Getting started with Non-Linear Least-Squares Fitting

The lmfit package is designed to provide simple tools to help you build of 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.

To do a non-linear least-squares fit of a model to data or for a variety of other optimization problems, 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 is to be minimized in the least-squares sense. For many data fitting processes, the least-squares 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.

In a traditional non-linear fit, one writes an objective function that takes the variable values and calculates the residual \(y^{\rm meas}_i - y_i^{\rm model}({\bf{v}})\), or the residual 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 example, one might write an objective function like this:

def residual(vars, x, data, eps_data):
    amp = vars[0]
    phaseshift = vars[1]
    freq = vars[2]
    decay = vars[3]

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

    return (data-model)/eps_data

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

from scipy.optimize import leastsq
vars = [10.0, 0.2, 3.0, 0.007]
out = leastsq(residual, vars, args=(x, data, eps_data))

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 – vars[0] is the amplitude, vars[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, those 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 really do solely to the use of traditional arrays of variables, as matches closely the implementation of the Fortran code. The lmfit module overcomes these shortcomings by using a core reason for using Python – objects. The key concept for lmfit is to use Parameter objects instead of plain floating point numbers as the variables for the fit. By using Parameter objects (or the closely related Parameters – a dictionary of Parameter objects), one can

1. not care about the order of variables, but refer to Parameters by meaningful names.
2. place bounds on Parameters as attributes, without worrying about order.
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 as:

from lmfit import minimize, Parameters

def residual(params, x, data, eps_data):
    amp = params['amp'].value
    pshift = params['phase'].value
    freq = params['frequency'].value
    decay = params['decay'].value

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

    return (data-model)/eps_data

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, eps_data))

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 hold additional attributes to modify the value during the fit. For example, Parameters can be fixed or bounded. This can be done when being defined:

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 parameters value) or after being defined by setting the corresponding attributes after they have been created:

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

Importantly, our function to be minimized remains unchanged.

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 by the user of the objective function instead of just by the author of the objective function.

Finally, in addition to the Parameters approach to fitting data, lmfit allows you to easily switch optimization methods without rewriting your objective function, and provides tools for writing fitting reports and for better determining the confidence levels for Parameters.