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| Non-Linear Least-Squares Minimization and Curve-Fitting for Python | Introduction | Parameters | Models |
As shown in the previous chapter, a simple fit can be performed with the minimize() function. For more sophisticated modeling, the Minimizer class can be used to gain a bit more control, especially when using complicated constraints.
The minimize function takes a objective function (the function that calculates the array to be minimized), a Parameters ordered dictionary, and several optional arguments. See Writing a Fitting Function for details on writing the function to minimize.
find values for the params so that the sum-of-squares of the array returned from function is minimized.
| Parameters: |
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| Returns: | Minimizer object, which can be used to inspect goodness-of-fit statistics, or to re-run fit. |
On output, the params will be updated with best-fit values and, where appropriate, estimated uncertainties and correlations. See Goodness-of-Fit and estimated uncertainty and correlations for further details.
If provided, the iter_cb function should take arguments of params, iter, resid, *args, **kws, where params will have the current parameter values, iter the iteration, resid the current residual array, and *args and **kws as passed to the objective function.
An important component of a fit is writing a function to be minimized – the objective function. Since this function will be called by other routines, there are fairly stringent requirements for its call signature and return value. In principle, your function can be any python callable, but it must look like this:
calculate objective residual to be minimized from parameters.
| Parameters: |
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| Returns: | residual array (generally data-model) to be minimized in the least-squares sense. |
| Return type: | numpy array. The length of this array cannot change between calls. |
A common use for the positional and keyword arguments would be to pass in other data needed to calculate the residual, including such things as the data array, dependent variable, uncertainties in the data, and other data structures for the model calculation.
The objective function should return the value to be minimized. For the Levenberg-Marquardt algorithm from leastsq(), this returned value must be an array, with a length greater than or equal to the number of fitting variables in the model. For the other methods, the return value can either be a scalar or an array. If an array is returned, the sum of squares of the array will be sent to the underlying fitting method, effectively doing a least-squares optimization of the return values.
Since the function will be passed in a dictionary of Parameters, it is advisable to unpack these to get numerical values at the top of the function. A simple way to do this is with Parameters.valuesdict(), as with:
def residual(pars, x, data=None, eps=None):
# unpack parameters:
# extract .value attribute for each parameter
parvals = pars.valuesdict()
period = parvals['period']
shift = parvals['shift']
decay = parvals['decay']
if abs(shift) > pi/2:
shift = shift - sign(shift)*pi
if abs(period) < 1.e-10:
period = sign(period)*1.e-10
model = parvals['amp'] * sin(shift + x/period) * exp(-x*x*decay*decay)
if data is None:
return model
if eps is None:
return (model - data)
return (model - data)/eps
In this example, x is a positional (required) argument, while the data array is actually optional (so that the function returns the model calculation if the data is neglected). Also note that the model calculation will divide x by the value of the ‘period’ Parameter. It might be wise to ensure this parameter cannot be 0. It would be possible to use the bounds on the Parameter to do this:
params['period'] = Parameter(value=2, min=1.e-10)
but putting this directly in the function with:
if abs(period) < 1.e-10:
period = sign(period)*1.e-10
is also a reasonable approach. Similarly, one could place bounds on the decay parameter to take values only between -pi/2 and pi/2.
By default, the Levenberg-Marquardt algorithm is used for fitting. While often criticized, including the fact it finds a local minima, this approach has some distinct advantages. These include being fast, and well-behaved for most curve-fitting needs, and making it easy to estimate uncertainties for and correlations between pairs of fit variables, as discussed in Goodness-of-Fit and estimated uncertainty and correlations.
Alternative algorithms can also be used by providing the method keyword to the minimize() function or use the corresponding method name from the Minimizer class as listed in the Table of Supported Fitting Methods.
Table of Supported Fitting Methods:
Fitting Meth method arg to minimize() Minimizer method method arg to scalar_minimize() Levenberg-Marquardt leastsq leastsq() Not available Nelder-Mead nelder fmin() Nelder-Mead L-BFGS-B lbfgsb lbfgsb() L-BFGS-B Powell powell Powell Conjugate Gradient cg CG Newton-CG newton Newton-CG COBYLA cobyla COBYLA COBYLA cobyla COBYLA Truncated Newton tnc TNC Trust Newton-CGn trust-ncg trust-ncg Dogleg dogleg dogleg Sequential Linear Squares Programming slsqp SLSQP Differential Evolution differential_evolution differential_evolution
Note
The objective function for the Levenberg-Marquardt method must return an array, with more elements than variables. All other methods can return either a scalar value or an array.
Warning
Much of this documentation assumes that the Levenberg-Marquardt method is the method used. Many of the fit statistics and estimates for uncertainties in parameters discussed in Goodness-of-Fit and estimated uncertainty and correlations are done only for this method.
On a successful fit using the leastsq method, several goodness-of-fit statistics and values related to the uncertainty in the fitted variables will be calculated. These are all encapsulated in the Minimizer object for the fit, as returned by minimize(). The values related to the entire fit are stored in attributes of the Minimizer object, as shown in Table of Fit Results while those related to each fitted variables are stored as attributes of the corresponding Parameter.
Table of Fit Results: These values, including the standard Goodness-of-Fit statistics, are all attributes of the Minimizer object returned by minimize().
| Minimizer Attribute | Description / Formula |
|---|---|
| nfev | number of function evaluations |
| success | boolean (True/False) for whether fit succeeded. |
| errorbars | boolean (True/False) for whether uncertainties were estimated. |
| message | message about fit success. |
| ier | integer error value from scipy.optimize.leastsq() |
| lmdif_message | message from scipy.optimize.leastsq() |
| nvarys | number of variables in fit \(N_{\rm varys}\) |
| ndata | number of data points: \(N\) |
| nfree ` | degrees of freedom in fit: \(N - N_{\rm varys}\) |
| residual | residual array (return of func(): \({\rm Resid}\) |
| chisqr | chi-square: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\) |
| redchi | reduced chi-square: \(\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}\) |
| var_map | list of variable parameter names for rows/columns of covar |
| covar | covariance matrix (with rows/columns using var_map |
Note that the calculation of chi-square and reduced chi-square assume that the returned residual function is scaled properly to the uncertainties in the data. For these statistics to be meaningful, the person writing the function to be minimized must scale them properly.
After a fit using using the leastsq() method has completed successfully, standard errors for the fitted variables and correlations between pairs of fitted variables are automatically calculated from the covariance matrix. The standard error (estimated \(1\sigma\) error-bar) go into the stderr attribute of the Parameter. The correlations with all other variables will be put into the correl attribute of the Parameter – a dictionary with keys for all other Parameters and values of the corresponding correlation.
In some cases, it may not be possible to estimate the errors and correlations. For example, if a variable actually has no practical effect on the fit, it will likely cause the covariance matrix to be singular, making standard errors impossible to estimate. Placing bounds on varied Parameters makes it more likely that errors cannot be estimated, as being near the maximum or minimum value makes the covariance matrix singular. In these cases, the errorbars attribute of the fit result (Minimizer object) will be False.
For full control of the fitting process, you’ll want to create a Minimizer object, or at least use the one returned from the minimize() function.
creates a Minimizer, for fine-grain access to fitting methods and attributes.
| Parameters: |
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| Returns: | Minimizer object, which can be used to inspect goodness-of-fit statistics, or to re-run fit. |
The Minimizer object has a few public methods:
perform fit with Levenberg-Marquardt algorithm. Keywords will be passed directly to scipy.optimize.leastsq(). By default, numerical derivatives are used, and the following arguments are set:
leastsq() arg Default Value Description xtol 1.e-7 Relative error in the approximate solution ftol 1.e-7 Relative error in the desired sum of squares maxfev 2000*(nvar+1) maximum number of function calls (nvar= # of variables) Dfun None function to call for Jacobian calculation
perform fit with L-BFGS-B algorithm. Keywords will be passed directly to scipy.optimize.fmin_l_bfgs_b().
lbfgsb() arg Default Value Description factr 1000.0 approx_grad True calculate approximations of gradient maxfun 2000*(nvar+1) maximum number of function calls (nvar= # of variables)
Warning
lbfgsb() is deprecated. Use minimize() with method='lbfgsb'.
perform fit with Nelder-Mead downhill simplex algorithm. Keywords will be passed directly to scipy.optimize.fmin().
fmin() arg Default Value Description ftol 1.e-4 function tolerance xtol 1.e-4 parameter tolerance maxfun 5000*(nvar+1) maximum number of function calls (nvar= # of variables)
Warning
fmin() is deprecated. Use minimize() with method='nelder'.
perform fit with any of the scalar minimization algorithms supported by scipy.optimize.minimize().
scalar_minimize() arg Default Value Description method Nelder-Mead fitting method tol 1.e-7 fitting and parameter tolerance hess None Hessian of objective function
prepares and initializes model and Parameters for subsequent fitting. This routine prepares the conversion of Parameters into fit variables, organizes parameter bounds, and parses, checks and “compiles” constrain expressions.
This is called directly by the fitting methods, and it is generally not necessary to call this function explicitly. An exception is when you would like to call your function to minimize prior to running one of the minimization routines, for example, to calculate the initial residual function. In that case, you might want to do something like:
myfit = Minimizer(my_residual, params, fcn_args=(x,), fcn_kws={'data':data})
myfit.prepare_fit()
init = my_residual(p_fit, x)
pylab.plot(x, init, 'b--')
myfit.leastsq()
That is, this method should be called prior to your fitting function being called.
generate and return text of report of best-fit values, uncertainties, and correlations from fit.
| Parameters: |
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If the first argument is a Minimizer object, as returned from minimize(), the report will include some goodness-of-fit statistics.
print text of report from fit_report().
An example fit with report would be
#!/usr/bin/env python
#<examples/doc_withreport.py>
from __future__ import print_function
from lmfit import Parameters, minimize, fit_report
from numpy import random, linspace, pi, exp, sin, sign
p_true = Parameters()
p_true.add('amp', value=14.0)
p_true.add('period', value=5.46)
p_true.add('shift', value=0.123)
p_true.add('decay', value=0.032)
def residual(pars, x, data=None):
vals = pars.valuesdict()
amp = vals['amp']
per = vals['period']
shift = vals['shift']
decay = vals['decay']
if abs(shift) > pi/2:
shift = shift - sign(shift)*pi
model = amp * sin(shift + x/per) * exp(-x*x*decay*decay)
if data is None:
return model
return (model - data)
n = 1001
xmin = 0.
xmax = 250.0
random.seed(0)
noise = random.normal(scale=0.7215, size=n)
x = linspace(xmin, xmax, n)
data = residual(p_true, x) + noise
fit_params = Parameters()
fit_params.add('amp', value=13.0)
fit_params.add('period', value=2)
fit_params.add('shift', value=0.0)
fit_params.add('decay', value=0.02)
out = minimize(residual, fit_params, args=(x,), kws={'data':data})
fit = residual(fit_params, x)
print(fit_report(fit_params))
#<end of examples/doc_withreport.py>
which would write out:
[[Variables]]
amp: 13.9121944 +/- 0.141202 (1.01%) (init= 13)
decay: 0.03264538 +/- 0.000380 (1.16%) (init= 0.02)
period: 5.48507044 +/- 0.026664 (0.49%) (init= 2)
shift: 0.16203677 +/- 0.014056 (8.67%) (init= 0)
[[Correlations]] (unreported correlations are < 0.100)
C(period, shift) = 0.797
C(amp, decay) = 0.582