Curve fitting¶

Curve fitting in ERLabPy largely relies on lmfit, a flexible curve fitting library for Python, and xarray-lmfit, a compatibility layer between xarray objects and lmfit models.

ERLabPy also provides optional integration of lmfit models with iminuit , which is a Python interface to the Minuit C++ library developed at CERN.

Note

If you are new to lmfit or xarray-lmfit, visit the lmfit documentation and the xarray-lmfit user guide first!

In this tutorial, we begin with some convenient functions that ERLabPy provides for common tasks such as Fermi edge fitting. Next, we will introduce some models that are available in ERLabPy. Finally, we will show how to use iminuit with lmfit models.

import lmfit
import matplotlib.pyplot as plt
import numpy as np
import xarray as xr

import erlab.analysis as era
import erlab.plotting as eplt
from erlab.io.exampledata import generate_gold_edge

Fermi edge fitting¶

Functions related to the Fermi edge are available in erlab.analysis.gold. To fit a polynomial to a Fermi edge, you can use erlab.analysis.gold.poly().

Hint

The interactive Fermi edge fitting tool erlab.interactive.goldtool() can be used to generate the code below interactively.

gold = generate_gold_edge(temp=100, seed=1)

result = era.gold.poly(
    gold,
    angle_range=(-15, 15),
    eV_range=(-0.2, 0.2),
    temp=100.0,
    vary_temp=False,
    bkg_slope=False,
    degree=2,
    plot=True,
)

../_images/37d6e16fe02a2ae41d18b9ec0a4392a3acb650b18e8040b2acdbf57c759264de.svg

The resulting polynomial can be used to correct the Fermi edge with erlab.analysis.gold.correct_with_edge():

corrected = era.gold.correct_with_edge(gold, result)

corrected.qplot(cmap="Greys")  # Plot the corrected data
eplt.fermiline()  # Annotate the Fermi level

<matplotlib.lines.Line2D at 0x719cedbca0c0>

../_images/0a93fb43bd28079bd70f4f9868fdb82329af46a92dcf878df5558d84af2e7458.svg

Pre-defined models¶

Creating composite models with different prefixes every time can be cumbersome, so ERLabPy provides some pre-defined models in erlab.analysis.fit.models.

Before fitting, let us generate a Gaussian peak on a linear background:

# Generate toy data
x = np.linspace(0, 10, 50)
y = -0.1 * x + 2 + 3 * np.exp(-((x - 5) ** 2) / (2 * 1**2))

# Add some noise with fixed seed for reproducibility
rng = np.random.default_rng(5)
yerr = np.full_like(x, 0.3)
y = rng.normal(y, yerr)

# Plot the data
plt.errorbar(x, y, yerr, fmt="o")

<ErrorbarContainer object of 3 artists>

../_images/c52b4b7adfe9e1a1a305b8774630d6128f75bc76fd3c5678d19fcdf69e094326.svg

Fitting multiple peaks¶

One example is MultiPeakModel, which works like a composite model of multiple Gaussian or Lorentzian peaks.

By supplying keyword arguments, you can specify the number of peaks, their shapes, whether to multiply with a Fermi-Dirac distribution, the shape of the background, and whether to convolve the result with experimental resolution.

For a detailed explanation of all the arguments, see its documentation.

The model can be constructed as follows:

model = era.fit.models.MultiPeakModel(
    npeaks=1, peak_shapes=["gaussian"], fd=False, background="linear", convolve=False
)
params = model.make_params(p0_center=5.0, p0_width=0.2, p0_height=3.0)
params

Parameters
name	value	initial value	min	max	vary	expression
p0_center	5.00000000	5.0	-inf	inf	True
p0_width	0.20000000	0.2	0.00000000	inf	True
p0_height	3.00000000	3.0	0.00000000	inf	True
const_bkg	0.00000000	None	-inf	inf	True
lin_bkg	0.00000000	None	-inf	inf	True
p0_sigma	0.08493218	None	-inf	inf	False	p0_width / (2 * sqrt(2 * log(2)))
p0_amplitude	0.10164911	None	-inf	inf	False	p0_height * p0_sigma / sqrt(2 * pi)

We can now fit the model to the toy data:

result = model.fit(y, x=x, params=params, weights=1 / yerr)
_ = result.plot()

/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/uncertainties/core.py:1024: UserWarning: Using UFloat objects with std_dev==0 may give unexpected results.
  warn("Using UFloat objects with std_dev==0 may give unexpected results.")
/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/IPython/core/events.py:82: UserWarning: There are no gridspecs with layoutgrids. Possibly did not call parent GridSpec with the "figure" keyword
  func(*args, **kwargs)
/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/IPython/core/pylabtools.py:170: UserWarning: There are no gridspecs with layoutgrids. Possibly did not call parent GridSpec with the "figure" keyword
  fig.canvas.print_figure(bytes_io, **kw)

../_images/870ba984634dffb99093a46dc558758164a992b3a6e194e5dba21063e4c927d9.svg

We can also plot components.

comps = result.eval_components()
plt.errorbar(x, y, yerr, fmt="o", zorder=-1, alpha=0.3)
plt.plot(x, result.eval(), label="Best fit")
plt.plot(x, comps["1Peak_p0"], "--", label="Peak")
plt.plot(x, comps["1Peak_bkg"], "--", label="Background")
plt.legend()

<matplotlib.legend.Legend at 0x719cedb97560>

../_images/3db418a8023e01d57f18d09b032582a2c09a1226eb4545962f1b054e6d541adf.svg

Now, let us try fitting MDCs cut from simulated data with multiple Lorentzian peaks, convolved with a common instrumental resolution.

from erlab.io.exampledata import generate_data

data = generate_data(seed=1).T
cut = data.qsel(ky=0.3)
cut.qplot(colorbar=True)

<matplotlib.image.AxesImage at 0x719ced964ad0>

../_images/191a9150d2f79a07143665233d72e1c9a80cba25822596c68a81e8bb5ba51971.svg

mdc = cut.qsel(eV=0.0)
mdc.qplot()

[<matplotlib.lines.Line2D at 0x719ce47f18b0>]

../_images/78b8234df0bb1cf7f57862aa89d49d5a94d68e692ed47a0aff704316ec7c9b82.svg

First, we define the model and set the initial parameters.

model = era.fit.models.MultiPeakModel(
    npeaks=2, peak_shapes=["lorentzian"], fd=False, background="linear", convolve=True
)

params = model.make_params(
    p0_center=-0.5,
    p1_center=0.5,
    p0_width=0.03,
    p1_width=0.03,
    lin_bkg={"value": 0.0, "vary": False},
    const_bkg=0.0,
    resolution=0.03,
)
params

Parameters
name	value	initial value	min	max	vary	expression
p0_center	-0.50000000	-0.5	-inf	inf	True
p0_width	0.03000000	0.03	0.00000000	inf	True
p0_height	-inf	None	0.00000000	inf	True
p1_center	0.50000000	0.5	-inf	inf	True
p1_width	0.03000000	0.03	0.00000000	inf	True
p1_height	-inf	None	0.00000000	inf	True
const_bkg	0.00000000	0.0	-inf	inf	True
lin_bkg	0.00000000	0.0	-inf	inf	False
resolution	0.03000000	0.03	0.00000000	inf	True
p0_sigma	0.01500000	None	-inf	inf	False	p0_width / 2
p0_amplitude	-inf	None	-inf	inf	False	p0_height * p0_sigma * pi
p1_sigma	0.01500000	None	-inf	inf	False	p1_width / 2
p1_amplitude	-inf	None	-inf	inf	False	p1_height * p1_sigma * pi

Then, we can fit the model to the data using xarray.DataArray.xlm.modelfit() from xarray-lmfit:

result = mdc.xlm.modelfit("kx", model=model, params=params)
_ = result.modelfit_results.item().plot()

/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/uncertainties/core.py:1024: UserWarning: Using UFloat objects with std_dev==0 may give unexpected results.
  warn("Using UFloat objects with std_dev==0 may give unexpected results.")
/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/IPython/core/events.py:82: UserWarning: There are no gridspecs with layoutgrids. Possibly did not call parent GridSpec with the "figure" keyword
  func(*args, **kwargs)
/home/docs/checkouts/readthedocs.org/user_builds/erlabpy/envs/v3.12.0/lib/python3.12/site-packages/IPython/core/pylabtools.py:170: UserWarning: There are no gridspecs with layoutgrids. Possibly did not call parent GridSpec with the "figure" keyword
  fig.canvas.print_figure(bytes_io, **kw)

../_images/b83c85ec27fcdba9cb13cb6494cae39143e9e620470d08565c1b303112fe5693.svg

Fitting background models¶

xarray.Dataset.xlm.modelfit() and xarray.DataArray.xlm.modelfit() works with any lmfit model, including background models from lmfitxps. If you have lmfitxps installed, you can use the ShirleyBG model to iteratively fit a Shirley background to the data:

from lmfitxps.models import ShirleyBG
from lmfit.models import GaussianModel

darr.xlm.modelfit("alpha", GaussianModel() + ShirleyBG())

Visualizing fits¶

Note

If you are viewing this documentation online, the plots will not be interactive. Run the code locally to try it out.

If hvplot is installed, we can visualize the fit results interactively with the xarray.Dataset.qshow() accessor.

To plot the data with the fit and fit components:

result_ds.qshow(plot_components=True)

To plot each parameter as a function of the coordinate:

result_ds.qshow.params()

Parallelization¶

For non-dask objects, you can achieve joblib-based parallelization:

For non-dask Datasets, basic parallelization across multiple data variables can be achieved with the parallel argument to xarray.Dataset.xlm.modelfit().
For parallelizing fits on a single DataArray along a dimension with many points, the xarray.DataArray.parallel_fit() accessor can be used. This method is similar to xarray.DataArray.xlm.modelfit(), but requires the name of the dimension to parallelize over instead of the coordinates to fit along. For example, to parallelize the fit in the previous example, you can use the following code:
```
gold_selected.parallel_fit(
    dim="alpha",
    model=FermiEdgeModel(),
    params={"temp": {"value": 100.0, "vary": False}},
    guess=True,
)
```
Note
- Note that the initial run will take a long time due to the overhead of creating parallel workers. Subsequent calls will run faster, since joblib’s default backend will try to reuse the workers.
- The accessor has some intrinsic overhead due to post-processing. If you need the best performance, handle the parallelization yourself with joblib and lmfit.Model.fit.

Saving and loading fits¶

See the xarray-lmfit documentation for details on saving and loading fit results.

Using `iminuit`¶

Note

This part requires the optional iminuit dependency.

iminuit is a powerful Python interface to the Minuit C++ library developed at CERN. To learn more, see the iminuit documentation.

ERLabPy provides a thin wrapper around iminuit.Minuit that allows you to use lmfit models with iminuit. The example below conducts the same fit as the previous one, but using iminuit.

model = era.fit.models.MultiPeakModel(
    npeaks=2, peak_shapes=["lorentzian"], fd=False, convolve=True
)

m = era.fit.minuit.Minuit.from_lmfit(
    model,
    mdc,
    mdc.kx,
    p0_center=-0.5,
    p1_center=0.5,
    p0_width=0.03,
    p1_width=0.03,
    p0_height=1000,
    p1_height=1000,
    lin_bkg={"value": 0.0, "vary": False},
    const_bkg=0.0,
    resolution=0.03,
)

m.migrad()
m.minos()
m.hesse()

Migrad
FCN = 938.4 (χ²/ndof = 3.9)	Nfcn = 1333
EDM = 0.000132 (Goal: 0.0002)	time = 0.2 sec
Valid Minimum	Below EDM threshold (goal x 10)
No parameters at limit	Below call limit
Hesse ok	Covariance accurate

	Name	Value	Hesse Error	Minos Error-	Minos Error+	Limit-	Limit+	Fixed
0	p0_center	-518.935e-3	0.017e-3	-0.017e-3	0.017e-3
1	p0_width	30.38e-3	0.11e-3	-0.11e-3	0.11e-3	0
2	p0_height	1.0804e3	0.0031e3	-0.0031e3	0.0031e3	0
3	p1_center	519.207e-3	0.017e-3	-0.017e-3	0.017e-3
4	p1_width	30.32e-3	0.11e-3	-0.11e-3	0.11e-3	0
5	p1_height	1.0912e3	0.0031e3	-0.0031e3	0.0031e3	0
6	const_bkg	0.87	0.08	-0.08	0.08
7	lin_bkg	0.0	0.1					yes
8	resolution	26.89e-3	0.12e-3	-0.12e-3	0.12e-3	0

	p0_center		p0_width		p0_height		p1_center		p1_width		p1_height		const_bkg		resolution
Error	-0.017e-3	0.017e-3	-0.11e-3	0.11e-3	-3.1	3.1	-0.017e-3	0.017e-3	-0.11e-3	0.11e-3	-3.1	3.1	-0.08	0.08	-0.12e-3	0.12e-3
Valid	True	True	True	True	True	True	True	True	True	True	True	True	True	True	True	True
At Limit	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False
Max FCN	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False
New Min	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False	False

	p0_center	p0_width	p0_height	p1_center	p1_width	p1_height	const_bkg	lin_bkg	resolution
p0_center	2.97e-10	0	53.08e-9	-0	-0.01e-9 (-0.004)	235.92e-9 (0.004)	-0.66e-9	0	0.01e-9 (0.005)
p0_width	0	1.27e-08	-339.751e-6 (-0.974)	0.02e-9 (0.012)	0.010e-6 (0.792)	-283.577e-6 (-0.803)	-4.642e-6 (-0.505)	0	-0.012e-6 (-0.878)
p0_height	53.08e-9	-339.751e-6 (-0.974)	9.57	-649.92e-9 (-0.012)	-280.127e-6 (-0.804)	8 (0.822)	0.112 (0.443)	0	327.678e-6 (0.904)
p1_center	-0	0.02e-9 (0.012)	-649.92e-9 (-0.012)	2.92e-10	0.02e-9 (0.008)	-493.21e-9 (-0.009)	-4.85e-9 (-0.003)	0	-0.03e-9 (-0.014)
p1_width	-0.01e-9 (-0.004)	0.010e-6 (0.792)	-280.127e-6 (-0.804)	0.02e-9 (0.008)	1.27e-08	-344.032e-6 (-0.975)	-4.639e-6 (-0.504)	0	-0.012e-6 (-0.881)
p1_height	235.92e-9 (0.004)	-283.577e-6 (-0.803)	8 (0.822)	-493.21e-9 (-0.009)	-344.032e-6 (-0.975)	9.81	0.113 (0.443)	0	332.495e-6 (0.906)
const_bkg	-0.66e-9	-4.642e-6 (-0.505)	0.112 (0.443)	-4.85e-9 (-0.003)	-4.639e-6 (-0.504)	0.113 (0.443)	0.00666	0.000	4.120e-6 (0.431)
lin_bkg	0	0	0	0	0	0	0.000	0	0
resolution	0.01e-9 (0.005)	-0.012e-6 (-0.878)	327.678e-6 (0.904)	-0.03e-9 (-0.014)	-0.012e-6 (-0.881)	332.495e-6 (0.906)	4.120e-6 (0.431)	0	1.37e-08

../_images/3e293f8962a4a463c3657db314c40a405195e4a3b0162c95e430909b36d6237e.svg

You can also use the interactive fitting interface provided by iminuit.

Note

Requires ipywidgets to be installed.
If you are viewing this documentation online, changing the sliders won’t change the plot. run the code locally to try it out.

m.interactive()

Curve fitting¶

Fermi edge fitting¶

Pre-defined models¶

Fitting multiple peaks¶

Fitting across multiple dimensions¶

Fitting multidimensional models¶

Fitting background models¶

Visualizing fits¶

Parallelization¶

Saving and loading fits¶

Using `iminuit`¶

Curve fitting¶

Fermi edge fitting¶

Pre-defined models¶

Fitting multiple peaks¶

Fitting across multiple dimensions¶

Fitting multidimensional models¶

Fitting background models¶

Visualizing fits¶

Parallelization¶

Saving and loading fits¶

Using iminuit¶

Using `iminuit`¶