Source code for gammapy.detect.test_statistics

# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Functions to compute TS images."""
import logging
import contextlib
import warnings
from functools import partial
from multiprocessing import Pool
import numpy as np
from scipy.optimize import newton, brentq
from astropy.convolution import CustomKernel, Kernel2D
from ..stats import cash, cash_sum_cython
from ..utils.array import shape_2N, symmetric_crop_pad_width
from ._test_statistics_cython import (
    _amplitude_bounds_cython,
    _f_cash_root_cython,
    _x_best_leastsq,
)

__all__ = ["TSMapEstimator"]

log = logging.getLogger(__name__)

FLUX_FACTOR = 1e-12
MAX_NITER = 20
RTOL = 1e-3


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[1] // 2
    y_width = shape[0] // 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]


def f_cash(x, counts, background, model):
    """Wrapper for cash statistics, that defines the model function.

    Parameters
    ----------
    x : float
        Model amplitude.
    counts : `~numpy.ndarray`
        Count image slice, where model is defined.
    background : `~numpy.ndarray`
        Background image slice, where model is defined.
    model : `~numpy.ndarray`
        Source template (multiplied with exposure).
    """
    return cash_sum_cython(
        counts.ravel(), (background + x * FLUX_FACTOR * model).ravel()
    )


[docs]class TSMapEstimator: """ Compute TS map using different optimization methods. The map is computed fitting by a single parameter amplitude fit. The fit is simplified by finding roots of the the derivative of the fit statistics using various root finding algorithms. The approach is sescribed in Appendix A in Stewart (2009). Parameters ---------- method : str ('root') The following options are available: * ``'root brentq'`` (default) Fit amplitude by finding the roots of the the derivative of the fit statistics using the brentq method. * ``'root newton'`` Fit amplitude by finding the roots of the the derivative of the fit statistics using Newton's method. * ``'leastsq iter'`` Fit the amplitude by an iterative least square fit, that can be solved analytically. error_method : ['covar', 'conf'] Error estimation method. error_sigma : int (1) Sigma for flux error. ul_method : ['covar', 'conf'] Upper limit estimation method. ul_sigma : int (2) Sigma for flux upper limits. n_jobs : int Number of parallel jobs to use for the computation. 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 amplitude fit. 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 amplitude. References ---------- [Stewart2009]_ """ def __init__( self, method="root brentq", error_method="covar", error_sigma=1, ul_method="covar", ul_sigma=2, n_jobs=1, threshold=None, rtol=0.001, ): if method not in ["root brentq", "root newton", "leastsq iter"]: raise ValueError("Not a valid method: '{}'".format(method)) if error_method not in ["covar", "conf"]: raise ValueError("Not a valid error method '{}'".format(error_method)) self.parameters = { "method": method, "error_method": error_method, "error_sigma": error_sigma, "ul_method": ul_method, "ul_sigma": ul_sigma, "n_jobs": n_jobs, "threshold": threshold, "rtol": rtol, }
[docs] @staticmethod def flux_default(maps, kernel): """Estimate default flux map using a given kernel. Parameters ---------- maps : dict Input sky maps. Requires "counts", "background" and "exposure" maps. kernel : `astropy.convolution.Kernel2D` Source model kernel. Returns ------- flux_approx : `gammapy.maps.WcsNDMap` Approximate flux map. """ flux = (maps["counts"].data - maps["background"].data) / maps["exposure"].data flux = maps["counts"].copy(data=flux / np.sum(kernel.array ** 2)) return flux.convolve(kernel.array)
[docs] @staticmethod def mask_default(maps, kernel): """Compute default mask where to estimate TS values. Parameters ---------- maps : dict Input sky maps. Requires "background" and "exposure". kernel : `astropy.convolution.Kernel2D` Source model kernel. Returns ------- mask : `gammapy.maps.WcsNDMap` Mask map. """ mask = np.zeros(maps["exposure"].data.shape, dtype=int) # mask boundary slice_x = slice(kernel.shape[1] // 2, -kernel.shape[1] // 2 + 1) slice_y = slice(kernel.shape[0] // 2, -kernel.shape[0] // 2 + 1) mask[slice_y, slice_x] = 1 # positions where exposure == 0 are not processed mask &= maps["exposure"].data > 0 # 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 mask[maps["background"].data == 0] = 0 return maps["exposure"].copy(data=mask.astype("int"))
[docs] @staticmethod def sqrt_ts(map_ts): 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. """ with np.errstate(invalid="ignore", divide="ignore"): ts = map_ts.data sqrt_ts = np.where(ts > 0, np.sqrt(ts), -np.sqrt(-ts)) return map_ts.copy(data=sqrt_ts)
[docs] def run(self, maps, kernel, which="all", downsampling_factor=None): """ Run TS map estimation. Requires "counts", "exposure" and "background" map to run. Parameters ---------- maps : dict Input sky maps. kernel : `astropy.convolution.Kernel2D` or 2D `~numpy.ndarray` Source model kernel. which : list of str or 'all' Which maps to compute. 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. Returns ------- maps : dict Result maps. """ p = self.parameters if (np.array(kernel.shape) > np.array(maps["counts"].data.shape)).any(): raise ValueError( "Kernel shape larger than map shape, please adjust" " size of the kernel" ) if downsampling_factor: maps_downsampled = {} shape = maps["counts"].data.shape pad_width = symmetric_crop_pad_width(shape, shape_2N(shape))[0] for name, map_ in maps.items(): preserve_counts = name in ["counts", "background", "exclusion"] maps_downsampled[name] = map_.pad(pad_width).downsample( downsampling_factor, preserve_counts=preserve_counts ) maps = maps_downsampled if not isinstance(kernel, Kernel2D): kernel = CustomKernel(kernel) if which == "all": which = ["ts", "sqrt_ts", "flux", "flux_err", "flux_ul", "niter"] result = {} for name in which: data = np.nan * np.ones_like(maps["counts"].data) result[name] = maps["counts"].copy(data=data) mask = self.mask_default(maps, kernel) if "mask" in maps: mask.data &= maps["mask"].data if p["threshold"] or p["method"] == "root newton": flux = self.flux_default(maps, kernel).data else: flux = None # prepare dtype for cython methods counts = maps["counts"].data.astype(float) background = maps["background"].data.astype(float) exposure = maps["exposure"].data.astype(float) # Compute null statistics per pixel for the whole image c_0 = cash(counts, background) error_method = p["error_method"] if "flux_err" in which else "none" ul_method = p["ul_method"] if "flux_ul" in which else "none" wrap = partial( _ts_value, counts=counts, exposure=exposure, background=background, c_0=c_0, kernel=kernel, flux=flux, method=p["method"], error_method=error_method, threshold=p["threshold"], error_sigma=p["error_sigma"], ul_method=ul_method, ul_sigma=p["ul_sigma"], rtol=p["rtol"], ) x, y = np.where(mask.data) positions = list(zip(x, y)) with contextlib.closing(Pool(processes=p["n_jobs"])) as pool: log.info("Using {} jobs to compute TS map.".format(p["n_jobs"])) results = pool.map(wrap, positions) pool.join() # Set TS values at given positions j, i = zip(*positions) for name in ["ts", "flux", "niter"]: result[name].data[j, i] = [_[name] for _ in results] if "flux_err" in which: result["flux_err"].data[j, i] = [_["flux_err"] for _ in results] if "flux_ul" in which: result["flux_ul"].data[j, i] = [_["flux_ul"] for _ in results] # Compute sqrt(TS) values if "sqrt_ts" in which: result["sqrt_ts"] = self.sqrt_ts(result["ts"]) if downsampling_factor: for name in which: order = 0 if name == "niter" else 1 result[name] = result[name].upsample( factor=downsampling_factor, preserve_counts=False, order=order ) result[name] = result[name].crop(crop_width=pad_width) return result
def __repr__(self): p = self.parameters info = self.__class__.__name__ info += "\n\nParameters:\n\n" for key in p: info += "\t{key:13s}: {value}\n".format(key=key, value=p[key]) return info
def _ts_value( position, counts, exposure, background, c_0, kernel, flux, method, error_method, error_sigma, ul_method, ul_sigma, threshold, rtol, ): """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 flux : `~numpy.ndarray` Flux 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. """ # Get data slices counts_ = _extract_array(counts, kernel.shape, position) background_ = _extract_array(background, kernel.shape, position) exposure_ = _extract_array(exposure, kernel.shape, position) c_0_ = _extract_array(c_0, kernel.shape, position) model = exposure_ * kernel._array c_0 = c_0_.sum() if threshold is not None: with np.errstate(invalid="ignore", divide="ignore"): amplitude = flux[position] c_1 = f_cash(amplitude / FLUX_FACTOR, counts_, background_, model) # Don't fit if pixel significance is low if c_0 - c_1 < threshold: result = {} result["ts"] = (c_0 - c_1) * np.sign(amplitude) result["flux"] = amplitude result["niter"] = 0 result["flux_err"] = np.nan result["flux_ul"] = np.nan return result if method == "root brentq": amplitude, niter = _root_amplitude_brentq( counts_, background_, model, rtol=rtol ) elif method == "root newton": amplitude, niter = _root_amplitude( counts_, background_, model, flux[position], rtol=rtol ) elif method == "leastsq iter": amplitude, niter = _leastsq_iter_amplitude( counts_, background_, model, rtol=rtol ) else: raise ValueError("Invalid method: {}".format(method)) with np.errstate(invalid="ignore", divide="ignore"): c_1 = f_cash(amplitude, counts_, background_, model) result = {} result["ts"] = (c_0 - c_1) * np.sign(amplitude) result["flux"] = amplitude * FLUX_FACTOR result["niter"] = niter if error_method == "covar": flux_err = _compute_flux_err_covar(amplitude, counts_, background_, model) result["flux_err"] = flux_err * error_sigma elif error_method == "conf": flux_err = _compute_flux_err_conf( amplitude, counts_, background_, model, c_1, error_sigma ) result["flux_err"] = FLUX_FACTOR * flux_err if ul_method == "covar": result["flux_ul"] = result["flux"] + ul_sigma * result["flux_err"] elif ul_method == "conf": flux_ul = _compute_flux_err_conf( amplitude, counts_, background_, model, c_1, ul_sigma ) result["flux_ul"] = FLUX_FACTOR * flux_ul + result["flux"] return result def _leastsq_iter_amplitude(counts, background, model, maxiter=MAX_NITER, rtol=RTOL): """Fit amplitude using an iterative least squares algorithm. Parameters ---------- counts : `~numpy.ndarray` Slice of counts image background : `~numpy.ndarray` Slice of background image model : `~numpy.ndarray` Model template to fit. maxiter : int Maximum number of iterations. rtol : float Relative flux error. Returns ------- amplitude : float Fitted flux amplitude. niter : int Number of function evaluations needed for the fit. """ bounds = _amplitude_bounds_cython(counts, background, model) amplitude_min, amplitude_max, amplitude_min_total = bounds if not counts.sum() > 0: return amplitude_min_total, 0 weights = np.ones(model.shape) x_old = 0 for i in range(maxiter): x = _x_best_leastsq(counts, background, model, weights) if abs((x - x_old) / x) < rtol: return max(x / FLUX_FACTOR, amplitude_min_total), i + 1 else: weights = x * model + background x_old = x return max(x / FLUX_FACTOR, amplitude_min_total), MAX_NITER def _root_amplitude(counts, background, model, flux, rtol=RTOL): """Fit amplitude by finding roots using newton algorithm. See Appendix A Stewart (2009). Parameters ---------- counts : `~numpy.ndarray` Slice of count image background : `~numpy.ndarray` Slice of background image model : `~numpy.ndarray` Model template to fit. flux : float Starting value for the fit. Returns ------- amplitude : float Fitted flux amplitude. niter : int Number of function evaluations needed for the fit. """ args = (counts, background, model) with warnings.catch_warnings(): warnings.simplefilter("ignore") try: return ( newton( _f_cash_root_cython, flux, args=args, maxiter=MAX_NITER, tol=rtol ), 0, ) except RuntimeError: # Where the root finding fails NaN is set as amplitude return np.nan, MAX_NITER def _root_amplitude_brentq(counts, background, model, rtol=RTOL): """Fit amplitude by finding roots using Brent algorithm. See Appendix A Stewart (2009). Parameters ---------- counts : `~numpy.ndarray` Slice of count image background : `~numpy.ndarray` Slice of background image model : `~numpy.ndarray` Model template to fit. Returns ------- amplitude : float Fitted flux amplitude. niter : int Number of function evaluations needed for the fit. """ # Compute amplitude bounds and assert counts > 0 bounds = _amplitude_bounds_cython(counts, background, model) amplitude_min, amplitude_max, amplitude_min_total = bounds if not counts.sum() > 0: return amplitude_min_total, 0 args = (counts, background, model) with warnings.catch_warnings(): warnings.simplefilter("ignore") try: result = brentq( _f_cash_root_cython, amplitude_min, amplitude_max, args=args, maxiter=MAX_NITER, full_output=True, rtol=rtol, ) return max(result[0], amplitude_min_total), result[1].iterations except (RuntimeError, ValueError): # Where the root finding fails NaN is set as amplitude return np.nan, MAX_NITER def _compute_flux_err_covar(x, counts, background, model): """ Compute amplitude errors using inverse 2nd derivative method. """ with np.errstate(invalid="ignore", divide="ignore"): stat = (model ** 2 * counts) / (background + x * FLUX_FACTOR * model) ** 2 return np.sqrt(1.0 / stat.sum()) def _compute_flux_err_conf(amplitude, counts, background, model, c_1, error_sigma): """ Compute amplitude errors using likelihood profile method. """ def ts_diff(x, counts, background, model): return (c_1 + error_sigma ** 2) - f_cash(x, counts, background, model) args = (counts, background, model) amplitude_max = amplitude + 1e4 with warnings.catch_warnings(): warnings.simplefilter("ignore") try: result = brentq( ts_diff, amplitude, amplitude_max, args=args, maxiter=MAX_NITER, rtol=1e-3, ) return result - amplitude except (RuntimeError, ValueError): # Where the root finding fails NaN is set as amplitude return np.nan