In [1]:
import matplotlib.pyplot as plt
%matplotlib inline
# try:
# import seaborn as sns
# except ImportError:
# print("No seaborn installed. Oh well.")
import numpy as np
# import pandas as pd
import astropy.io.fits as fits
from astropy.table import Table
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
import os.path
# from clarsach.respond import RMF, ARF
In [2]:
datadir = "./"
ui.load_data(id="p1", filename=datadir+"RXTE_HEXTE_ClusterA.fak")
In [3]:
d = ui.get_data("p1")
In [4]:
# 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
print(bin_lo)
counts = d.counts
exposure = d.exposure
channel = d.channel
RXTE does not have BIN_LO and BIN_HI set. Because of course it doesn't.
In [5]:
plt.figure(figsize=(10,7))
plt.plot(channel, counts)
Out[5]:
RXTE HEXTE has an ARF, just like a regular X-ray telescope (and unlike RXTE/PCA, which doesn't):
In [6]:
arf_list = fits.open(datadir+"rxte_hexte_00may26_pwa.arf")
In [7]:
arf_list[1].data.field("ENERG_LO")[:10]
Out[7]:
In [8]:
arf_list[1].data.columns
Out[8]:
In [9]:
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")
In [10]:
energ_lo.shape
Out[10]:
Let's also load the rmf:
In [11]:
rmf_list = fits.open(datadir+"rxte_hexte_97mar20c_pwa.rmf")
rmf_list[1].header["TUNIT1"]
Out[11]:
In [12]:
rmf_list.info()
In [13]:
rmf_list["EBOUNDS"].columns
Out[13]:
In [14]:
energ_lo = rmf_list["MATRIX"].data.field("ENERG_LO")
energ_hi = rmf_list["MATRIX"].data.field("ENERG_HI")
bin_lo = rmf_list["EBOUNDS"].data.field("E_MIN")
bin_hi = rmf_list["EBOUNDS"].data.field("E_MAX")
bin_mid = bin_lo + (bin_hi - bin_lo)/2.
Ok, so all of our files have the same units. This is good.
Let's also get the ARF and RMF from sherpa:
In [15]:
arf = d.get_arf()
In [16]:
rmf = d.get_rmf()
rmf
Out[16]:
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)
In [19]:
m = pl.calculate(ener_lo=energ_lo, ener_hi=energ_hi)
In [20]:
plt.figure()
plt.loglog(energ_lo, m)
Out[20]:
In [21]:
m_arf = arf.apply_arf(m)
m_rmf = rmf.apply_rmf(m_arf)
In [22]:
plt.figure()
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_rmf*1e5, label="model with RMF")
plt.legend()
Out[22]:
Let's save model with ARF and RMF applied:
In [23]:
np.savetxt("../data/rxte_hexte_m_arf.txt", np.array([energ_lo, m_arf]).T)
np.savetxt("../data/rxte_hexte_m_rmf.txt", np.array([bin_mid, m_rmf]).T)
In [24]:
class PoissonLikelihood(object):
def __init__(self, x_low, x_high, y, model, arf=None, rmf=None, exposure=1.0):
self.x_low = x_low
self.x_high = x_high
self.y = y
self.model = model
self.arf = arf
self.rmf = rmf
self.exposure = exposure
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 self.arf is not None:
m_arf = self.arf.apply_arf(mean_model)
else:
m_arf = mean_model
if self.rmf is not None:
ymodel = self.rmf.apply_rmf(m_arf)
else:
ymodel = mean_model
ymodel += 1e-20
ymodel *= self.exposure
# compute the log-likelihood
loglike = np.sum(-ymodel + self.y*np.log(ymodel) \
- scipy_gammaln(self.y + 1.))
if np.isfinite(loglike):
return loglike
else:
return -1.e16
def __call__(self, pars):
l = -self.evaluate(pars)
#print(l)
return l
In [25]:
loglike = PoissonLikelihood(energ_lo, energ_hi, counts, pl, arf=arf, rmf=rmf, exposure=exposure)
loglike([1.0, 2.0])
Out[25]:
In [26]:
from scipy.optimize import minimize
In [27]:
opt = minimize(loglike, [1.0, 2.0], method="powell")
In [28]:
opt
Out[28]:
In [29]:
m_fit = pl.calculate(energ_lo, energ_hi)
m_fit_arf = arf.apply_arf(m_fit)
m_fit_rmf = rmf.apply_rmf(m_fit_arf) * exposure
In [30]:
plt.figure()
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_fit_rmf, label="model with ARF")
plt.legend()
Out[30]:
The sherpa stuff works! Good!
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 write a function that applies the ARF, which is just straightforward element-wise multiplication:
In [31]:
def apply_arf(spec, specresp, exposure=1.0):
"""
Apply the anxilliary response to
a spectrum.
Parameters
----------
spec : numpy.ndarray
The source spectrum in flux units
specresp: numpy.ndarray
The response
exposure : float
The exposure of the observation
Returns
-------
spec_arf : numpy.ndarray
The spectrum with the response applied.
"""
spec_arf = spec*specresp*exposure
return spec_arf
In [32]:
m_arf = apply_arf(m, arf.specresp)
In [33]:
m_arf_sherpa = arf.apply_arf(m)
In [34]:
np.allclose(m_arf, m_arf_sherpa)
Out[34]:
In [35]:
plt.plot(m_arf, label="Clarsach ARF")
plt.plot(m_arf_sherpa, label="Sherpa ARF")
plt.legend()
Out[35]:
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 [36]:
def rmf_fold(spec, rmf, nchannels=None):
#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 [37]:
def rmf_fold_vector(spec, rmf, nchannels=None):
"""
Fold the spectrum through the redistribution matrix.
Parameters
----------
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)
#print('len(nchannels): ' + str(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 [38]:
# not vectorized version
m_rmf = rmf_fold(m_arf, rmf)
%timeit m_rmf = rmf_fold(m_arf, rmf)
In [39]:
# vectorized version
m_rmf_v = rmf_fold_vector(m_arf, rmf)
%timeit m_rmf_v = rmf_fold_vector(m_arf, rmf)
In [40]:
# C++ (sherpa) version
m_rmf2 = rmf.apply_rmf(m_arf)
%timeit m_rmf2 = rmf.apply_rmf(m_arf)
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 [41]:
np.allclose(m_rmf_v[:len(m_rmf2)], m_rmf2)
Out[41]:
In [42]:
np.allclose(m_rmf[:len(m_rmf2)], m_rmf2)
Out[42]:
In [43]:
class RMF(object):
def __init__(self, filename):
self._load_rmf(filename)
pass
def _load_rmf(self, filename):
"""
Load an RMF from a FITS file.
Parameters
----------
filename : str
The file name with the RMF file
Attributes
----------
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 = fits.open(filename)
# get all the extension names
extnames = np.array([h.name for h in hdulist])
# figure out the right extension to use
if "MATRIX" in extnames:
h = hdulist["MATRIX"]
elif "SPECRESP MATRIX" in extnames:
h = hdulist["SPECRESP MATRIX"]
data = h.data
hdr = h.header
hdulist.close()
# 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 = self.__get_tlmin(h)
# flatten the variable-length arrays
self.n_grp, self.f_chan, self.n_chan, self.matrix = \
self.__flatten_arrays(n_grp, f_chan, n_chan, matrix)
return
def __get_tlmin(self, h):
"""
Get the tlmin keyword for `F_CHAN`.
Parameters
----------
h : an astropy.io.fits.hdu.table.BinTableHDU object
The extension containing the `F_CHAN` column
Returns
-------
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 = np.int(list(hdr.items())[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])
# some matrices actually have more elements
# than groups in `n_grp`, so we'll only pick out
# those values that have a correspondence in
# n_grp
f_chan_new = []
n_chan_new = []
for i,t in enumerate(nz_idx):
if t:
n = n_grp[i]
f = f_chan[i]
nc = n_chan[i]
f_chan_new.append(f[:n])
n_chan_new.append(nc[:n])
n_chan_flat = np.hstack(n_chan_new)
f_chan_flat = np.hstack(f_chan_new)
# if n_chan is zero, we'll remove those as well.
nz_idx2 = (n_chan_flat > 0)
n_chan_flat = n_chan_flat[nz_idx2]
f_chan_flat = f_chan_flat[nz_idx2]
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
set
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.
Parameters
----------
spec : numpy.ndarray
The (model) spectrum to be folded
Returns
-------
counts : numpy.ndarray
The (model) spectrum after folding, in
counts/s/channel
"""
# an empty array for the output counts
counts = np.zeros_like(spec)
# 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(spec.shape[0]):
# 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):
current_num_chans = int(self.n_chan[k])
if current_num_chans == 0:
k += 1
resp_idx += current_num_chans
continue
else:
# 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
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] += self.matrix[resp_idx:resp_idx+current_num_chans] * \
np.float(source_bin_i)
# iterate the response index for next round
resp_idx += current_num_chans
return counts[:self.detchans]
Let's make an object of that class and then make another model with the RMF applied:
In [44]:
rmf_new = RMF(datadir+"rxte_hexte_97mar20c_pwa.rmf")
m_rmf = rmf_new.apply_rmf(m_arf)
Hooray! What does the total spectrum look like?
In [45]:
plt.figure(figsize=(10, 6))
plt.plot(m_rmf)
#plt.plot(m_rmf_v)
Out[45]:
There is, of course, also an implementation in clarsach:
In [46]:
from clarsach import respond
In [47]:
rmf_c = respond.RMF(datadir+"rxte_hexte_97mar20c_pwa.rmf")
In [48]:
m_rmf_c = rmf_new.apply_rmf(m_arf)
In [49]:
np.allclose(m_rmf_c, m_rmf2)
Out[49]:
In [50]:
plt.figure()
plt.plot(bin_mid, counts, label="data")
plt.plot(bin_mid, m_rmf_c*exposure, label="model", alpha=0.5)
Out[50]:
And it works! Woo!