# Fitting X-ray Absorbance Spectra¶

When numerical methods are insufficient, it may be necessary to fit the pixel spectra with a model function and extract parameters from the model. The core XanesFrameset class has methods for common fitting-related use-cases, such as using curves to approximate L3 and K edges, and linear combination fitting of standard spectra. If the pre-rolled options are not enough, arbitary callables can be created and fit against the data. The following is an example for fitting a single L-edge spectrum:

Es = np.linspace(845, 865, num=1000)
l3 = xp.fitting.L3Curve(Es, num_peaks=2)

# Create initial guess, matching l3.param_names
p0 = (1.1, 853, 0.6, 1, 855, 0.6, 0.15, 854, 10, 3)

# Now do the fitting
result = xp.fitting.fit_spectra(observations=obs, func=l3, p0=p0)
params, residuals = result

# Plot the resulting fit and original data
predicted = l3(*params)
plt.plot(Es, obs)
plt.plot(Es, predicted, linestyle=':')


Todo

Come up with a better illustration for fitting.

Note

If no p0 is given to fit_spectra(), xanespy will attempt to guess starting parameters. Not every callable in xanespy.fitting supports this feature. If trying to fit spectra without p0 raises a GuessParamsError, then p0 is required.

## Linear Combination Fitting¶

Often times the observed spectrum is a linear combination of spectral sources from known standards. This works best when the standards is stable and have been isolated and measured on the same instrument as the observed data.

In order the fit linear combinations of source, use the fit_linear_combinations() method:

# Load your previously imported data
fs = xp.XanesFrameset(...)

# Prepare the sources for fitting
source1 = ...
source2 = ...

# The sources must have the same number of points as the data
assert len(source1) == fs.num_energies
assert len(source2) == fs.num_energies

# Now execute the fitting
results = fs.fit_linear_combinations(sources=[source1, source2])
fits, residuals = results


Todo

Create a figure to illustrate LC fitting.

This method will create three new HDF5 datasets:

• linear_combination_parameters (maps)
• linear_combination_residuals (maps)
• linear_combination_sources (arrays)

The naming prefix can be controlled by passing the name parameter to fit_linear_combinations() method. If more control is needed, the xanespy.fitting.LinearCombination class can be subclassed and given to the fit_spectra() method as described below.

## Shortcuts for Common Use-Cases¶

The XanesFrameset class has several shortcuts for common fitting tasks. Fitting the spectra with a K-edge spectra can be done easily with the fit_kedge() method. Linear combinations of existing functions can be easily fit using fit_linear_combinations().

## Rolling Your Own Fit Function¶

If none of the options suit your needs, you can create a callable that produces the curve you wish to fit given a number of parameters, then pass this to the fit_spectra() method. In the simplest case this can be a simple function:

import numpy as np
import xanespy as xp

# Define the function we wish to fit against
def sin_curve(scale, frequency, phase):
theta = np.linspace(0, 2*np.pi, num=100)
out = scale * np.sin(frequency * theta(phase))
return out

fs = xp.XanesFrameset(...)
# Come up with an initial guess
pnames = ('scale', 'frequency', 'phase')
p0 = (0, 1, 0)
fs.fit_spectra(func=sin_curve, p0=p0, pnames=pnames, name='sin_curve')


In many cases, static information (such as the list of energies) is needed to construct the curve. This can be given to a class’s constructor and the algorithm itself placed in the __call__ method. This is illustrated below by fitting a variable number of sine waves, making a sort of horribly inefficient fourier transform. Since the number of sine waves is not known at import-time, the use of star-arguments makes the result more dynamic. Adding the param_names saves us the trouble of passing it in every time. Providing a guess_params method allows fit_spectra() to automatically guess the parameters for each spectrum before fitting.

import random

import xanespy as xp
import numpy as np

# Define a new callable for passing to the fitting function
class SineCurves(xp.fitting.Curve):
def __init__(self, theta, num_sines=1):
self.theta = theta
self.num_sines = num_sines

def __call__(self, *params):
out = np.zeros_like(self.theta)
# Iterate on the parameters in groups of 3
for i in xrange(0, len(params), 3):
scale, freq, phase = params[i:i+3]
# Add another sin wave to the total curve
out += scale * np.sin((self.theta-phase) * frequency)
return out

@property
def param_names(self):
# Build a list of 2 params for each sine wave
names = []
for num in range(self.num_sines):
names.append('scale%d' % num)
names.append('frequency%d' % num)
names.append('phase%d' % num)
return names

def guess_params(self, intensities, edge, named_tuple=True):
p0 = []
for i in range(self.num_sines):
p0 += [1, 2*i+1, 0]
# Convert to named tuple for user convenience (optional)
if named_tuple:
Params = namedtuple('Params', self.param_names)
p0 = Params(*p0)
else:
p0 = tuple(p0)
return p0

# Create the actual callable object
theta = np.linspace(0, 2*pi, num=100)
sines = SineCurves(theta=theta, num_sines=3)
# Load the data and do the fitting
fs = xp.XanesFrameset(...)
fs.fit_spectra(func=sines, p0=p0, name='sine_curve_fit')
`