Responses for Chandra/HETG

Let's see if we can figure out responses for the Chandra/HETG. This notebook will require sherpa, because we're using this for comparison purposes!

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
    import seaborn as sns
except ImportError:
    print("No seaborn installed. Oh well.")
import numpy as np 
import pandas as pd
import as fits
import sherpa.astro.ui as ui
import astropy.modeling.models as models
from astropy.modeling.fitting import _fitter_to_model_params
from scipy.special import gammaln as scipy_gammaln

from clarsach.respond import RMF, ARF

WARNING: imaging routines will not be available, 
failed to import sherpa.image.ds9_backend due to 
'RuntimeErr: DS9Win unusable: Could not find ds9 on your PATH'
WARNING: failed to import WCS module; WCS routines will not be available
WARNING: failed to import sherpa.astro.xspec; XSPEC models will not be available

Let's load some HETG simulations of a simple power law. The input values for that spectrum are:

  • norm = 1
  • ph_index = 2

In [2]:
datadir = "./"

Note: sherpa will only load the responses correctly if the data is in the same folder as this notebook. Go figure.

In [3]:
ui.load_data(id="p1", filename=datadir+"fake_heg_p1.pha")

WARNING: systematic errors were not found in file './fake_heg_p1.pha'
statistical errors were found in file './fake_heg_p1.pha' 
but not used; to use them, re-read with use_errors=True
read ARF file arfs/aciss_heg1_cy19.garf
read RMF file rmfs/aciss_heg1_cy19.grmf

Let's take a look at that data:

In [4]:
d = ui.get_data("p1")

In [5]:
# conversion factor from keV to angstrom
c = 12.3984191

# This is the data in angstrom
bin_lo = d.bin_lo
bin_hi = d.bin_hi
bin_mid = bin_lo + (bin_hi - bin_lo) / 2.0

counts = d.counts

In [6]:
plt.plot(bin_mid, counts)

[<matplotlib.lines.Line2D at 0x116398748>]

Let's load the ARF from a file:

In [7]:
hdulist ="fake_heg_p1.pha")

In [8]:


In [9]:
exposure = hdulist[1].header["EXPOSURE"]

In [10]:
bin_lo = hdulist[1].data.field("BIN_LO")
bin_hi = hdulist[1].data.field("BIN_HI")
channels = hdulist[1].data.field("CHANNEL")
counts = hdulist[1].data.field("COUNTS")

In [11]:
arf_list ="arfs/aciss_heg-1_cy19.garf")

In [12]:


In [13]:
specresp = arf_list[1].data.field("SPECRESP")
energ_lo = arf_list[1].data.field("ENERG_LO")
energ_hi = arf_list[1].data.field("ENERG_HI")
phafrac = arf_list[1].data.field("PHAFRAC")

In [14]:

    name = 'ENERG_LO'; format = '1E'; unit = 'keV'
    name = 'ENERG_HI'; format = '1E'; unit = 'keV'
    name = 'SPECRESP'; format = '1E'; unit = 'cm**2'
    name = 'BIN_LO'; format = '1E'; unit = 'angstrom'
    name = 'BIN_HI'; format = '1E'; unit = 'angstrom'
    name = 'FRACEXPO'; format = '1E'
    name = 'PHAFRAC'; format = '1E'

Let's also load the rmf:

In [15]:
rmf_list ="rmfs/aciss_heg1_cy19.grmf")


Ok, so all of our files have the same units. This is good, because this is not always the case for Chandra/HETG.

Note: We need to fix that in the code!

Let's load the ARF and the RMF from sherpa:

In [16]:
arf = d.get_arf()
specresp = arf.specresp

rmf = d.get_rmf()

Now we can make a model spectrum we can play around with! clarsach has an implementation of a Powerlaw model that matches the ISIS one that created the data:

In [17]:
from clarsach.models.powerlaw import Powerlaw

In [18]:
pl = Powerlaw(norm=1.0, phoindex=2.0)

Now we can produce a model with the same resolution as the HETG:

In [57]:
m = pl.calculate(ener_lo=energ_lo, ener_hi=energ_hi)

In [58]:
plt.plot(bin_mid, m)

[<matplotlib.lines.Line2D at 0x115fcf6a0>]

This looks pretty weird! But remember, those are the integral of the power law over the bins! Let's apply the rmf and arf to this model:

In [59]:
m_arf = m*specresp*1.e5

m_rmf = rmf.apply_rmf(m_arf)

In [60]:
np.savetxt("../data/chandra_hetg_m_rmf.txt", np.array([bin_mid, m_rmf]).T)

In [61]:
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_rmf, label="model with ARF")

<matplotlib.legend.Legend at 0x116bc5668>

Okay, that looks pretty good!

Building a Model

Let's build a model for fitting a model to the data. We're going to do nothing fancy here, just a simple Poisson likelihood. It's similar to the one in the TestAthenaXIFU notebook, except it uses the Clarsach powerlaw function and integrates over bins.

In [23]:
class PoissonLikelihood(object):
    def __init__(self, x_low, x_high, y, model, arf=None, rmf=None):
        self.x_low = x_low
        self.x_high = x_high
        self.y = y
        self.model = model
        self.arf = arf
        self.rmf = rmf

    def evaluate(self, pars):
        # store the new parameters in the model

        self.model.norm = pars[0]
        self.model.phoindex = pars[1]
        # evaluate the model at the positions x
        mean_model = self.model.calculate(self.x_low, self.x_high)

        # run the ARF and RMF calculations
        if arf is not None and rmf is not None:
            m_arf = arf.apply_arf(mean_model)*arf.exposure
            ymodel = rmf.apply_rmf(m_arf)
            ymodel = mean_model
        ymodel += 1e-20
        # compute the log-likelihood
        loglike = np.sum(-ymodel + self.y*np.log(ymodel) \
               - scipy_gammaln(self.y + 1.))

        if np.isfinite(loglike):
            return loglike
            return -1.e16

    def __call__(self, pars):
        l = -self.evaluate(pars)
        return l

In [24]:
loglike = PoissonLikelihood(bin_lo, bin_hi, counts, pl, arf=arf, rmf=rmf)
loglike([1.0, 2.0])


Let's fit that:

In [25]:
from scipy.optimize import minimize

In [26]:
opt = minimize(loglike, [1.0, 2.0])

In [27]:

      fun: 21421.497920481364
 hess_inv: array([[  3.93325864e-03,   3.99093102e-04],
       [  3.99093102e-04,   4.07296844e-05]])
      jac: array([-0.00097656,  0.00463867])
  message: 'Desired error not necessarily achieved due to precision loss.'
     nfev: 187
      nit: 19
     njev: 44
   status: 2
  success: False
        x: array([ 10.1011883,   1.9971106])

In [28]:
m_fit = pl.calculate(bin_lo, bin_hi)
m_fit_arf = arf.apply_arf(m_fit) * arf.exposure
m_fit_rmf = rmf.apply_rmf(m_fit_arf)

In [29]:
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_fit_rmf, label="model with ARF")

<matplotlib.legend.Legend at 0x1092b8b38>

Woo! It works! There's something funny with the exposure I need to ask Lia about!

Making Our Own ARF/RMF Classes

Ideally, we don't want to be dependent on the sherpa implementation. We're now going to write our own implementations, and then test them on the model above.

Let's first write a function for applying the ARF. This is easy, because the ARF just requires a multiplication with the input spectrum.

In [45]:
def apply_arf(spec, specresp, exposure=1.0):
    Apply the anxilliary response to 
    a spectrum. 
    spec : numpy.ndarray
        The source spectrum in flux units
    specresp: numpy.ndarray
        The response 
    exposure : float
        The exposure of the observation
    spec_arf : numpy.ndarray
        The spectrum with the response applied.
    spec_arf = spec*specresp*exposure
    return spec_arf

Let's try it!

In [63]:
m_arf = apply_arf(m, arf.specresp)

We can compare that with the result from sherpa:

In [64]:
m_arf_sherpa = arf.apply_arf(m)*1e5

In [66]:

array([ 0.01688494,  0.01498274,  0.01315201,  0.01144996,  0.00988449,
        0.00843824,  0.00712234,  0.00593137,  0.00484785,  0.00387135])

In [67]:
np.savetxt("../data/chandra_hetg_m_arf.txt", np.array([bin_mid, m_arf_sherpa]).T, 

In [70]:
np.allclose(m_arf*1e5, m_arf_sherpa)


In [ ]:

In [51]:

array([ 0.01688504,  0.01498289,  0.01315159,  0.01145029,  0.00988428,
        0.00843817,  0.00712239,  0.00593143,  0.00484788,  0.00387131])

In [52]:

array([ 0.01688504,  0.01498289,  0.01315159,  0.01145029,  0.00988428,
        0.00843817,  0.00712239,  0.00593143,  0.00484788,  0.00387131])

It works! There's also a version in the new clarsach package, which we should test and compare as well:

In [34]:
from clarsach.respond import ARF

In [53]:
arf_c = ARF(datadir+"arfs/aciss_heg1_cy19.garf")
m_arf_c = arf_c.apply_arf(m)

In [55]:
np.allclose(m_arf_sherpa, m_arf_c*1e5)


It works!

Next, let's look at the RMF, which is more complex. This requires a matrix multiplication. However, the response matrices are compressed to remove zeros and save space in memory, so they require a little more complex fiddling. Here's an implementation that is basically almost a line-by-line translation of the C++ code:

In [37]:
def rmf_fold(spec, rmf):
    #current_num_groups = 0
    #current_num_chans = 0
    nchannels = spec.shape[0]
    resp_idx = 0
    first_chan_idx = 0
    num_chans_idx =0
    counts_idx = 0
    counts = np.zeros(nchannels)
    for i in range(nchannels):
        source_bin_i = spec[i]
        current_num_groups = rmf.n_grp[i]
        while current_num_groups:
            counts_idx = int(rmf.f_chan[first_chan_idx] - rmf.offset)
            current_num_chans = rmf.n_chan[num_chans_idx]
            first_chan_idx += 1
            num_chans_idx +=1 
            while current_num_chans:
                counts[counts_idx] += rmf.matrix[resp_idx] * source_bin_i
                counts_idx += 1
                resp_idx += 1
                current_num_chans -= 1
            current_num_groups -= 1
    return counts

Let's see if we can make this more vectorized:

In [38]:
def rmf_fold_vector(spec, rmf):
    Fold the spectrum through the redistribution matrix.
    spec : numpy.ndarray
        The (model) spectrum to be folded
    rmf : sherpa.RMFData object
        The object with the RMF data
    # get the number of channels in the data
    nchannels = spec.shape[0]
    # an empty array for the output counts
    counts = np.zeros(nchannels)
    # index for n_chan and f_chan incrementation
    k = 0
    # index for the response matrix incrementation
    resp_idx = 0

    # loop over all channels
    for i in range(nchannels):
        # this is the current bin in the flux spectrum to 
        # be folded
        source_bin_i = spec[i]
        # get the current number of groups 
        current_num_groups = rmf.n_grp[i]
        for j in range(current_num_groups):
            counts_idx = int(rmf.f_chan[k] - rmf.offset)
            current_num_chans = int(rmf.n_chan[k])
            k += 1
            counts[counts_idx:counts_idx+current_num_chans] += rmf.matrix[resp_idx:resp_idx+current_num_chans] * source_bin_i
            resp_idx += current_num_chans
    return counts

Let's time the different implementations and compare them to the sherpa version (which is basically a wrapper around the C++):

In [39]:
# not vectorized version
m_rmf = rmf_fold(m_arf, rmf)
%timeit m_rmf = rmf_fold(m_arf, rmf)

1 loop, best of 3: 692 ms per loop

In [40]:
# vectorized version
m_rmf_v = rmf_fold_vector(m_arf, rmf)
%timeit m_rmf_v = rmf_fold_vector(m_arf, rmf)

10 loops, best of 3: 35.6 ms per loop

In [41]:
# C++ (sherpa) version
m_rmf2 = rmf.apply_rmf(m_arf)
%timeit m_rmf2 = rmf.apply_rmf(m_arf)

1000 loops, best of 3: 826 µs per loop

So my vectorized implementation is ~20 times slower than the sherpa version, and the non-vectorized version is really slow. But are they all the same?

In [42]:
np.allclose(m_rmf_v, m_rmf2)


In [43]:
np.allclose(m_rmf, m_rmf2)


They are! It looks like for this particular Chandra/HETG data set, it's working!

Making an ARF/RMF Class

the ARF and RMF code would live well in a class, so let's wrap it into a class:

In [44]:
class RMF(object):
    def __init__(self, filename):
    def _load_rmf(self, filename):
        Load an RMF from a FITS file.
        filename : str
            The file name with the RMF file
        n_grp : numpy.ndarray
            the Array with the number of channels in each 
            channel set
        f_chan : numpy.ndarray
            The starting channel for each channel group;
            If an element i in n_grp > 1, then the resulting 
            row entry in f_chan will be a list of length n_grp[i];
            otherwise it will be a single number
        n_chan : numpy.ndarray
            The number of channels in each channel group. The same 
            logic as for f_chan applies
        matrix : numpy.ndarray
            The redistribution matrix as a flattened 1D vector
        energ_lo : numpy.ndarray
            The lower edges of the energy bins
        energ_hi : numpy.ndarray
            The upper edges of the energy bins
        detchans : int
            The number of channels in the detector
        # open the FITS file and extract the MATRIX extension
        # which contains the redistribution matrix and 
        # anxillary information
        hdulist =
        h = hdulist["MATRIX"]
        data =
        hdr = h.header
        # extract + store the attributes described in the docstring 
        n_grp = np.array(data.field("N_GRP"))
        f_chan = np.array(data.field('F_CHAN'))
        n_chan = np.array(data.field("N_CHAN"))
        matrix = np.array(data.field("MATRIX"))
        self.energ_lo = np.array(data.field("ENERG_LO"))
        self.energ_hi = np.array(data.field("ENERG_HI"))
        self.detchans = hdr["DETCHANS"]
        self.offset = 1#  self.__get_tlmin(h)
        self.n_grp, self.f_chan, self.n_chan, self.matrix = \
                self.__flatten_arrays(n_grp, f_chan, n_chan, matrix)
    def __get_tlmin(self, h):
        Get the tlmin keyword for `F_CHAN`.
        h : an object
            The extension containing the `F_CHAN` column
        tlmin : int
            The tlmin keyword
        # get the header
        hdr = h.header
        # get the keys of all 
        keys = np.array(list(hdr.keys()))
        # find the place where the tlmin keyword is defined
        t = np.array(["TLMIN" in k for k in keys])
        # get the index of the TLMIN keyword
        tlmin_idx = np.hstack(np.where(t))[0]
        # get the corresponding value
        tlmin =[tlmin_idx][1])
        return tlmin
    def __flatten_arrays(self, n_grp, f_chan, n_chan, matrix):
        # find all non-zero groups
        nz_idx = (n_grp > 0)
        # stack all non-zero rows in the matrix
        matrix_flat = np.hstack(matrix[nz_idx])
        # stack all nonzero rows in n_chan and f_chan
        n_chan_flat = np.hstack(n_chan[nz_idx])
        f_chan_flat = np.hstack(f_chan[nz_idx])
        return n_grp, f_chan_flat, n_chan_flat, matrix_flat
    def apply_rmf(self, spec):
        Fold the spectrum through the redistribution matrix.

        The redistribution matrix is saved as a flattened 1-dimensional 
        vector to save space. In reality, for each entry in the flux 
        vector, there exists one or more sets of channels that this 
        flux is redistributed into. The additional arrays `n_grp`, 
        `f_chan` and `n_chan` store this information:
            * `n_group` stores the number of channel groups for each 
              energy bin
            * `f_chan` stores the *first channel* that each channel 
              for each channel set 
            * `n_chan` stores the number of channels in each channel 
        As a result, for a given energy bin i, we need to look up the 
        number of channel sets in `n_grp` for that energy bin. We 
        then need to loop over the number of channel sets. For each 
        channel set, we look up the first channel into which flux 
        will be distributed as well as the number of channels in the 
        group. We then need to also loop over the these channels and 
        actually use the corresponding elements in the redistribution 
        matrix to redistribute the photon flux into channels. 
        All of this is basically a big bookkeeping exercise in making 
        sure to get the indices right.
        spec : numpy.ndarray
            The (model) spectrum to be folded

        counts : numpy.ndarray
            The (model) spectrum after folding, in 

        # get the number of channels in the data
        nchannels = spec.shape[0]

        # an empty array for the output counts
        counts = np.zeros(nchannels)

        # index for n_chan and f_chan incrementation
        k = 0

        # index for the response matrix incrementation
        resp_idx = 0

        # loop over all channels
        for i in range(nchannels):

            # this is the current bin in the flux spectrum to 
            # be folded
            source_bin_i = spec[i]

            # get the current number of groups 
            current_num_groups = self.n_grp[i]

            # loop over the current number of groups
            for j in range(current_num_groups):
                # get the right index for the start of the counts array 
                # to put the data into
                counts_idx = int(self.f_chan[k] - self.offset)
                # this is the current number of channels to use
                current_num_chans = int(self.n_chan[k])

                # iterate k for next round
                k += 1
                # add the flux to the subarray of the counts array that starts with 
                # counts_idx and runs over current_num_chans channels
                counts[counts_idx:counts_idx+current_num_chans] = counts[counts_idx:counts_idx+current_num_chans] + \
                                                                  self.matrix[resp_idx:resp_idx+current_num_chans] * \
                # iterate the response index for next round
                resp_idx += current_num_chans

        return counts

Let's make an object of that class and then make another model with the RMF applied:

In [45]:
rmf_new = RMF(datadir+"rmfs/aciss_heg1_cy19.grmf")
m_rmf = rmf_new.apply_rmf(m_arf)

Does this produce the same result as the vectorized function (it should)?

In [46]:
np.allclose(m_rmf, m_rmf_v)


Hooray! What does the total spectrum look like?

In [47]:

[<matplotlib.lines.Line2D at 0x108ee7e80>]

There is, of course, also an implementation in clarsach:

In [48]:
from clarsach import respond

In [49]:
rmf_c = respond.RMF(datadir+"rmfs/aciss_heg1_cy19.grmf")

In [50]:
m_rmf_c = rmf_new.apply_rmf(m_arf)

In [51]:
np.allclose(m_rmf_c, m_rmf_v)


It all works! Hooray! I think we've got a solution for Chandra/HETG!

Let's make a final plot:

In [52]:
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_rmf_c, label="model")

[<matplotlib.lines.Line2D at 0x109dd47f0>]

It all works! Woo!