None,
):
self.rtol = rtol
self.n_sigma = n_sigma
self.n_sigma_ul = n_sigma_ul
self.selection_optional = selection_optional
self.max_niter = max_niter
self.ts_threshold = ts_threshold
def estimate_best_fit(self, dataset):
"""Optimize for a single parameter"""
result = {}
# Compute norm bounds and assert counts > 0
norm_min, norm_max, norm_min_total = dataset.norm_bounds
if not dataset.counts.sum() > 0:
norm, niter = norm_min_total, 0
else:
with warnings.catch_warnings():
warnings.simplefilter("ignore")
try:
result_fit = scipy.optimize.brentq(
f=dataset.stat_derivative,
a=norm_min,
b=norm_max,
maxiter=self.max_niter,
full_output=True,
rtol=self.rtol,
)
norm = max(result_fit[0], norm_min_total)
niter = result_fit[1].iterations
except (RuntimeError, ValueError):
# Where the root finding fails NaN is set as norm
norm, niter = norm_min_total, self.max_niter
stat = dataset.stat_sum(norm=norm)
stat_null = dataset.stat_sum(norm=0)
result["ts"] = stat_null - stat
result["norm"] = norm
result["niter"] = niter
with np.errstate(invalid="ignore", divide="ignore"):
result["norm_err"] = (
np.sqrt(1 / dataset.stat_2nd_derivative(norm)) * self.n_sigma
)
result["stat"] = stat
return result
def _confidence(self, dataset, n_sigma, result, positive):
stat_best = result["stat"]
norm = result["norm"]
norm_err = result["norm_err"]
def ts_diff(x):
return (stat_best + n_sigma ** 2) - dataset.stat_sum(x)
if positive:
min_norm = norm
max_norm = norm + 1e2 * norm_err
factor = 1
else:
min_norm = norm - 1e2 * norm_err
max_norm = norm
factor = -1
with warnings.catch_warnings():
warnings.simplefilter("ignore")
try:
result_fit = scipy.optimize.brentq(
ts_diff, min_norm, max_norm, maxiter=self.max_niter, rtol=self.rtol,
)
return (result_fit - norm) * factor
except (RuntimeError, ValueError):
# Where the root finding fails NaN is set as norm
return np.nan
def estimate_ul(self, dataset, result):
""""""
flux_ul = self._confidence(
dataset=dataset, n_sigma=self.n_sigma_ul, result=result, positive=True
)
return {"norm_ul": flux_ul}
def estimate_errn_errp(self, dataset, result):
"""
Compute norm errors using likelihood profile method.
"""
flux_errn = self._confidence(
dataset=dataset, result=result, n_sigma=self.n_sigma, positive=False
)
flux_errp = self._confidence(
dataset=dataset, result=result, n_sigma=self.n_sigma, positive=True
)
return {"norm_errn": flux_errn, "norm_errp": flux_errp}
def estimate_default(self, dataset):
norm = dataset.norm_guess
stat = dataset.stat_sum(norm=norm)
stat_null = dataset.stat_sum(norm=0)
ts = stat_null - stat
with np.errstate(invalid="ignore", divide="ignore"):
norm_err = np.sqrt(1 / dataset.stat_2nd_derivative(norm)) * self.n_sigma
return {"norm": norm, "ts": ts, "norm_err": norm_err, "stat": stat, "niter": 0}
def run(self, dataset):
""""""
if self.ts_threshold is not None:
result = self.estimate_default(dataset)
if result["ts"] > self.ts_threshold:
result = self.estimate_best_fit(dataset)
else:
result = self.estimate_best_fit(dataset)
if "ul" in self.selection_optional:
result.update(self.estimate_ul(dataset, result))
if "errn-errp" in self.selection_optional:
result.update(self.estimate_errn_errp(dataset, result))
return result
def _ts_value(position, counts, exposure, background, kernel, norm, flux_estimator):
"""Compute TS value at a given pixel position.
Uses approach described in Stewart (2009).
Parameters
----------
position : tuple (i, j)
Pixel position.
counts : `~numpy.ndarray`
Counts image
background : `~numpy.ndarray`
Background image
exposure : `~numpy.ndarray`
Exposure image
kernel : `astropy.convolution.Kernel2D`
Source model kernel
norm : `~numpy.ndarray`
Norm image. The flux value at the given pixel position is used as
starting value for the minimization.
Returns
-------
TS : float
TS value at the given pixel position.
"""
dataset = SimpleMapDataset.from_arrays(
counts=counts,
background=background,
exposure=exposure,
kernel=kernel,
position=position,
norm=norm,
)
return flux_estimator.run(dataset)