Source code for gammapy.estimators.map.ts

# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Functions to compute test statistic images."""
import warnings
from itertools import repeat
import numpy as np
import scipy.optimize
from astropy.coordinates import Angle
from astropy.utils import lazyproperty
import gammapy.utils.parallel as parallel
from gammapy.datasets.map import MapEvaluator
from gammapy.datasets.utils import get_nearest_valid_exposure_position
from gammapy.maps import Map, Maps
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 gammapy.utils.pbar import progress_bar
from gammapy.utils.roots import find_roots
from ..core import Estimator
from ..utils import estimate_exposure_reco_energy
from .core import FluxMaps

__all__ = ["TSMapEstimator"]


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, parallel.ParallelMixin): r"""Compute test statistic 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 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, optional If the test statistic value corresponding to the initial flux estimate is not above this threshold, the optimizing step is omitted to save computing time. Default is None. rtol : float Relative precision of the flux estimate. Used as a stopping criterion for the norm fit. Default is 0.01. selection_optional : list of str, optional Which maps to compute besides TS, sqrt(TS), flux and symmetric error on flux. Available options are: * "all": all the optional steps are executed * "errn-errp": estimate asymmetric error on flux. * "ul": estimate upper limits on flux. Default is None so the optional steps are not executed. energy_edges : list of `~astropy.units.Quantity`, optional Edges of the target maps energy bins. The resulting bin edges won't be exactly equal to the input ones, but rather the closest values to the energy axis edges of the parent dataset. Default is None: apply the estimator in each energy bin of the parent dataset. For further explanation see :ref:`estimators`. 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. Default is one, unless `~gammapy.utils.parallel.N_JOBS_DEFAULT` was modified. The number of jobs limited to the number of physical CPUs. parallel_backend : {"multiprocessing", "ray"} Which backend to use for multiprocessing. Defaults to `~gammapy.utils.parallel.BACKEND_DEFAULT`. 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. Examples -------- >>> import astropy.units as u >>> from gammapy.estimators import TSMapEstimator >>> from gammapy.datasets import MapDataset >>> from gammapy.modeling.models import (SkyModel, PowerLawSpectralModel,PointSpatialModel) >>> spatial_model = PointSpatialModel() >>> spectral_model = PowerLawSpectralModel(amplitude="1e-22 cm-2 s-1 keV-1", index=2) >>> model = SkyModel(spatial_model=spatial_model, spectral_model=spectral_model) >>> dataset = MapDataset.read("$GAMMAPY_DATA/fermi-3fhl-gc/fermi-3fhl-gc.fits.gz") >>> estimator = TSMapEstimator( ... model, kernel_width="1 deg", energy_edges=[10, 100] * u.GeV, downsampling_factor=4 ... ) >>> maps = estimator.run(dataset) >>> print(maps) FluxMaps -------- <BLANKLINE> geom : WcsGeom axes : ['lon', 'lat', 'energy'] shape : (400, 200, 1) quantities : ['ts', 'norm', 'niter', 'norm_err', 'npred', 'npred_excess', 'stat', 'stat_null', 'success'] ref. model : pl n_sigma : 1 n_sigma_ul : 2 sqrt_ts_threshold_ul : 2 sed type init : likelihood References ---------- [Stewart2009]_ """ tag = "TSMapEstimator" _available_selection_optional = ["errn-errp", "ul"] def __init__( self, model=None, kernel_width=None, downsampling_factor=None, n_sigma=1, n_sigma_ul=2, threshold=None, rtol=0.01, selection_optional=None, energy_edges=None, sum_over_energy_groups=True, n_jobs=None, parallel_backend=None, ): if kernel_width is not None: kernel_width = Angle(kernel_width) self.kernel_width = 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.parallel_backend = parallel_backend 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", "norm", "niter", "norm_err", "npred", "npred_excess", "stat", "stat_null", "success", ] if "errn-errp" in self.selection_optional: selection += ["norm_errp", "norm_errn"] if "ul" in self.selection_optional: selection += ["norm_ul"] return selection
[docs] def estimate_kernel(self, dataset): """Get the convolution kernel for the input dataset. Convolves the model with the IRFs at the center of the dataset, or at the nearest position with non-zero exposure. Parameters ---------- dataset : `~gammapy.datasets.MapDataset` Input dataset. Returns ------- kernel : `~gammapy.maps.Map` Kernel map. """ geom = dataset.exposure.geom if self.kernel_width is not None: geom = geom.to_odd_npix(max_radius=self.kernel_width / 2) model = self.model.copy() model.spatial_model.position = geom.center_skydir # Creating exposure map with the mean non-null exposure exposure = Map.from_geom(geom, unit=dataset.exposure.unit) position = get_nearest_valid_exposure_position( dataset.exposure, geom.center_skydir ) exposure_position = dataset.exposure.to_region_nd_map(position) if not np.any(exposure_position.data): raise ValueError( "No valid exposure. Impossible to compute kernel for TS Map." ) exposure.data[...] = exposure_position.data # We use global evaluation mode to not modify the geometry evaluator = MapEvaluator(model=model) evaluator.update( exposure=exposure, psf=dataset.psf, edisp=dataset.edisp, geom=dataset.counts.geom, mask=dataset.mask_image, ) kernel = evaluator.compute_npred() kernel.data /= kernel.data.sum() return kernel
[docs] def estimate_flux_default(self, dataset, kernel=None, exposure=None): """Estimate default flux map using a given kernel. Parameters ---------- dataset : `~gammapy.datasets.MapDataset` Input dataset. kernel : `~gammapy.maps.WcsNDMap` Source model kernel. exposure : `~gammapy.maps.WcsNDMap` Exposure map on reconstructed energy. Returns ------- flux : `~gammapy.maps.WcsNDMap` Approximate flux map. """ if exposure is None: exposure = estimate_exposure_reco_energy(dataset, self.model.spectral_model) if kernel is None: kernel = self.estimate_kernel(dataset=dataset) kernel = kernel.data / np.sum(kernel.data**2) with np.errstate(invalid="ignore", divide="ignore"): flux = (dataset.counts - dataset.npred()) / exposure flux.data = np.nan_to_num(flux.data) flux.quantity = flux.quantity.to("1 / (cm2 s)") flux = flux.convolve(kernel) return flux.sum_over_axes()
[docs] @staticmethod def estimate_mask_default(dataset): """Compute default mask where to estimate test statistic values. Parameters ---------- dataset : `~gammapy.datasets.MapDataset` Input dataset. Returns ------- mask : `WcsNDMap` Mask map. """ geom = dataset.counts.geom.to_image() mask = np.ones(geom.data_shape, dtype=bool) 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_pad_width(self, dataset, kernel=None): """Estimate pad width of the dataset. Parameters ---------- dataset : `MapDataset` Input MapDataset. kernel : `WcsNDMap` Source model kernel. Returns ------- pad_width : tuple Padding width. """ if kernel is None: kernel = self.estimate_kernel(dataset=dataset) geom = dataset.counts.geom.to_image() geom_kernel = kernel.geom.to_image() pad_width = np.array(geom_kernel.data_shape) // 2 if self.downsampling_factor and self.downsampling_factor > 1: shape = tuple(np.array(geom.data_shape) + 2 * pad_width) pad_width = symmetric_crop_pad_width(geom.data_shape, shape_2N(shape))[0] return tuple(pad_width)
[docs] def estimate_fit_input_maps(self, dataset): """Estimate fit input maps. Parameters ---------- dataset : `MapDataset` Map dataset. Returns ------- maps : dict of `Map` Maps dictionary. """ # 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=dataset) flux = self.estimate_flux_default( dataset=dataset, kernel=kernel, 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 * mask.data).to_unit("") norm = (flux / flux_ref).to_unit("") return { "counts": counts, "background": background, "norm": norm, "mask": mask, "exposure": exposure_npred, "kernel": kernel, }
[docs] def estimate_flux_map(self, dataset): """Estimate flux and test statistic maps for single dataset. Parameters ---------- dataset : `~gammapy.datasets.MapDataset` Map dataset. """ maps = self.estimate_fit_input_maps(dataset=dataset) x, y = np.where(np.squeeze(maps["mask"].data)) positions = list(zip(x, y)) inputs = zip( positions, repeat(maps["counts"].data.astype(float)), repeat(maps["exposure"].data.astype(float)), repeat(maps["background"].data.astype(float)), repeat(maps["kernel"].data), repeat(maps["norm"].data), repeat(self._flux_estimator), ) results = parallel.run_multiprocessing( _ts_value, inputs, pool_kwargs=dict(processes=self.n_jobs), task_name="TS map", ) result = {} j, i = zip(*positions) geom = maps["counts"].geom.squash(axis_name="energy") for name in self.selection_all: m = Map.from_geom(geom=geom, data=np.nan, unit="") m.data[0, j, i] = [_[name] for _ in results] result[name] = m return result
[docs] def run(self, dataset): """ Run test statistic map estimation. Requires a MapDataset with counts, exposure and background_model properly set to run. Notes ----- The progress bar can be displayed for this function. 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 dataset.stat_type != "cash": raise TypeError(f"{type(dataset)} is not a valid type for {self.__class__}") dataset_models = dataset.models pad_width = self.estimate_pad_width(dataset=dataset) dataset = dataset.pad(pad_width, name=dataset.name) dataset = dataset.downsample(self.downsampling_factor, name=dataset.name) energy_axis = self._get_energy_axis(dataset=dataset) results = [] for energy_min, energy_max in progress_bar( energy_axis.iter_by_edges, desc="Energy bins" ): dataset_sliced = dataset.slice_by_energy( energy_min=energy_min, energy_max=energy_max, name=dataset.name ) if self.sum_over_energy_groups: dataset_sliced = dataset_sliced.to_image(name=dataset.name) if dataset_models is not None: models_sliced = dataset_models._slice_by_energy( energy_min=energy_min, energy_max=energy_max, sum_over_energy_groups=self.sum_over_energy_groups, ) dataset_sliced.models = models_sliced result = self.estimate_flux_map(dataset_sliced) results.append(result) maps = Maps() for name in self.selection_all: m = Map.from_stack(maps=[_[name] for _ in results], axis_name="energy") order = 0 if name in ["niter", "success"] else 1 m = m.upsample( factor=self.downsampling_factor, preserve_counts=False, order=order ) maps[name] = m.crop(crop_width=pad_width) maps["success"].data = maps["success"].data.astype(bool) meta = {"n_sigma": self.n_sigma, "n_sigma_ul": self.n_sigma_ul} return FluxMaps( data=maps, reference_model=self.model, gti=dataset.gti, meta=meta, )
# 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): """Statistics sum.""" return cash_sum_cython(self.counts.ravel(), self.npred(norm).ravel()) def stat_derivative(self, norm): """Statistics derivative.""" return f_cash_root_cython( norm, self.counts.ravel(), self.background.ravel(), self.model.ravel() ) def stat_2nd_derivative(self, norm): """Statistics 2nd derivative.""" term_top = self.model**2 * self.counts term_bottom = (self.background + norm * self.model) ** 2 mask = term_bottom == 0 return (term_top / term_bottom)[~mask].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): """Estimate best fit norm parameter. Parameters ---------- dataset : `SimpleMapDataset` Simple map dataset. Returns ------- result : dict Result dictionary including 'norm' and 'norm_err'. """ # 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, success = norm_min_total, 0, True else: with warnings.catch_warnings(): warnings.simplefilter("ignore") try: # here we do not use avoid find_roots for performance 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 success = result_fit[1].converged except (RuntimeError, ValueError): norm, niter, success = norm_min_total, self.max_niter, False with np.errstate(invalid="ignore", divide="ignore"): norm_err = np.sqrt(1 / dataset.stat_2nd_derivative(norm)) * self.n_sigma stat = dataset.stat_sum(norm=norm) stat_null = dataset.stat_sum(norm=0) return { "norm": norm, "norm_err": norm_err, "niter": niter, "ts": stat_null - stat, "stat": stat, "stat_null": stat_null, "success": success, } 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") roots, res = find_roots( ts_diff, [min_norm], [max_norm], nbin=1, maxiter=self.max_niter, rtol=self.rtol, ) # Where the root finding fails NaN is set as norm return (roots[0] - norm) * factor def estimate_ul(self, dataset, result): """Compute upper limit using likelihood profile method. Parameters ---------- dataset : `SimpleMapDataset` Simple map dataset. Returns ------- result : dict Result dictionary including 'norm_ul'. """ flux_ul = result["norm"] + 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. Parameters ---------- dataset : `SimpleMapDataset` Simple map dataset. Returns ------- result : dict Result dictionary including 'norm_errp' and 'norm_errn'. """ 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): """Estimate default norm. Parameters ---------- dataset : `SimpleMapDataset` Simple map dataset. Returns ------- result : dict Result dictionary including 'norm', 'norm_err' and "niter". """ norm = dataset.norm_guess with np.errstate(invalid="ignore", divide="ignore"): norm_err = np.sqrt(1 / dataset.stat_2nd_derivative(norm)) * self.n_sigma stat = dataset.stat_sum(norm=norm) stat_null = dataset.stat_sum(norm=0) return { "norm": norm, "norm_err": norm_err, "niter": 0, "ts": stat_null - stat, "stat": stat, "stat_null": stat_null, "success": True, } def run(self, dataset): """Run flux estimator. Parameters ---------- dataset : `SimpleMapDataset` Simple map dataset. Returns ------- result : dict Result dictionary. """ 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) norm = result["norm"] result["npred"] = dataset.npred(norm=norm).sum() result["npred_excess"] = result["npred"] - dataset.npred(norm=0).sum() 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 test statistic 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 Test statistic 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)