Getting started with Non-Linear Least-Squares Fitting

The lmfit package provides simple tools to help you build complex fitting models for non-linear least-squares problems and apply these models to real data. This section gives an overview of the concepts and describes how to set up and perform simple fits. Some basic knowledge of Python, NumPy, and modeling data are assumed – this is not a tutorial on why or how to perform a minimization or fit data, but is rather aimed at explaining how to use lmfit to do these things.

In order to do a non-linear least-squares fit of a model to data or for any other optimization problem, the main task is to write an objective function that takes the values of the fitting variables and calculates either a scalar value to be minimized or an array of values that are to be minimized, typically in the least-squares sense. For many data fitting processes, the latter approach is used, and the objective function should return an array of (data-model), perhaps scaled by some weighting factor such as the inverse of the uncertainty in the data. For such a problem, the chi-square (\(\chi^2\)) statistic is often defined as:

\[\chi^2 = \sum_i^{N} \frac{[y^{\rm meas}_i - y_i^{\rm model}({\bf{v}})]^2}{\epsilon_i^2}\]

where \(y_i^{\rm meas}\) is the set of measured data, \(y_i^{\rm model}({\bf{v}})\) is the model calculation, \({\bf{v}}\) is the set of variables in the model to be optimized in the fit, and \(\epsilon_i\) is the estimated uncertainty in the data, respectively.

In a traditional non-linear fit, one writes an objective function that takes the variable values and calculates the residual array \(y^{\rm meas}_i - y_i^{\rm model}({\bf{v}})\), or the residual array scaled by the data uncertainties, \([y^{\rm meas}_i - y_i^{\rm model}({\bf{v}})]/{\epsilon_i}\), or some other weighting factor.

As a simple concrete example, one might want to model data with a decaying sine wave, and so write an objective function like this:

from numpy import exp, sin

def residual(variables, x, data, uncertainty):
    """Model a decaying sine wave and subtract data."""
    amp = variables[0]
    phaseshift = variables[1]
    freq = variables[2]
    decay = variables[3]

    model = amp * sin(x*freq + phaseshift) * exp(-x*x*decay)

    return (data-model) / uncertainty

To perform the minimization with scipy.optimize, one would do this:

from numpy import linspace, random
from scipy.optimize import leastsq

# generate synthetic data with noise
x = linspace(0, 100)
noise = random.normal(size=x.size, scale=0.2)
data = 7.5 * sin(x*0.22 + 2.5) * exp(-x*x*0.01) + noise

# generate experimental uncertainties
uncertainty = abs(0.16 + random.normal(size=x.size, scale=0.05))

variables = [10.0, 0.2, 3.0, 0.007]
out = leastsq(residual, variables, args=(x, data, uncertainty))

Though it is wonderful to be able to use Python for such optimization problems, and the SciPy library is robust and easy to use, the approach here is not terribly different from how one would do the same fit in C or Fortran. There are several practical challenges to using this approach, including:

  1. The user has to keep track of the order of the variables, and their meaning – variables[0] is the amplitude, variables[2] is the frequency, and so on, although there is no intrinsic meaning to this order.

  2. If the user wants to fix a particular variable (not vary it in the fit), the residual function has to be altered to have fewer variables, and have the corresponding constant value passed in some other way. While reasonable for simple cases, this quickly becomes a significant work for more complex models, and greatly complicates modeling for people not intimately familiar with the details of the fitting code.

  3. There is no simple, robust way to put bounds on values for the variables, or enforce mathematical relationships between the variables. In fact, the optimization methods that do provide bounds, require bounds to be set for all variables with separate arrays that are in the same arbitrary order as variable values. Again, this is acceptable for small or one-off cases, but becomes painful if the fitting model needs to change.

These shortcomings are due to the use of traditional arrays to hold the variables, which matches closely the implementation of the underlying Fortran code, but does not fit very well with Python’s rich selection of objects and data structures. The key concept in lmfit is to define and use Parameter objects instead of plain floating point numbers as the variables for the fit. Using Parameter objects (or the closely related Parameters – a dictionary of Parameter objects), allows one to:

  1. forget about the order of variables and refer to Parameters by meaningful names.

  2. place bounds on Parameters as attributes, without worrying about preserving the order of arrays for variables and boundaries.

  3. fix Parameters, without having to rewrite the objective function.

  4. place algebraic constraints on Parameters.

To illustrate the value of this approach, we can rewrite the above example for the decaying sine wave as:

from numpy import exp, sin

from lmfit import minimize, Parameters

def residual(params, x, data, uncertainty):
    amp = params['amp']
    phaseshift = params['phase']
    freq = params['frequency']
    decay = params['decay']

    model = amp * sin(x*freq + phaseshift) * exp(-x*x*decay)

    return (data-model) / uncertainty

params = Parameters()
params.add('amp', value=10)
params.add('decay', value=0.007)
params.add('phase', value=0.2)
params.add('frequency', value=3.0)

out = minimize(residual, params, args=(x, data, uncertainty))

At first look, we simply replaced a list of values with a dictionary, accessed by name – not a huge improvement. But each of the named Parameter in the Parameters object holds additional attributes to modify the value during the fit. For example, Parameters can be fixed or bounded. This can be done during definition:

params = Parameters()
params.add('amp', value=10, vary=False)
params.add('decay', value=0.007, min=0.0)
params.add('phase', value=0.2)
params.add('frequency', value=3.0, max=10)

where vary=False will prevent the value from changing in the fit, and min=0.0 will set a lower bound on that parameter’s value. It can also be done later by setting the corresponding attributes after they have been created:

params['amp'].vary = False
params['decay'].min = 0.10

Importantly, our objective function remains unchanged. This means the objective function can simply express the parametrized phenomenon to be modeled, and is separate from the choice of parameters to be varied in the fit.

The params object can be copied and modified to make many user-level changes to the model and fitting process. Of course, most of the information about how your data is modeled goes into the objective function, but the approach here allows some external control; that is, control by the user performing the fit, instead of by the author of the objective function.

Finally, in addition to the Parameters approach to fitting data, lmfit allows switching optimization methods without changing the objective function, provides tools for generating fitting reports, and provides a better determination of Parameters confidence levels.