Best fit tools

Using leastsq()

Running an iterative best fit process with leastsq() involves the following steps:

• designing a fit model function having the signature f(x, ...), where x is an array of values of the independent variable and all remaining parameters are the parameters to be fitted

• defining the sequence of initial values for all parameters to be fitted. You can usually start with [1., 1., ...] if you don’t have a better estimate. The algorithm is robust enough to converge to a solution most of the times.

• setting constraints (optional)

Data required to perform a fit is:

• an array of x values (abscissa, independent variable)

• an array of y data points

• the sigma array of uncertainties associated to each data point. This is optional, by default each data point is assigned a weight of 1.

Default (unweighted) fit

Let’s demonstrate this process in a short example, using synthetic data. We generate an array of synthetic data using a polynomial function of degree 4, and try to use leastsq() to find back the function parameters.

import numpy
from silx.math.fit import leastsq

# create some synthetic polynomial data
x = numpy.arange(1000)
y = 2.4 * x**4 - 10. * x**3 + 15.2 * x**2 - 24.6 * x + 150.

# define our fit function: a generic polynomial of degree 4
def poly4(x, a, b, c, d, e):
    return a * x**4 + b * x**3 + c * x**2 + d * x + e

# The fit is an iterative process that requires an initial
# estimation of the parameters. Let's just use 1s.
initial_parameters = numpy.array([1., 1., 1., 1., 1.])

# Run fit
fitresult = leastsq(model=poly4,
                    xdata=x,
                    ydata=y,
                    p0=initial_parameters,
                    full_output=True)

# leastsq with full_output=True returns 3 objets
optimal_parameters, covariance, infodict = fitresult
# the first object is an array with the fitted parameters
a, b, c, d, e = optimal_parameters

print("Fit took %d iterations" % infodict["niter"])
print("Reduced chi-square: %f" % infodict["reduced_chisq"])
print("Theoretical parameters:\n\t" +
      "a=2.4, b=-10, c=15.2, d=-24.6, e=150")
print("Optimal parameters for y2 fitting:\n\t" +
      "a=%f, b=%f, c=%f, d=%f, e=%f" % (a, b, c, d, e))

The output of the above program is:

Fit took 35 iterations
Reduced chi-square: 682592.670690
Theoretical parameters:
    a=2.4, b=-10, c=15.2, d=-24.6, e=150
Optimal parameters for y fitting:
    a=2.400000, b=-9.999665, c=14.970422, d=31.683448, e=-3216.131136

Note

The exact results may vary depending on your Python version.

We can see that the fitting result is poor. In particular, parameters d and e are very poorly fitted.

This is due to the fact that data points with large values have a stronger influence in the fitting process. In our examples, as x increases, y increases fast. The influence of the weighting and how to solve this issue is explained in more details in the next section.

In the meantime, if you simply limit the x range to deal with smaller y values, you can notice that the fit result becomes perfect.

In our example, replacing x by:

x = numpy.arange(100)

produces the following result:

Fit took 9 iterations
Reduced chi-square: 0.000000
Theoretical parameters:
    a=2.4, b=-10, c=15.2, d=-24.6, e=150
Optimal parameters for y fitting:
    a=2.400000, b=-10.000000, c=15.200000, d=-24.600000, e=150.000000

Weighted fit

Since the fitting algorithm minimizes the sum of squared differences between input and calculated data, points with higher y values have a greater weight in the fitting process. A way to improve the fit, is to attach uncertainties to the data. The larger the uncertainty on a data sample, the smaller its weight in the least-square problem.

It is important to set the uncertainties correctly, or you risk to introduce a bias either toward the lower values or toward the higher values in your data.

The common approach in event counting experiments is to use the squareroot of the data values as the uncertainty values (assuming a Poissonian law). Let’s apply it to the previous example:

sigma = numpy.sqrt(y)

# Fit y
fitresult = leastsq(model=poly4,
                    xdata=x,
                    ydata=y,
                    sigma=sigma,
                    p0=initial_parameters,
                    full_output=True)

This results in a great improvement:

Weighted fit took 6 iterations
Reduced chi-square: 0.000000
Theoretical parameters:
    a=2.4, b=-10, c=15.2, d=-24.6, e=150
Optimal parameters for y fitting:
    a=2.400000, b=-10.000000, c=15.200000, d=-24.600000, e=150.000000

The resulting fit is perfect. The fit converged even faster than when we limited the x range to 0 – 100.

To use a real world example, when fitting x-ray fluorescence spectroscopy data, this common approach means that we take the variance of each channel to be the number of counts in that channel. That corresponds to assuming a normal distribution. The true distribution being a Poisson distribution, the Gaussian distribution is a good approximation for channels with high number of counts, but the approximation is not valid when the number of counts in a channel is low.

Therefore, in spectra where the overall statistics is very low, a weighted fit can lead the fitting process to fit the background considering the peaks as outliers, because the fit procedure will consider a channel with 1 count has a 100 times higher weight than a channel with 100 counts.

Constrained fit

Let’s revert to our unweighted fit and try out with different approaches for improving the fit.

The leastsq() function provides a way to set constraints on parameters. You can for instance enforce that a given parameter must remain equal to its initial value, or define an acceptable range for it to vary, or decide that a parameter must be equal to another parameter multiplied by a certain factor. This is very useful for cases in which you have enough knowledge to make reasonable assumptions on some parameters.

In our case, we will set constraints on d and e. We will require d to stay in the range between -25 and -24, and fix e to 150.

Replace the call to leastsq() by the following lines:

# Define constraints
cons = [[0, 0, 0],          # a: no constraint
        [0, 0, 0],          # b: no constraint
        [0, 0, 0],          # c: no constraint
        [2, -25., -23.],    # -25 < d < -24
        [3, 0, 0]]          # e is fixed to initial value
fitresult = leastsq(poly4, x, y,
                    # initial values must be consistent with constraints
                    p0=[1., 1., 1., -24., 150.],
                    constraints=cons,
                    full_output=True)

The output of this program is:

Constrained fit took 100 iterations
Reduced chi-square: 3.749280
Theoretical parameters:
    a=2.4, b=-10, c=15.2, d=-24.6, e=150
Optimal parameters for y fitting:
    a=2.400000, b=-9.999999, c=15.199648, d=-24.533014, e=150.000000

The chi-square value is substantially improved and the results are much better, at the cost of more iterations.

Using FitManager

A FitManager is a tool that provides a way of handling fit functions, associating estimation functions to estimate the initial parameters, modify the configuration parameters for the fit (enabling or disabling weights…) or for the estimation function, and choosing a background model.

It provides an abstraction layer on top of leastsq().

Weighted polynomial fit

The following program accomplishes the same weighted fit of a polynomial as in the previous tutorial (See Weighted fit)

import numpy
from silx.math.fit.fitmanager import FitManager

# Create synthetic data with a sum of gaussian functions
x = numpy.arange(1000).astype(numpy.float)
y = 2.4 * x**4 - 10. * x**3 + 15.2 * x**2 - 24.6 * x + 150.

# define our fit function: a generic polynomial of degree 4
def poly4(x, a, b, c, d, e):
    return a * x**4 + b * x**3 + c * x**2 + d * x + e

# define an estimation function to that returns initial parameters
# and constraints
def esti(x, y):
    p0 = numpy.array([1., 1., 1., 1., 1.])
    cons = numpy.zeros(shape=(5, 3))
    return p0, cons

# Fitting
fit = FitManager()
fit.setdata(x=x, y=y)

fit.addtheory("polynomial",
              function=poly4,
              # any list of 5 parameter names would be OK
              parameters=["A", "B", "C", "D", "E"],
              estimate=esti)
fit.settheory('polynomial')
fit.configure(WeightFlag=True)
fit.estimate()
fit.runfit()

print("\n\nFit took %d iterations" % fit.niter)
print("Reduced chi-square: %f" % fit.chisq)
print("Theoretical parameters:\n\t" +
      "a=2.4, b=-10, c=15.2, d=-24.6, e=150")
a, b, c, d, e = (param['fitresult'] for param in fit.fit_results)
print("Optimal parameters for y2 fitting:\n\t" +
      "a=%f, b=%f, c=%f, d=%f, e=%f" % (a, b, c, d, e))

The result is the same as in the weighted leastsq() example, as expected:

Fit took 6 iterations
Reduced chi-square: 0.000000
Theoretical parameters:
    a=2.4, b=-10, c=15.2, d=-24.6, e=150
Optimal parameters for y2 fitting:
    a=2.400000, b=-10.000000, c=15.200000, d=-24.600000, e=150.000000

Fitting Gaussians

The FitManager object is especially useful for fitting multi-peak Gaussian-shaped spectra. The silx module silx.math.fit.fittheories contains fit functions and their associated estimation functions that are specifically designed for this purpose.

These fit functions can handle a varying number of parameters defining a variable number of peaks, and the estimation functions use a peak detection algorithm to determine how many initial parameters must be returned.

For the sake of example, let’s test the multi-peak fitting on synthetic data, generated using another silx module: silx.math.fit.functions.

import numpy
from silx.math.fit.functions import sum_gauss
from silx.math.fit import fittheories
from silx.math.fit.fitmanager import FitManager

# Create synthetic data with a sum of gaussian functions
x = numpy.arange(1000).astype(numpy.float)

# height, center x, fwhm
p = [1000, 100., 250,     # 1st peak
     255, 690., 45,       # 2nd peak
     1500, 800.5, 95]     # 3rd peak

y = sum_gauss(x, *p)

# Fitting
fit = FitManager()
fit.setdata(x=x, y=y)
fit.loadtheories(fittheories)
fit.settheory('Gaussians')
fit.estimate()
fit.runfit()

print("Searched parameters = %s" % p)
print("Obtained parameters : ")
dummy_list = []
for param in fit.fit_results:
    print(param['name'], ' = ', param['fitresult'])
    dummy_list.append(param['fitresult'])
print("chisq = ", fit.chisq)

The result of this program is:

Searched parameters = [1000, 100.0, 250, 255, 690.0, 45, 1500, 800.5, 95]
Obtained parameters :
('Height1', ' = ', 1000.0)
('Position1', ' = ', 100.0)
('FWHM1', ' = ', 250.0)
('Height2', ' = ', 255.0)
('Position2', ' = ', 690.0)
('FWHM2', ' = ', 44.999999999999993)
('Height3', ' = ', 1500.0)
('Position3', ' = ', 800.5)
('FWHM3', ' = ', 95.000000000000014)
('chisq = ', 0.0)

In addition to Gaussians, we could have fitted several other similar types of function: asymmetric Gaussian functions, Lorentzian functions, pseudo-voigt functions or hypermet tailing functions.

The loadtheories() method can also be used to load user defined functions. Instead of a module, a path to a Python source file can be given as a parameter. This source file must adhere to certain conventions, as explained in the documentation of silx.math.fit.fittheories and silx.math.fit.fittheory.FitTheory.

Background subtraction

FitManager provides a few standard background theories, for cases when a background signal is superimposed on the multi-peak spectrum.

For instance, let’s add a linear background to our synthetic data, and see how FitManager handles the fitting.

In our previous example, redefine y as follows:

p = [1000, 100., 250,
     255, 690., 45,
     1500, 800.5, 95]
y = sum_gauss(x, *p)
# add a synthetic linear background
y += 0.13 * x + 100.

Before the line fit.estimate(), add the following line:

fit.setbackground('Linear')

The result becomes:

Searched parameters = [1000, 100.0, 250, 255, 690.0, 45, 1500, 800.5, 95]
Obtained parameters :
('Constant', ' = ', 100.00000000000001)
('Slope', ' = ', 0.12999999999999998)
('Height1', ' = ', 1000.0)
('Position1', ' = ', 100.0)
('FWHM1', ' = ', 249.99999999999997)
('Height2', ' = ', 255.00000000000003)
('Position2', ' = ', 690.0)
('FWHM2', ' = ', 44.999999999999993)
('Height3', ' = ', 1500.0)
('Position3', ' = ', 800.5)
('FWHM3', ' = ', 95.0)
('chisq = ', 3.1789004676997597e-27)

The available background theories are: Linear, Constant and Strip.

The strip background is a popular background model that can compute and subtract any background shape as long as its curvature is significantly lower than the peaks’ curvature. In other words, as long as the background signal is significantly smoother than the actual signal, it can be easily computed.

The main parameters required by the strip function are the strip width w and the number of iterations. At each iteration, if the content of channel i, y(i) is above the average of the contents of the channels at a distance w (measured in channel units) i.e., y(i-w) and y(i+w), y(i) is replaced by the average value of the neighbouring channels. At the end of the process one is left with something that resembles a spectrum in which the peaks have been “stripped”.

The following example illustrates the strip background removal process:

from silx.sx import plot
from silx.gui import qt
import numpy
from silx.math.fit.filters import strip
from silx.math.fit.functions import sum_gauss

x = numpy.arange(5000)
# (height1, center1, fwhm1, ...) 5 peaks
params1 = (50, 500, 100,
           20, 2000, 200,
           50, 2250, 100,
           40, 3000, 75,
           23, 4000, 150)
y0 = sum_gauss(x, *params1)

# random values between [-1;1]
noise = 2 * numpy.random.random(5000) - 1
# make it +- 5%
noise *= 0.05

# 2 gaussians with very large fwhm, as background signal
actual_bg = sum_gauss(x, 15, 3500, 3000, 5, 1000, 1500)

# Add 5% random noise to gaussians and add background
y = y0 * (1 + noise) + actual_bg

# compute strip background model
strip_bg = strip(y, w=5, niterations=5000)

# plot results
app = qt.QApplication([])
plot(x, y, x, actual_bg, x, strip_bg)
plot(x, y, x, (y - strip_bg))
app.exec_()

	Data with background in black (y), actual background in red, computed strip background in green
	Data with background in blue, data after subtracting the strip background in black

The strip also removes the statistical noise, so the computed strip background will be slightly lower than the actual background. This can be solved by smoothing the data prior to the strip computation.

See the PyMca documentation for more information on the strip background.

To configure the strip background model of FitManager, use its configure() method to modify the following parameters:

• StripWidth: strip width parameter w, mentioned earlier

• StripNIterations: number of iterations

• StripThresholdFactor: if this parameter is left to its default value 1, the algorithm behaves as explained earlier: y(i) is compared to the average of y(i-1) and y(i+1). If this factor is set to another value, f, y(i) is compared to the average multiplied by f.

• SmoothStrip: if this parameter is set to True, smoothing is performed prior to stripping.

These parameters can be modified like this:

# ...
fit.settheory('Strip')
fit.configure(StripWidth=5,
              StripNIterations=5000,
              StripThresholdFactor=1.1,
              SmoothStrip=True)
# ...

Using a strip background has performance implications. You should try to keep the number of iterations as low as possible if you need to run batch fitting using this model. Increasing the strip width can help reducing the number of iterations, with the risk of underestimating the background signal.

Using FitWidget

Simple usage

FitWidget is a graphical interface to FitManager.

import numpy
from silx.gui import qt
from silx.gui.fit import FitWidget
from silx.math.fit.functions import sum_gauss

x = numpy.arange(2000).astype(numpy.float)
constant_bg = 3.14

# gaussian parameters: height, position, fwhm
p = numpy.array([1000, 100., 30.0,
                 500, 300., 25.,
                 1700, 500., 35.,
                 750, 700., 30.0,
                 1234, 900., 29.5,
                 302, 1100., 30.5,
                 75, 1300., 75.])
y = sum_gauss(x, *p) + constant_bg

a = qt.QApplication([])

w = FitWidget()
w.setData(x=x, y=y)
w.show()

a.exec_()

Executing this code opens the following widget.

The functions you can choose from the widget are the standard Gaussian-shaped functions from silx.math.fit.fittheories. At the top of the list, you will find the Add Function(s) option, that allows loading the user defined fit theories from a .py source file.

After selecting the Constant background model and clicking the Estimate button, the widget displays this:

7 peaks have been detected, and their parameters estimated. Also, the estimation function defines some constraints (positive height and full-width at half-maximum).

You can modify the values in the estimation column of the table, to use different initial fit parameters.

The individual constraints can be modified prior to fitting. It is also possible to modify the constraints globally by clicking the Configure button to open a configuration dialog. To get help on the meaning of the various parameters, hover the mouse cursor on the corresponding check box or entry widget to display a tooltip help message.

The other configuration tabs can be modified to change the peak search parameters and the strip background parameters prior to estimating them. After closing the configuration dialog, you must re-run the estimation by clicking the Estimate button.

After all configuration parameters and all constraints are set according to your preferences, you can click the Start Fit button. This runs the fit and displays the results in the Fit Value column of the table.

Customising the functions

The FitWidget can be initialised with a non-standard FitManager to customise the available functions.

from silx.gui import qt
from silx.math.fit import FitManager
from silx.gui.fit import FitWidget

def linearfun(x, a, b):
    return a * x + b

# create synthetic data for the example
x = list(range(0, 100))
y = [linearfun(x_, 2.0, 3.0) for x_ in x]

# we need to create a custom fit manager and add our theory
myfitmngr = FitManager()
myfitmngr.setdata(x, y)
myfitmngr.addtheory("my linear function",
                    function=linearfun,
                    parameters=["a", "b"])

a = qt.QApplication([])

# our fit widget can now use our custom fit manager
fw = FitWidget(fitmngr=myfitmngr)
fw.show()

a.exec_()

In our previous example, we didn’t load a customised FitManager, therefore, the fit widget automatically initialised the default fit manager and loaded the default custom Gaussian functions.

This time, we initialised our own FitManager and loaded our own function, so only this function is presented as an option in the GUI.

Our custom function does not provide an associated estimation function, so the default estimation function of the FitManager is used. This default estimation function returns an array of ones with the same length as the list of parameter names, and sets all constraints to FREE.