# The implementation of the MBACK function

Here is the entire mback function:

def mback(energy, mu, group=None, order=3, z=None, edge='K', e0=None, emin=None, emax=None,
whiteline=None, form='mback', tables='cl', fit_erfc=False, return_f1=False,
_larch=None):
"""
Match mu(E) data for tabulated f"(E) using the MBACK algorithm and,
optionally,  the Lee & Xiang extension

energy, mu:    arrays of energy and mu(E)
order:         order of polynomial [3]
group:         output group (and input group for e0)
z:             Z number of absorber
edge:          absorption edge (K, L3)
e0:            edge energy
emin:          beginning energy for fit
emax:          ending energy for fit
whiteline:     exclusion zone around white lines
form:          'mback' or 'lee'
tables:        'cl' or 'chantler'
fit_erfc:      True to float parameters of error function

References:
* MBACK (Weng, Waldo, Penner-Hahn): http://dx.doi.org/10.1086/303711
* Lee and Xiang: http://dx.doi.org/10.1088/0004-637X/702/2/970
* Cromer-Liberman: http://dx.doi.org/10.1063/1.1674266
* Chantler: http://dx.doi.org/10.1063/1.555974
"""
order=int(order)
if order < 1: order = 1 # set order of polynomial
if order > MAXORDER: order = MAXORDER

### implement the First Argument Group convention
energy, mu, group = parse_group_args(energy, members=('energy', 'mu'),
defaults=(mu,), group=group,
fcn_name='mback')
group = set_xafsGroup(group, _larch=_larch)

if e0 is None:              # need to run find_e0:
e0 = xray_edge(z, edge, _larch=_larch)[0]
if e0 is None:
e0 = group.e0
if e0 is None:
find_e0(energy, mu, group=group)

### theta is an array used to exclude the regions <emin, >emax, and
### around white lines, theta=0.0 in excluded regions, theta=1.0 elsewhere
(i1, i2) = (0, len(energy)-1)
if emin != None: i1 = index_of(energy, emin)
if emax != None: i2 = index_of(energy, emax)
theta = np.ones(len(energy)) # default: 1 throughout
theta[0:i1]  = 0
theta[i2:-1] = 0
if whiteline:
pre     = 1.0*(energy<e0)
post    = 1.0*(energy>e0+float(whiteline))
theta   = theta * (pre + post)
if edge.lower().startswith('l'):
l2      = xray_edge(z, 'L2', _larch=_larch)[0]
l2_pre  = 1.0*(energy<l2)
l2_post = 1.0*(energy>l2+float(whiteline))
theta   = theta * (l2_pre + l2_post)

## this is used to weight the pre- and post-edge differently as
## defined in the MBACK paper
weight1 = 1*(energy<e0)
weight2 = 1*(energy>e0)
weight  = np.sqrt(sum(weight1))*weight1 + np.sqrt(sum(weight2))*weight2

## get the f'' function from CL or Chantler
if tables.lower() == 'chantler':
f1 = f1_chantler(z, energy, _larch=_larch)
f2 = f2_chantler(z, energy, _larch=_larch)
else:
(f1, f2) = f1f2(z, energy, edge=edge, _larch=_larch)
group.f2=f2
if return_f1: group.f1=f1

n = edge
if edge.lower().startswith('l'): n = 'L'
params = Group(s      = Parameter(1, vary=True, _larch=_larch),     # scale of data
xi     = Parameter(50, vary=fit_erfc, min=0, _larch=_larch), # width of erfc
em     = Parameter(xray_line(z, n, _larch=_larch)[0], vary=False, _larch=_larch),
e0     = Parameter(e0, vary=False, _larch=_larch),   # abs. edge energy
en     = energy,
mu     = mu,
f2     = group.f2,
weight = weight,
theta  = theta,
form   = form,
_larch = _larch)
if fit_erfc:
params.a = Parameter(1, vary=True,  _larch=_larch) # amplitude of erfc
else:
params.a = Parameter(0, vary=False, _larch=_larch) # amplitude of erfc

for i in range(order): # polynomial coefficients
setattr(params, 'c%d' % i, Parameter(0, vary=True, _larch=_larch))

fit = Minimizer(match_f2, params, _larch=_larch, toler=1.e-5)
fit.leastsq()

eoff = energy - params.e0.value
normalization_function = params.a.value*erfc((energy-params.em.value)/params.xi.value) + params.c0.value
for i in range(MAXORDER):
j = i+1
attr = 'c%d' % j
if hasattr(params, attr):
normalization_function = normalization_function + getattr(getattr(params, attr),'value') * eoff**j

group.fpp = params.s*mu - normalization_function
group.mback_params = params


That's kind of long. Let's break it down into pieces.

## The function signature

def mback(energy, mu, group=None, order=3, z=None, edge='K', e0=None, emin=None, emax=None,
whiteline=None, form='mback', tables='cl', fit_erfc=False, return_f1=False,
_larch=None):


The first three lines provide the signature of the function. This sets the input interface for the function and defines default values for most of the input parameters. Function signatures are no different in Larch than in normal python.

The next several lines are the document string for the mback function. This can be read at any time from the larch command line by print _xafs.mback.__doc__.

## Managing the function arguments

The next three lines enforce the type (integer) and value (1<order<6) of the order argument, which sets the order of the Legendre polynomial used in the normalization.

    order=int(order)
if order < 1: order = 1 # set order of polynomial
if order > MAXORDER: order = MAXORDER


The next four lines enforce Larch's First Argument Group convention, which is a way of tersely specifying the data Group on which the function is to operate.

    energy, mu, group = parse_group_args(energy, members=('energy', 'mu'),
defaults=(mu,), group=group,
fcn_name='mback')
group = set_xafsGroup(group, _larch=_larch)


This allows the user to specify a Group and have Larch use the .energy and .mu attributes of the group in the function. While this is fine, my personal preference is not to use this convention. For one thing, it seems more clear to me explicitly to specify the energy and mu arrays. For another, it allows the user to use Group attributes with names other than .energy and .mu as the arguments for the mback function. Those attribute names are not guaranteed when using Larch's read_ascii function or some of its other IO functionality.

The next six lines are used to set the e0 argument if it is not provided.

    if e0 is None:              # need to run find_e0:
e0 = xray_edge(z, edge, _larch=_larch)[0]
if e0 is None:
e0 = group.e0
if e0 is None:
find_e0(energy, mu, group=group)


The default is to use the tabulated value for the specified z and edge. If that doesn't work and no e0 value is set for the input data group, Larch's find_e0 function will be run to determine a guess for e0 based on the content of the data.

## Excluding portions of the data

In certain situations, it is useful to exclude data from the MBACK fit. The emin and emax arguments allow you to exclude data from the beginning or end of the data range. That is used, for instance, if the data contain an edge step from another element, which is a data feature that this implementation of the MBACK algorithm is not designed to handle.

Another use of the exclusion array is to exclude the region around a white line from the fit. In some cases, the white line carries so much spectral weight that it skews the fit results, causing the matched data to be "slanted" relative to the tabulated data.

    (i1, i2) = (0, len(energy)-1)
if emin != None: i1 = index_of(energy, emin)
if emax != None: i2 = index_of(energy, emax)
theta = np.ones(len(energy)) # default: 1 throughout
theta[0:i1]  = 0
theta[i2:-1] = 0
if whiteline:
pre     = 1.0*(energy<e0)
post    = 1.0*(energy>e0+float(whiteline))
theta   = theta * (pre + post)
if edge.lower().startswith('l'):
l2      = xray_edge(z, 'L2', _larch=_larch)[0]
l2_pre  = 1.0*(energy<l2)
l2_post = 1.0*(energy>l2+float(whiteline))
theta   = theta * (l2_pre + l2_post)


The exclusion array, theta is initialized as an array for which each element is 1.0 and of the same length as the data. That is, it is intended to be interpreted on the same energy grid as the data. For regions of energy to be excluded from the fit, theta is set to 0.0. This array can then be multiplied by the minimization function, which has the effect of removing data in the exclusion regions from the determination of the parameters.

## Weights for the fit

In the minimization function given above, the function is split into two regions, the first n1 data points and all the rest. Often the pre-edge region contains far fewer data points than the region above the edge. To make sure that the normalization function does a good job of matching the pre-edge data, a different weighting is used. That is the purpose of the n11 and n21 terms in the minimization function. The weight array accommodates this feature of the MBACK algorithm.

    weight1 = 1*(energy<e0)
weight2 = 1*(energy>e0)
weight  = np.sqrt(sum(weight1))*weight1 + np.sqrt(sum(weight2))*weight2


## Tabulated cross section data

The next seven lines gather tabulated values for bare-atom cross sections from either the Cromer-Liberman or Chantler tables. These arrays are generated on the data grid, broadened by the core-hole lifetime, and placed in the data group. If the return_cl flag is set to True, the real part of the energy-dependent cross-section is also placed in the data group. (The real part is needed by the diffkk plugin, which relies upon this plugin for normalization).

    if tables.lower() == 'chantler':
f1 = f1_chantler(z, energy, _larch=_larch)
f2 = f2_chantler(z, energy, _larch=_larch)
else:
(f1, f2) = f1f2(z, energy, edge=edge, _larch=_larch)
group.f2=f2
if return_f1: group.f1=f1


Placing these arrays in the data group is an important feature of this plugin. Remember that a Group is simply a container for symbols. It should be a useful container. By placing the cross section data in the group, the user is easily able to make plots of the normalized data along with the normalization standard.

A plugin which operates on a data Group will almost always put useful and relevant arrays, constants, and other symbols in the data Group.

## Setting up the fit parameters

The next 19 lines establish the parameters of the fit.

First, we set a throw-away parameter based on the input edge argument.

    n = edge
if edge.lower().startswith('l'): n = 'L'


This will be used to set the fixed value of Eem in the normalization function.

Then we make a Group called params, which is our container for holding all the variables, constants, and arrays required to evaluate the minimization function.

    params = Group(s      = Parameter(1, vary=True, _larch=_larch),     # scale of data
xi     = Parameter(50, vary=fit_erfc, min=0, _larch=_larch), # width of erfc
em     = Parameter(xray_line(z, n, _larch=_larch)[0], vary=False, _larch=_larch), # erfc centroid
e0     = Parameter(e0, vary=False, _larch=_larch),   # abs. edge energy
en     = energy,
mu     = mu,
f2     = group.f2,
weight = weight,
theta  = theta,
form   = form,
_larch = _larch)


The first parameter set is the scale for the measured data, s. This is made into a Larch Parameter and flagged as a variable.

If the complementary error function is included in the fit by setting fit_erfc to True, then the width ξ is marked as variable parameter. The complementary error function centroid Eem and the edge energy are set as constants of the fit. Eem is set to the centroid of the emission lines associated with the absorption edge. Note that ξ is constrained never to be negative.

en, mu, f2, weight, and theta are the arrays needed to evaluate the minimization function. It is convenient to place these arrays in the params Group. form is a string. If it is set to lee then the modification to the minimization function given by Lee and Xiang (see Eq. 3) will be made.

Next the a parameter -- the amplitude of the complementary error function is set to 0 (which has the effect of setting the entire complementary error function to 0) if the fit_erfc input parameter is False. Otherwise, a is guessed

    if fit_erfc:
params.a = Parameter(1, vary=True,  _larch=_larch) # amplitude of erfc
else:
params.a = Parameter(0, vary=False, _larch=_larch) # amplitude of erfc


Finally, the parameters of the Legendre polynomial are guessed:

    for i in range(order): # polynomial coefficients
setattr(params, 'c%d' % i, Parameter(0, vary=True, _larch=_larch))


This is a tricky bit. For each order of polynomial specified in the input arguments, a parameter is guessed. We want parameters with names like c1, c2, and so on. Iteratively constructing parameter names like that requires using python's setattr function.

First the parameter name is made using the string formatting operator. Like the parameters defined above, the Larch's Parameter function is used and the parameter is defined to be a variable of the fit. The setattr function then makes a parameter symbol, gives it a name like c1 or c2, and places it in the params Group.

## Performing the minimization

Finally, the minimization happens.

    fit = Minimizer(match_f2, params, _larch=_larch, toler=1.e-5)
fit.leastsq()


A Minimizer object is made. This takes as arguments the name of the objective function (see the next section) and the name of the Group containing the parameters of the fit. Once the Minimizer object is made, the fit is performed by the call to leastsq().

## Finishing up

Once the fit is finished, we reconstruct the normalization function using the best fit values as the sum of the complementary error function and the Legendre polynomial.

    eoff = energy - params.e0.value
normalization_function = params.a.value*erfc((energy-params.em.value)/params.xi.value) + params.c0.value
for i in range(MAXORDER):
j = i+1
attr = 'c%d' % j
if hasattr(params, attr):
normalization_function  = normalization_function + getattr(getattr(params, attr), 'value') * eoff**j

group.fpp = params.s*mu - normalization_function
group.mback_params = params


The hasattr function is used to restrict the order of the polynomial to be the same as was used in the fit. If the order was 2 in the fit, there will not be a symbol in params called c3. hasattr returns true only for symbols in params. The repeated use of getattr digs the best fit value for each Legendre polynomial coefficient out of params. That's a bit unwieldy, but is required when probing the contents of a Group in a computed manner, as shown here.

Finally, the matched data are placed in the data Group, as is the params Group which allows the user access ot the parameters of the fit.