Performing Fits and Analyzing Outputs

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 or comparing results from related fits.

The minimize() function

The minimize() function is a wrapper around Minimizer for running an optimization problem. It takes an objective function (the function that calculates the array to be minimized), a Parameters object, and several optional arguments. See Writing a Fitting Function for details on writing the objective.

minimize(fcn, params, method='leastsq', args=None, kws=None, scale_covar=True, iter_cb=None, reduce_fcn=None, **fit_kws)

Perform a fit of a set of parameters by minimizing an objective (or cost) function using one one of the several available methods.

The minimize function takes a objective function to be minimized, a dictionary (Parameters) containing the model parameters, and several optional arguments.

Parameters:
  • fcn (callable) – Objective function to be minimized. When method is leastsq or least_squares, the objective function should return an array of residuals (difference between model and data) to be minimized in a least-squares sense. With the scalar methods the objective function can either return the residuals array or a single scalar value. The function must have the signature: fcn(params, *args, **kws)
  • params (Parameters) – Contains the Parameters for the model.
  • method (str, optional) –

    Name of the fitting method to use. Valid values are:

    • ‘leastsq’: Levenberg-Marquardt (default)
    • ‘least_squares’: Least-Squares minimization, using Trust Region Reflective method by default
    • ‘differential_evolution’: differential evolution
    • ‘brute’: brute force method
    • nelder‘: Nelder-Mead
    • ‘lbfgsb’: L-BFGS-B
    • ‘powell’: Powell
    • ‘cg’: Conjugate-Gradient
    • ‘newton’: Newton-Congugate-Gradient
    • ‘cobyla’: Cobyla
    • ‘tnc’: Truncate Newton
    • ‘trust-ncg’: Trust Newton-Congugate-Gradient
    • ‘dogleg’: Dogleg
    • ‘slsqp’: Sequential Linear Squares Programming

    In most cases, these methods wrap and use the method of the same name from scipy.optimize, or use scipy.optimize.minimize with the same method argument. Thus ‘leastsq‘ will use scipy.optimize.leastsq, while ‘powell‘ will use scipy.optimize.minimizer(...., method=’powell’)

    For more details on the fitting methods please refer to the SciPy docs.

  • args (tuple, optional) – Positional arguments to pass to fcn.
  • kws (dict, optional) – Keyword arguments to pass to fcn.
  • iter_cb (callable, optional) – Function to be called at each fit iteration. This function should have the signature iter_cb(params, iter, resid, *args, **kws), where 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.
  • scale_covar (bool, optional) – Whether to automatically scale the covariance matrix (leastsq only).
  • reduce_fcn (str or callable, optional) – Function to convert a residual array to a scalar value for the scalar minimizers. See notes in Minimizer.
  • **fit_kws (dict, optional) – Options to pass to the minimizer being used.
Returns:

Object containing the optimized parameter and several goodness-of-fit statistics.

Return type:

MinimizerResult

Changed in version 0.9.0: Return value changed to MinimizerResult.

Notes

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.

A common use for args and kws 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.

On output, params will be unchanged. The best-fit values, and where appropriate, estimated uncertainties and correlations, will all be contained in the returned MinimizerResult. See MinimizerResult – the optimization result for further details.

This function is simply a wrapper around Minimizer and is equivalent to:

fitter = Minimizer(fcn, params, fcn_args=args, fcn_kws=kws,
                   iter_cb=iter_cb, scale_covar=scale_covar, **fit_kws)
fitter.minimize(method=method)

Writing a Fitting 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:

func(params, *args, **kws):

Calculate objective residual to be minimized from parameters.

Parameters:
  • params (Parameters) – Parameters.
  • args – Positional arguments. Must match args argument to minimize().
  • kws – Keyword arguments. Must match kws argument to minimize().
Returns:

Residual array (generally data-model) to be minimized in the least-squares sense.

Return type:

numpy.ndarray. 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 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 shown below:

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.

Choosing Different Fitting Methods

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 MinimizerResult – the optimization result.

Alternative algorithms can also be used by providing the method keyword to the minimize() function or Minimizer.minimize() class as listed in the Table of Supported Fitting Methods.

Table of Supported Fitting Methods:

Fitting Method method arg to minimize() or Minimizer.minimize()
Levenberg-Marquardt leastsq or least_squares
Nelder-Mead nelder
L-BFGS-B lbfgsb
Powell powell
Conjugate Gradient cg
Newton-CG newton
COBYLA cobyla
Truncated Newton tnc
Dogleg dogleg
Sequential Linear Squares Programming slsqp
Differential Evolution differential_evolution
Brute force method brute

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 used. Many of the fit statistics and estimates for uncertainties in parameters discussed in MinimizerResult – the optimization result are done only for this method.

MinimizerResult – the optimization result

New in version 0.9.0.

An optimization with minimize() or Minimizer.minimize() will return a MinimizerResult object. This is an otherwise plain container object (that is, with no methods of its own) that simply holds the results of the minimization. These results will include several pieces of informational data such as status and error messages, fit statistics, and the updated parameters themselves.

Importantly, the parameters passed in to Minimizer.minimize() will be not be changed. To to find the best-fit values, uncertainties and so on for each parameter, one must use the MinimizerResult.params attribute. For example, to print the fitted values, bounds and other parameters attributes in a well formatted text tables you can execute:

result.params.pretty_print()

with results being a MinimizerResult object. Note that the method pretty_print() accepts several arguments for customizing the output (e.g., column width, numeric format, etcetera).

class MinimizerResult(**kws)

The results of a minimization.

Minimization results include data such as status and error messages, fit statistics, and the updated (i.e., best-fit) parameters themselves in the params attribute.

The list of (possible) MinimizerResult attributes is given below:

params

Parameters – The best-fit parameters resulting from the fit.

status

int – Termination status of the optimizer. Its value depends on the underlying solver. Refer to message for details.

var_names

list – Ordered list of variable parameter names used in optimization, and useful for understanding the values in init_vals and covar.

covar

numpy.ndarray – Covariance matrix from minimization (leastsq only), with rows and columns corresponding to var_names.

init_vals

list – List of initial values for variable parameters using var_names.

init_values

dict – Dictionary of initial values for variable parameters.

nfev

int – Number of function evaluations.

success

bool – True if the fit succeeded, otherwise False.

errorbars

bool – True if uncertainties were estimated, otherwise False.

message

str – Message about fit success.

ier

int – Integer error value from scipy.optimize.leastsq (leastsq only).

lmdif_message

str – Message from scipy.optimize.leastsq (leastsq only).

nvarys

int – Number of variables in fit: \(N_{\rm varys}\).

ndata

int – Number of data points: \(N\).

nfree

int – Degrees of freedom in fit: \(N - N_{\rm varys}\).

residual

numpy.ndarray – Residual array \({\rm Resid_i}\). Return value of the objective function when using the best-fit values of the parameters.

chisqr

float – Chi-square: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\).

redchi

float – Reduced chi-square: \(\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}\).

aic

float – Akaike Information Criterion statistic: \(N \ln(\chi^2/N) + 2 N_{\rm varys}\).

bic

float – Bayesian Information Criterion statistic: \(N \ln(\chi^2/N) + \ln(N) N_{\rm varys}\).

flatchain

pandas.DataFrame – A flatchain view of the sampling chain from the emcee method.

show_candidates()

Pretty_print() representation of candidates from the brute method.

Goodness-of-Fit Statistics

Table of Fit Results: These values, including the standard Goodness-of-Fit statistics, are all attributes of the MinimizerResult object returned by minimize() or Minimizer.minimize().
Attribute Name Description / Formula
nfev number of function evaluations
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, returned by the objective function: \(\{\rm Resid_i\}\)
chisqr chi-square: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\)
redchi reduced chi-square: \(\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}\)
aic Akaike Information Criterion statistic (see below)
bic Bayesian Information Criterion statistic (see below)
var_names ordered list of variable parameter names used for init_vals and covar
covar covariance matrix (with rows/columns using var_names)
init_vals list of initial values for variable parameters

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) goes 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.

Akaike and Bayesian Information Criteria

The MinimizerResult includes the traditional chi-square and reduced chi-square statistics:

\begin{eqnarray*} \chi^2 &=& \sum_i^N r_i^2 \\ \chi^2_\nu &=& = \chi^2 / (N-N_{\rm varys}) \end{eqnarray*}

where \(r\) is the residual array returned by the objective function (likely to be (data-model)/uncertainty for data modeling usages), \(N\) is the number of data points (ndata), and \(N_{\rm varys}\) is number of variable parameters.

Also included are the Akaike Information Criterion, and Bayesian Information Criterion statistics, held in the aic and bic attributes, respectively. These give slightly different measures of the relative quality for a fit, trying to balance quality of fit with the number of variable parameters used in the fit. These are calculated as:

\begin{eqnarray*} {\rm aic} &=& N \ln(\chi^2/N) + 2 N_{\rm varys} \\ {\rm bic} &=& N \ln(\chi^2/N) + \ln(N) N_{\rm varys} \\ \end{eqnarray*}

When comparing fits with different numbers of varying parameters, one typically selects the model with lowest reduced chi-square, Akaike information criterion, and/or Bayesian information criterion. Generally, the Bayesian information criterion is considered the most conservative of these statistics.

Using a Iteration Callback Function

An iteration callback function is a function to be called at each iteration, just after the objective function is called. The iteration callback allows user-supplied code to be run at each iteration, and can be used to abort a fit.

iter_cb(params, iter, resid, *args, **kws):

User-supplied function to be run at each iteration.

Parameters:
  • params (Parameters) – Parameters.
  • iter (int) – Iteration number.
  • resid (numpy.ndarray) – Residual array.
  • args – Positional arguments. Must match args argument to minimize()
  • kws – Keyword arguments. Must match kws argument to minimize()
Returns:

Residual array (generally data-model) to be minimized in the least-squares sense.

Return type:

None for normal behavior, any value like True to abort the fit.

Normally, the iteration callback would have no return value or return None. To abort a fit, have this function return a value that is True (including any non-zero integer). The fit will also abort if any exception is raised in the iteration callback. When a fit is aborted this way, the parameters will have the values from the last iteration. The fit statistics are not likely to be meaningful, and uncertainties will not be computed.

Using the Minimizer class

For full control of the fitting process, you will want to create a Minimizer object.

class Minimizer(userfcn, params, fcn_args=None, fcn_kws=None, iter_cb=None, scale_covar=True, nan_policy='raise', reduce_fcn=None, **kws)

A general minimizer for curve fitting and optimization.

Parameters:
  • userfcn (callable) –

    Objective function that returns the residual (difference between model and data) to be minimized in a least-squares sense. This function must have the signature:

    userfcn(params, *fcn_args, **fcn_kws)
    
  • params (Parameters) – Contains the Parameters for the model.
  • fcn_args (tuple, optional) – Positional arguments to pass to userfcn.
  • fcn_kws (dict, optional) – Keyword arguments to pass to userfcn.
  • iter_cb (callable, optional) –

    Function to be called at each fit iteration. This function should have the signature:

    iter_cb(params, iter, resid, *fcn_args, **fcn_kws)
    

    where params will have the current parameter values, iter the iteration, resid the current residual array, and *fcn_args and **fcn_kws are passed to the objective function.

  • scale_covar (bool, optional) – Whether to automatically scale the covariance matrix (leastsq only).
  • nan_policy (str, optional) –

    Specifies action if userfcn (or a Jacobian) returns NaN values. One of:

    • ‘raise’ : a ValueError is raised
    • ‘propagate’ : the values returned from userfcn are un-altered
    • ‘omit’ : non-finite values are filtered
  • reduce_fcn (str or callable, optional) –

    Function to convert a residual array to a scalar value for the scalar minimizers. Optional values are (where r is the residual array):

    • None : sum of squares of residual [default]
      = (r*r).sum()
    • ‘negentropy’ : neg entropy, using normal distribution
      = rho*log(rho).sum()`, where rho = exp(-r*r/2)/(sqrt(2*pi))
    • ‘neglogcauchy’: neg log likelihood, using Cauchy distribution
      = -log(1/(pi*(1+r*r))).sum()
    • callable : must take one argument (r) and return a float.
  • **kws (dict, optional) – Options to pass to the minimizer being used.

Notes

The objective function should return the value to be minimized. For the Levenberg-Marquardt algorithm from leastsq() or least_squares(), 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. If the objective function returns non-finite values then a ValueError will be raised because the underlying solvers cannot deal with them.

A common use for the fcn_args and fcn_kws 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 Minimizer object has a few public methods:

Minimizer.minimize(method='leastsq', params=None, **kws)

Perform the minimization.

Parameters:
  • method (str, optional) –

    Name of the fitting method to use. Valid values are:

    • ‘leastsq’: Levenberg-Marquardt (default)
    • ‘least_squares’: Least-Squares minimization, using Trust Region Reflective method by default
    • ‘differential_evolution’: differential evolution
    • ‘brute’: brute force method
    • nelder‘: Nelder-Mead
    • ‘lbfgsb’: L-BFGS-B
    • ‘powell’: Powell
    • ‘cg’: Conjugate-Gradient
    • ‘newton’: Newton-CG
    • ‘cobyla’: Cobyla
    • ‘tnc’: Truncate Newton
    • ‘trust-ncg’: Trust Newton-CGn
    • ‘dogleg’: Dogleg
    • ‘slsqp’: Sequential Linear Squares Programming

    In most cases, these methods wrap and use the method with the same name from scipy.optimize, or use scipy.optimize.minimize with the same method argument. Thus ‘leastsq‘ will use scipy.optimize.leastsq, while ‘powell‘ will use scipy.optimize.minimizer(...., method=’powell’)

    For more details on the fitting methods please refer to the SciPy docs.

  • params (Parameters, optional) – Parameters of the model to use as starting values.
  • **kws (optional) – Additional arguments are passed to the underlying minimization method.
Returns:

Object containing the optimized parameter and several goodness-of-fit statistics.

Return type:

MinimizerResult

Changed in version 0.9.0: Return value changed to MinimizerResult.

Minimizer.leastsq(params=None, **kws)

Use Levenberg-Marquardt minimization to perform a fit.

It assumes that the input Parameters have been initialized, and a function to minimize has been properly set up. When possible, this calculates the estimated uncertainties and variable correlations from the covariance matrix.

This method calls 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
Parameters:
Returns:

Object containing the optimized parameter and several goodness-of-fit statistics.

Return type:

MinimizerResult

Changed in version 0.9.0: Return value changed to MinimizerResult.

Minimizer.least_squares(params=None, **kws)

Use the least_squares (new in scipy 0.17) to perform a fit.

It assumes that the input Parameters have been initialized, and a function to minimize has been properly set up. When possible, this calculates the estimated uncertainties and variable correlations from the covariance matrix.

This method wraps scipy.optimize.least_squares, which has inbuilt support for bounds and robust loss functions.

Parameters:
Returns:

Object containing the optimized parameter and several goodness-of-fit statistics.

Return type:

MinimizerResult

Changed in version 0.9.0: Return value changed to MinimizerResult.

Minimizer.scalar_minimize(method='Nelder-Mead', params=None, **kws)

Scalar minimization using scipy.optimize.minimize.

Perform fit with any of the scalar minimization algorithms supported by scipy.optimize.minimize. Default argument values are:

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
Parameters:
  • method (str, optional) –

    Name of the fitting method to use. One of:

    • ‘Nelder-Mead’ (default)
    • ‘L-BFGS-B’
    • ‘Powell’
    • ‘CG’
    • ‘Newton-CG’
    • ‘COBYLA’
    • ‘TNC’
    • ‘trust-ncg’
    • ‘dogleg’
    • ‘SLSQP’
    • ‘differential_evolution’
  • params (Parameters, optional) – Parameters to use as starting point.
  • **kws (dict, optional) – Minimizer options pass to scipy.optimize.minimize.
Returns:

Object containing the optimized parameter and several goodness-of-fit statistics.

Return type:

MinimizerResult

Changed in version 0.9.0: Return value changed to MinimizerResult.

Notes

If the objective function returns a NumPy array instead of the expected scalar, the sum of squares of the array will be used.

Note that bounds and constraints can be set on Parameters for any of these methods, so are not supported separately for those designed to use bounds. However, if you use the differential_evolution method you must specify finite (min, max) for each varying Parameter.

Minimizer.prepare_fit(params=None)

Prepare parameters for fitting.

Prepares and initializes model and Parameters for subsequent fitting. This routine prepares the conversion of Parameters into fit variables, organizes parameter bounds, and parses, “compiles” and checks constrain expressions. The method also creates and returns a new instance of a MinimizerResult object that contains the copy of the Parameters that will actually be varied in the fit.

Parameters:params (Parameters, optional) – Contains the Parameters for the model; if None, then the Parameters used to initialize the Minimizer object are used.
Returns:
Return type:MinimizerResult

Notes

This method is called directly by the fitting methods, and it is generally not necessary to call this function explicitly.

Changed in version 0.9.0: Return value changed to MinimizerResult.

Minimizer.brute(params=None, Ns=20, keep=50)

Use the brute method to find the global minimum of a function.

The following parameters are passed to scipy.optimize.brute and cannot be changed:

brute() arg Value Description
full_output 1 Return the evaluation grid and the objective function’s values on it.
finish None No “polishing” function is to be used after the grid search.
disp False Do not print convergence messages (when finish is not None).

It assumes that the input Parameters have been initialized, and a function to minimize has been properly set up.

Parameters:
  • params (Parameters object, optional) – Contains the Parameters for the model. If None, then the Parameters used to initialize the Minimizer object are used.
  • Ns (int, optional) – Number of grid points along the axes, if not otherwise specified (see Notes).
  • keep (int, optional) – Number of best candidates from the brute force method that are stored in the candidates attribute. If ‘all’, then all grid points from scipy.optimize.brute are stored as candidates.
Returns:

Object containing the parameters from the brute force method. The return values (x0, fval, grid, Jout) from scipy.optimize.brute are stored as brute_<parname> attributes. The MinimizerResult also contains the candidates attribute and show_candidates() method. The candidates attribute contains the parameters and chisqr from the brute force method as a namedtuple, (‘Candidate’, [‘params’, ‘score’]), sorted on the (lowest) chisqr value. To access the values for a particular candidate one can use result.candidate[#].params or result.candidate[#].score, where a lower # represents a better candidate. The show_candidates(#) uses the pretty_print() method to show a specific candidate-# or all candidates when no number is specified.

Return type:

MinimizerResult

New in version 0.9.6.

Notes

The brute() method evalutes the function at each point of a multidimensional grid of points. The grid points are generated from the parameter ranges using Ns and (optional) brute_step. The implementation in scipy.optimize.brute requires finite bounds and the range is specified as a two-tuple (min, max) or slice-object (min, max, brute_step). A slice-object is used directly, whereas a two-tuple is converted to a slice object that interpolates Ns points from min to max, inclusive.

In addition, the brute() method in lmfit, handles three other scenarios given below with their respective slice-object:

  • lower bound (min) and brute_step are specified:
    range = (min, min + Ns * brute_step, brute_step).
  • upper bound (max) and brute_step are specified:
    range = (max - Ns * brute_step, max, brute_step).
  • numerical value (value) and brute_step are specified:
    range = (value - (Ns//2) * brute_step, value + (Ns//2) * brute_step, brute_step).

For more information, check the examples in examples/lmfit_brute.py.

Minimizer.emcee(params=None, steps=1000, nwalkers=100, burn=0, thin=1, ntemps=1, pos=None, reuse_sampler=False, workers=1, float_behavior='posterior', is_weighted=True, seed=None)

Bayesian sampling of the posterior distribution using emcee.

Bayesian sampling of the posterior distribution for the parameters using the emcee Markov Chain Monte Carlo package. The method assumes that the prior is Uniform. You need to have emcee installed to use this method.

Parameters:
  • params (Parameters, optional) – Parameters to use as starting point. If this is not specified then the Parameters used to initialize the Minimizer object are used.
  • steps (int, optional) – How many samples you would like to draw from the posterior distribution for each of the walkers?
  • nwalkers (int, optional) – Should be set so \(nwalkers >> nvarys\), where nvarys are the number of parameters being varied during the fit. “Walkers are the members of the ensemble. They are almost like separate Metropolis-Hastings chains but, of course, the proposal distribution for a given walker depends on the positions of all the other walkers in the ensemble.” - from the emcee webpage.
  • burn (int, optional) – Discard this many samples from the start of the sampling regime.
  • thin (int, optional) – Only accept 1 in every thin samples.
  • ntemps (int, optional) – If ntemps > 1 perform a Parallel Tempering.
  • pos (numpy.ndarray, optional) – Specify the initial positions for the sampler. If ntemps == 1 then pos.shape should be (nwalkers, nvarys). Otherwise, (ntemps, nwalkers, nvarys). You can also initialise using a previous chain that had the same ntemps, nwalkers and nvarys. Note that nvarys may be one larger than you expect it to be if your userfcn returns an array and is_weighted is False.
  • reuse_sampler (bool, optional) – If you have already run emcee on a given Minimizer object then it possesses an internal sampler attribute. You can continue to draw from the same sampler (retaining the chain history) if you set this option to True. Otherwise a new sampler is created. The nwalkers, ntemps, pos, and params keywords are ignored with this option. Important: the Parameters used to create the sampler must not change in-between calls to emcee. Alteration of Parameters would include changed min, max, vary and expr attributes. This may happen, for example, if you use an altered Parameters object and call the minimize method in-between calls to emcee.
  • workers (Pool-like or int, optional) – For parallelization of sampling. It can be any Pool-like object with a map method that follows the same calling sequence as the built-in map function. If int is given as the argument, then a multiprocessing-based pool is spawned internally with the corresponding number of parallel processes. ‘mpi4py’-based parallelization and ‘joblib’-based parallelization pools can also be used here. Note: because of multiprocessing overhead it may only be worth parallelising if the objective function is expensive to calculate, or if there are a large number of objective evaluations per step (ntemps * nwalkers * nvarys).
  • float_behavior (str, optional) –

    Specifies meaning of the objective function output if it returns a float. One of:

    • ‘posterior’ - objective function returns a log-posterior probability
    • ‘chi2’ - objective function returns \(\chi^2\)

    See Notes for further details.

  • is_weighted (bool, optional) – Has your objective function been weighted by measurement uncertainties? If is_weighted is True then your objective function is assumed to return residuals that have been divided by the true measurement uncertainty (data - model) / sigma. If is_weighted is False then the objective function is assumed to return unweighted residuals, data - model. In this case emcee will employ a positive measurement uncertainty during the sampling. This measurement uncertainty will be present in the output params and output chain with the name __lnsigma. A side effect of this is that you cannot use this parameter name yourself. Important this parameter only has any effect if your objective function returns an array. If your objective function returns a float, then this parameter is ignored. See Notes for more details.
  • seed (int or numpy.random.RandomState, optional) – If seed is an int, a new numpy.random.RandomState instance is used, seeded with seed. If seed is already a numpy.random.RandomState instance, then that numpy.random.RandomState instance is used. Specify seed for repeatable minimizations.
Returns:

MinimizerResult object containing updated params, statistics, etc. The updated params represent the median (50th percentile) of all the samples, whilst the parameter uncertainties are half of the difference between the 15.87 and 84.13 percentiles. The MinimizerResult also contains the chain, flatchain and lnprob attributes. The chain and flatchain attributes contain the samples and have the shape (nwalkers, (steps - burn) // thin, nvarys) or (ntemps, nwalkers, (steps - burn) // thin, nvarys), depending on whether Parallel tempering was used or not. nvarys is the number of parameters that are allowed to vary. The flatchain attribute is a pandas.DataFrame of the flattened chain, chain.reshape(-1, nvarys). To access flattened chain values for a particular parameter use result.flatchain[parname]. The lnprob attribute contains the log probability for each sample in chain. The sample with the highest probability corresponds to the maximum likelihood estimate.

Return type:

MinimizerResult

Notes

This method samples the posterior distribution of the parameters using Markov Chain Monte Carlo. To do so it needs to calculate the log-posterior probability of the model parameters, F, given the data, D, \(\ln p(F_{true} | D)\). This ‘posterior probability’ is calculated as:

\[\ln p(F_{true} | D) \propto \ln p(D | F_{true}) + \ln p(F_{true})\]

where \(\ln p(D | F_{true})\) is the ‘log-likelihood’ and \(\ln p(F_{true})\) is the ‘log-prior’. The default log-prior encodes prior information already known about the model. This method assumes that the log-prior probability is -numpy.inf (impossible) if the one of the parameters is outside its limits. The log-prior probability term is zero if all the parameters are inside their bounds (known as a uniform prior). The log-likelihood function is given by [1]:

\[\ln p(D|F_{true}) = -\frac{1}{2}\sum_n \left[\frac{(g_n(F_{true}) - D_n)^2}{s_n^2}+\ln (2\pi s_n^2)\right]\]

The first summand in the square brackets represents the residual for a given datapoint (\(g\) being the generative model, \(D_n\) the data and \(s_n\) the standard deviation, or measurement uncertainty, of the datapoint). This term represents \(\chi^2\) when summed over all data points. Ideally the objective function used to create lmfit.Minimizer should return the log-posterior probability, \(\ln p(F_{true} | D)\). However, since the in-built log-prior term is zero, the objective function can also just return the log-likelihood, unless you wish to create a non-uniform prior.

If a float value is returned by the objective function then this value is assumed by default to be the log-posterior probability, i.e. float_behavior is ‘posterior’. If your objective function returns \(\chi^2\), then you should use a value of ‘chi2’ for float_behavior. emcee will then multiply your \(\chi^2\) value by -0.5 to obtain the posterior probability.

However, the default behaviour of many objective functions is to return a vector of (possibly weighted) residuals. Therefore, if your objective function returns a vector, res, then the vector is assumed to contain the residuals. If is_weighted is True then your residuals are assumed to be correctly weighted by the standard deviation (measurement uncertainty) of the data points (res = (data - model) / sigma) and the log-likelihood (and log-posterior probability) is calculated as: -0.5 * numpy.sum(res**2). This ignores the second summand in the square brackets. Consequently, in order to calculate a fully correct log-posterior probability value your objective function should return a single value. If is_weighted is False then the data uncertainty, s_n, will be treated as a nuisance parameter and will be marginalized out. This is achieved by employing a strictly positive uncertainty (homoscedasticity) for each data point, \(s_n = \exp(\_\_lnsigma)\). __lnsigma will be present in MinimizerResult.params, as well as Minimizer.chain, nvarys will also be increased by one.

References

[1]http://dan.iel.fm/emcee/current/user/line/

Minimizer.emcee() - calculating the posterior probability distribution of parameters

Minimizer.emcee() can be used to obtain the posterior probability distribution of parameters, given a set of experimental data. An example problem is a double exponential decay. A small amount of Gaussian noise is also added in:

>>> import numpy as np
>>> import lmfit
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(1, 10, 250)
>>> np.random.seed(0)
>>> y = 3.0 * np.exp(-x / 2) - 5.0 * np.exp(-(x - 0.1) / 10.) + 0.1 * np.random.randn(len(x))
>>> plt.plot(x, y)
>>> plt.show()
_images/emcee_dbl_exp.png

Create a Parameter set for the initial guesses:

>>> p = lmfit.Parameters()
>>> p.add_many(('a1', 4.), ('a2', 4.), ('t1', 3.), ('t2', 3., True))

>>> def residual(p):
...     v = p.valuesdict()
...     return v['a1'] * np.exp(-x / v['t1']) + v['a2'] * np.exp(-(x - 0.1) / v['t2']) - y

Solving with minimize() gives the Maximum Likelihood solution.:

>>> mi = lmfit.minimize(residual, p, method='Nelder')
>>> lmfit.printfuncs.report_fit(mi.params, min_correl=0.5)
[[Variables]]
    a1:   2.98623688 (init= 4)
    a2:  -4.33525596 (init= 4)
    t1:   1.30993185 (init= 3)
    t2:   11.8240752 (init= 3)
[[Correlations]] (unreported correlations are <  0.500)
>>> plt.plot(x, y)
>>> plt.plot(x, residual(mi.params) + y, 'r')
>>> plt.show()
_images/emcee_dbl_exp2.png

However, this doesn’t give a probability distribution for the parameters. Furthermore, we wish to deal with the data uncertainty. This is called marginalisation of a nuisance parameter. emcee requires a function that returns the log-posterior probability. The log-posterior probability is a sum of the log-prior probability and log-likelihood functions. The log-prior probability is assumed to be zero if all the parameters are within their bounds and -np.inf if any of the parameters are outside their bounds.

>>> # add a noise parameter
>>> mi.params.add('f', value=1, min=0.001, max=2)
>>> # This is the log-likelihood probability for the sampling. We're going to estimate the
>>> # size of the uncertainties on the data as well.
>>> def lnprob(p):
...    resid = residual(p)
...    s = p['f']
...    resid *= 1 / s
...    resid *= resid
...    resid += np.log(2 * np.pi * s**2)
...    return -0.5 * np.sum(resid)

Now we have to set up the minimizer and do the sampling:

>>> mini = lmfit.Minimizer(lnprob, mi.params)
>>> res = mini.emcee(burn=300, steps=600, thin=10, params=mi.params)

Lets have a look at those posterior distributions for the parameters. This requires installation of the corner package:

>>> import corner
>>> corner.corner(res.flatchain, labels=res.var_names, truths=list(res.params.valuesdict().values()))
_images/emcee_triangle.png

The values reported in the MinimizerResult are the medians of the probability distributions and a 1 sigma quantile, estimated as half the difference between the 15.8 and 84.2 percentiles. The median value is not necessarily the same as the Maximum Likelihood Estimate. We’ll get that as well. You can see that we recovered the right uncertainty level on the data.:

>>> print("median of posterior probability distribution")
>>> print('------------------------------------------')
>>> lmfit.report_fit(res.params)
median of posterior probability distribution
------------------------------------------
[[Variables]]
    a1:   3.00975345 +/- 0.151034 (5.02%) (init= 2.986237)
    a2:  -4.35419204 +/- 0.127505 (2.93%) (init=-4.335256)
    t1:   1.32726415 +/- 0.142995 (10.77%) (init= 1.309932)
    t2:   11.7911935 +/- 0.495583 (4.20%) (init= 11.82408)
    f:    0.09805494 +/- 0.004256 (4.34%) (init= 1)
[[Correlations]] (unreported correlations are <  0.100)
    C(a2, t2)                    =  0.981
    C(a2, t1)                    = -0.927
    C(t1, t2)                    = -0.880
    C(a1, t1)                    = -0.519
    C(a1, a2)                    =  0.195
    C(a1, t2)                    =  0.146

>>> # find the maximum likelihood solution
>>> highest_prob = np.argmax(res.lnprob)
>>> hp_loc = np.unravel_index(highest_prob, res.lnprob.shape)
>>> mle_soln = res.chain[hp_loc]
>>> for i, par in enumerate(p):
...     p[par].value = mle_soln[i]

>>> print("\nMaximum likelihood Estimation")
>>> print('-----------------------------')
>>> print(p)
Maximum likelihood Estimation
-----------------------------
Parameters([('a1', <Parameter 'a1', 2.9943337359308981, bounds=[-inf:inf]>),
('a2', <Parameter 'a2', -4.3364489105166593, bounds=[-inf:inf]>),
('t1', <Parameter 't1', 1.3124544105342462, bounds=[-inf:inf]>),
('t2', <Parameter 't2', 11.80612160586597, bounds=[-inf:inf]>)])

>>> # Finally lets work out a 1 and 2-sigma error estimate for 't1'
>>> quantiles = np.percentile(res.flatchain['t1'], [2.28, 15.9, 50, 84.2, 97.7])
>>> print("2 sigma spread", 0.5 * (quantiles[-1] - quantiles[0]))
2 sigma spread 0.298878202908

Getting and Printing Fit Reports

fit_report(inpars, modelpars=None, show_correl=True, min_correl=0.1, sort_pars=False)

Generate a report of the fitting results.

The report contains the best-fit values for the parameters and their uncertainties and correlations.

Parameters:
  • inpars (Parameters) – Input Parameters from fit or MinimizerResult returned from a fit.
  • modelpars (Parameters, optional) – Known Model Parameters.
  • show_correl (bool, optional) – Whether to show list of sorted correlations (default is True).
  • min_correl (float, optional) – Smallest correlation in absolute value to show (default is 0.1).
  • sort_pars (bool or callable, optional) – Whether to show parameter names sorted in alphanumerical order. If False (default), then the parameters will be listed in the order they were added to the Parameters dictionary. If callable, then this (one argument) function is used to extract a comparison key from each list element.
Returns:

Multi-line text of fit report.

Return type:

string

An example using this to write out a fit 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})

print(fit_report(out))


#<end of examples/doc_withreport.py>

which would write out:

[[Fit Statistics]]
    # function evals   = 85
    # data points      = 1001
    # variables        = 4
    chi-square         = 498.812
    reduced chi-square = 0.500
    Akaike info crit   = -689.223
    Bayesian info crit = -669.587
[[Variables]]
    amp:      13.9121944 +/- 0.141202 (1.01%) (init= 13)
    period:   5.48507044 +/- 0.026664 (0.49%) (init= 2)
    shift:    0.16203676 +/- 0.014056 (8.67%) (init= 0)
    decay:    0.03264538 +/- 0.000380 (1.16%) (init= 0.02)
[[Correlations]] (unreported correlations are <  0.100)
    C(period, shift)             =  0.797
    C(amp, decay)                =  0.582
    C(amp, shift)                = -0.297
    C(amp, period)               = -0.243
    C(shift, decay)              = -0.182
    C(period, decay)             = -0.150