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# 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 function.

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

  Perform the minimization of the objective function.

  The minimize function takes an objective function to be minimized, a dictionary (`Parameters` ; Parameters) containing the model parameters, and several optional arguments including the fitting method.

  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

      - ’differential_evolution’: differential evolution

      - ’brute’: brute force method

      - ’basinhopping’: basinhopping

      - ’ampgo’: Adaptive Memory Programming for Global Optimization

      - ’nelder’: Nelder-Mead

      - ’lbfgsb’: L-BFGS-B

      - ’powell’: Powell

      - ’cg’: Conjugate-Gradient

      - ’newton’: Newton-CG

      - ’cobyla’: Cobyla

      - ’bfgs’: BFGS

      - ’tnc’: Truncated Newton

      - ’trust-ncg’: Newton-CG trust-region

      - ’trust-exact’: nearly exact trust-region

      - ’trust-krylov’: Newton GLTR trust-region

      - ’trust-constr’: trust-region for constrained optimization

      - ’dogleg’: Dog-leg trust-region

      - ’slsqp’: Sequential Linear Squares Programming

      - ’emcee’: Maximum likelihood via Monte-Carlo Markov Chain

      - ’shgo’: Simplicial Homology Global Optimization

      - ’dual_annealing’: Dual Annealing optimization

      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 params will have the current parameter values, iter the iteration number, 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 (default is True).

    - nan_policy ({'raise', 'propagate', 'omit'}, optional) –

      Specifies action if fcn (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. See Notes in Minimizer.

    - calc_covar (bool, optional) – Whether to calculate the covariance matrix (default is True) for solvers other than ‘leastsq’ and ‘least_squares’. Requires the numdifftools package to be installed.

    - max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). The default value depends on the fitting method.

    - **fit_kws (dict, optional) – Options to pass to the minimizer being used.

  Returns

    Object containing the optimized parameters 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,
                         nan_policy=nan_policy, reduce_fcn=reduce_fcn,
                         calc_covar=calc_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:

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 bounds on the `Parameter` to do this:

but putting this directly in the function with:

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. If you have the `numdifftools` package installed, lmfit will try to estimate the covariance matrix and determine parameter uncertainties and correlations if `calc_covar` is `True` (default).

> 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`
> BFGS | `bfgsb`
> Truncated Newton | `tnc`
> Newton CG trust-region | `trust-ncg`
> Exact trust-region | `trust-exact`
> Newton GLTR trust-region | `trust-krylov`
> Constrained trust-region | `trust-constr`
> Dogleg | `dogleg`
> Sequential Linear Squares Programming | `slsqp`
> Differential Evolution | `differential_evolution`
> Brute force method | `brute`
> Basinhopping | `basinhopping`
> Adaptive Memory Programming for Global Optimization | `ampgo`
> Simplicial Homology Global Ooptimization | `shgo`
> Dual Annealing | `dual_annealing`
> Maximum likelihood via Monte-Carlo Markov Chain | `emcee`

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. The Monte-Carlo Markov Chain or `emcee` method has two different operating methods when the objective function returns a scalar value. See the documentation for `emcee`.

Warning

Much of this documentation assumes that the Levenberg-Marquardt (`leastsq`) method is used. Many of the fit statistics and estimates for uncertainties in parameters discussed in MinimizerResult – the optimization result are done only unconditionally for this (and the `least_squares`) method. Lmfit versions newer than 0.9.11 provide the capability to use `numdifftools` to estimate the covariance matrix and calculate parameter uncertainties and correlations for other methods as well.

## `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 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 parameter 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

    The best-fit parameters resulting from the fit.

    Type

      Parameters

  status

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

    Type

      int

  var_names

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

    Type

      list

  covar

    Covariance matrix from minimization, with rows and columns corresponding to `var_names`.

    Type

      numpy.ndarray

  init_vals

    List of initial values for variable parameters using `var_names`.

    Type

      list

  init_values

    Dictionary of initial values for variable parameters.

    Type

      dict

  nfev

    Number of function evaluations.

    Type

      int

  success

    True if the fit succeeded, otherwise False.

    Type

      bool

  errorbars

    True if uncertainties were estimated, otherwise False.

    Type

      bool

  message

    Message about fit success.

    Type

      str

  call_kws

    Keyword arguments sent to underlying solver.

    Type

      dict

  ier

    Integer error value from scipy.optimize.leastsq (‘leastsq’ method only).

    Type

      int

  lmdif_message

    Message from scipy.optimize.leastsq (‘leastsq’ method only).

    Type

      str

  nvarys

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

    Type

      int

  ndata

    Number of data points: \(N\).

    Type

      int

  nfree

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

    Type

      int

  residual

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

    Type

      numpy.ndarray

  chisqr

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

    Type

      float

  redchi

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

    Type

      float

  aic

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

    Type

      float

  bic

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

    Type

      float

  flatchain

    A flatchain view of the sampling chain from the emcee method.

    Type

      pandas.DataFrame

  show_candidates()

    `pretty_print()` representation of candidates from the brute fitting 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
call_kws | dict of keyword arguments sent to underlying solver

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 the `leastsq()` or `least_squares()` method has completed successfully, standard errors for the fitted variables and correlations between pairs of fitted variables are automatically calculated from the covariance matrix. For other methods, the `calc_covar` parameter (default is `True`) in the `Minimizer` class determines whether or not to use the `numdifftools` package to estimate 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.

### Uncertainties in Variable Parameters, and their Correlations

As mentioned above, when a fit is complete the uncertainties for fitted Parameters as well as the correlations between pairs of Parameters are usually calculated. This happens automatically either when using the default `leastsq()` method, the `least_squares()` method, or for most other fitting methods if the highly-recommended `numdifftools` package is available. The estimated standard error (the \(1\sigma\) uncertainty) for each variable Parameter will be contained in the `stderr`, while the `correl` attribute for each Parameter will contain a dictionary of the correlation with each other variable Parameter.

These estimates of the uncertainties are done by inverting the Hessian matrix which represents the second derivative of fit quality for each variable parameter. There are situations for which the uncertainties cannot be estimated, which generally indicates that this matrix cannot be inverted because one of the fit is not actually sensitive to one of the variables. This can happen if a Parameter is stuck at an upper or lower bound, if the variable is simply not used by the fit, or if the value for the variable is such that it has no real influence on the fit.

In principle, the scale of the uncertainties in the Parameters is closely tied to the goodness-of-fit statistics chi-square and reduced chi-square (`chisqr` and `redchi`). The standard errors or \(1 \sigma\) uncertainties are those that increase chi-square by 1. Since a “good fit” should have `redchi` of around 1, this requires that the data uncertainties (and to some extent the sampling of the N data points) is correct. Unfortunately, it is often not the case that one has high-quality estimates of the data uncertainties (getting the data is hard enough!). Because of this common situation, the uncertainties reported and held in `stderr` are not those that increase chi-square by 1, but those that increase chi-square by reduced chi-square. This is equivalent to rescaling the uncertainty in the data such that reduced chi-square would be 1. To be clear, this rescaling is done by default because if reduced chi-square is far from 1, this rescaling often makes the reported uncertainties sensible, and if reduced chi-square is near 1 it does little harm. If you have good scaling of the data uncertainty and believe the scale of the residual array is correct, this automatic rescaling can be turned off using `scale_covar=False`.

Note that the simple (and fast!) approach to estimating uncertainties and correlations by inverting the second derivative matrix assumes that the components of the residual array (if, indeed, an array is used) are distributed around 0 with a normal (Gaussian distribution), and that a map of probability distributions for pairs would be elliptical – the size of the of ellipse gives the uncertainty itself and the eccentricity of the ellipse gives the correlation. This simple approach to assessing uncertainties ignores outliers, highly asymmetric uncertainties, or complex correlations between Parameters. In fact, it is not too hard to come up with problems where such effects are important. Our experience is that the automated results are usually the right scale and quite reasonable as initial estimates, but a more thorough exploration of the Parameter space using the tools described in Minimizer.emcee() - calculating the posterior probability distribution of parameters and An advanced example for evaluating confidence intervals can give a more complete understanding of the distributions and relations between Parameters.

## 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

    str

An example using this to write out a fit report would be:

which would give as output:

To be clear, you can get at all of these values from the fit result `out` and `out.params`. For example, a crude printout of the best fit variables and standard errors could be done as

## 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, calc_covar=True, max_nfev=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 number, 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 (default is True).

    - nan_policy ({'raise', 'propagate', 'omit}, optional) –

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

      - ’raise’ : a ValueError is raised (default)

      - ’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.

    - calc_covar (bool, optional) – Whether to calculate the covariance matrix (default is True) for solvers other than `'leastsq'` and `'least_squares'`. Requires the `numdifftools` package to be installed.

    - max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). The default value depends on the fitting method.

    - **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

      - ’differential_evolution’: differential evolution

      - ’brute’: brute force method

      - ’basinhopping’: basinhopping

      - ’ampgo’: Adaptive Memory Programming for Global Optimization

      - ’nelder’: Nelder-Mead

      - ’lbfgsb’: L-BFGS-B

      - ’powell’: Powell

      - ’cg’: Conjugate-Gradient

      - ’newton’: Newton-CG

      - ’cobyla’: Cobyla

      - ’bfgs’: BFGS

      - ’tnc’: Truncated Newton

      - ’trust-ncg’: Newton-CG trust-region

      - ’trust-exact’: nearly exact trust-region

      - ’trust-krylov’: Newton GLTR trust-region

      - ’trust-constr’: trust-region for constrained optimization

      - ’dogleg’: Dog-leg trust-region

      - ’slsqp’: Sequential Linear Squares Programming

      - ’emcee’: Maximum likelihood via Monte-Carlo Markov Chain

      - ’shgo’: Simplicial Homology Global Optimization

      - ’dual_annealing’: Dual Annealing optimization

      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 documentation.

    - 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 parameters and several goodness-of-fit statistics.

  Return type

    MinimizerResult

  Changed in version 0.9.0: Return value changed to `MinimizerResult`.

Minimizer.leastsq(params=None, max_nfev=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 and, 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
  Dfun | None | Function to call for Jacobian calculation

  Parameters

    - params (Parameters, optional) – Parameters to use as starting point.

    - max_nfev (int or None, optional) – Maximum number of function evaluations. Defaults to `2000*(nvars+1)`, where `nvars` is the number of variable parameters.

    - **kws (dict, optional) – Minimizer options to pass to scipy.optimize.leastsq.

  Returns

    Object containing the optimized parameters 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, max_nfev=None, **kws)

  Least-squares minimization using scipy.optimize.least_squares.

  This method wraps scipy.optimize.least_squares, which has built-in support for bounds and robust loss functions. By default it uses the Trust Region Reflective algorithm with a linear loss function (i.e., the standard least-squares problem).

  Parameters

    - params (Parameters, optional) – Parameters to use as starting point.

    - max_nfev (int or None, optional) – Maximum number of function evaluations. Defaults to `2000*(nvars+1)`, where `nvars` is the number of variable parameters.

    - **kws (dict, optional) – Minimizer options to pass to scipy.optimize.least_squares.

  Returns

    Object containing the optimized parameters 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, max_nfev=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’

      - ’BFGS’

      - ’TNC’

      - ’trust-ncg’

      - ’trust-exact’

      - ’trust-krylov’

      - ’trust-constr’

      - ’dogleg’

      - ’SLSQP’

      - ’differential_evolution’

    - params (Parameters, optional) – Parameters to use as starting point.

    - max_nfev (int or None, optional) – Maximum number of function evaluations. Defaults to `2000*(nvars+1)`, where `nvars` is the number of variable parameters.

    - **kws (dict, optional) – Minimizer options pass to scipy.optimize.minimize.

  Returns

    Object containing the optimized parameters 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, workers=1, max_nfev=None)

  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, 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.

    - workers (int or map-like callable, optional) – For parallel evaluation of the grid (see scipy.optimize.brute for more details).

    - max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). Defaults to `200000*(nvarys+1)`.

  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 :attr:`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()` method 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 evaluates 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_example.ipynb`.

Minimizer.basinhopping(params=None, max_nfev=None, **kws)

  Use the basinhopping algorithm to find the global minimum.

  This method calls scipy.optimize.basinhopping using the default arguments. The default minimizer is `BFGS`, but since lmfit supports parameter bounds for all minimizers, the user can choose any of the solvers present in scipy.optimize.minimize.

  Parameters

    - params (Parameters, optional) – Contains the Parameters for the model. If None, then the Parameters used to initialize the Minimizer object are used.

    - max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). Defaults to `200000*(nvarys+1)`.

    - **kws (dict, optional) – Minimizer options to pass to scipy.optimize.basinhopping.

  Returns

    Object containing the optimization results from the basinhopping algorithm.

  Return type

    MinimizerResult

  New in version 0.9.10.

Minimizer.ampgo(params=None, max_nfev=None, **kws)

  Find the global minimum of a multivariate function using AMPGO.

  AMPGO stands for ‘Adaptive Memory Programming for Global Optimization’ and is an efficient algorithm to find the global minimum.

  Parameters

    - params (Parameters, optional) – Contains the Parameters for the model. If None, then the Parameters used to initialize the Minimizer object are used.

    - max_nfev (int, optional) – Maximum number of total function evaluations. If None (default), the optimization will stop after totaliter number of iterations (see below)..

    - **kws (dict, optional) –

      Minimizer options to pass to the ampgo algorithm, the options are listed below:

          local: str, optional
              Name of the local minimization method. Valid options
              are:
              - `'L-BFGS-B'` (default)
              - `'Nelder-Mead'`
              - `'Powell'`
              - `'TNC'`
              - `'SLSQP'`
          local_opts: dict, optional
              Options to pass to the local minimizer (default is
              None).
          maxfunevals: int, optional
              Maximum number of function evaluations. If None
              (default), the optimization will stop after
              `totaliter` number of iterations (deprecated: use
              `max_nfev` instead).
          totaliter: int, optional
              Maximum number of global iterations (default is 20).
          maxiter: int, optional
              Maximum number of `Tabu Tunneling` iterations during
              each global iteration (default is 5).
          glbtol: float, optional
              Tolerance whether or not to accept a solution after a
              tunneling phase (default is 1e-5).
          eps1: float, optional
              Constant used to define an aspiration value for the
              objective function during the Tunneling phase (default
              is 0.02).
          eps2: float, optional
              Perturbation factor used to move away from the latest
              local minimum at the start of a Tunneling phase
              (default is 0.1).
          tabulistsize: int, optional
              Size of the (circular) tabu search list (default is 5).
          tabustrategy: {'farthest', 'oldest'}, optional
              Strategy to use when the size of the tabu list exceeds
              `tabulistsize`. It can be `'oldest'` to drop the oldest
              point from the tabu list or `'farthest'` (defauilt) to
              drop the element farthest from the last local minimum
              found.
          disp: bool, optional
              Set to True to print convergence messages (default is
              False).

  Returns

    Object containing the parameters from the ampgo method, with fit parameters, statistics and such. The return values (`x0`, `fval`, `eval`, `msg`, `tunnel`) are stored as `ampgo_<parname>` attributes.

  Return type

    MinimizerResult

  New in version 0.9.10.

  Notes

  The Python implementation was written by Andrea Gavana in 2014 (http://infinity77.net/global_optimization/index.html).

  The details of the AMPGO algorithm are described in the paper “Adaptive Memory Programming for Constrained Global Optimization” located here:

  http://leeds-faculty.colorado.edu/glover/fred%20pubs/416%20-%20AMP%20(TS)%20for%20Constrained%20Global%20Opt%20w%20Lasdon%20et%20al%20.pdf

Minimizer.shgo(params=None, max_nfev=None, **kws)

  Use the SHGO algorithm to find the global minimum.

  SHGO stands for “simplicial homology global optimization” and calls scipy.optimize.shgo using its default arguments.

  Parameters

    - params (Parameters, optional) – Contains the Parameters for the model. If None, then the Parameters used to initialize the Minimizer object are used.

    - max_nfev (int or None, optional) – Maximum number of function evaluations. Defaults to `200000*(nvars+1)`, where `nvars` is the number of variable parameters.

    - **kws (dict, optional) – Minimizer options to pass to the SHGO algorithm.

  Returns

    Object containing the parameters from the SHGO method. The return values specific to scipy.optimize.shgo (`x`, `xl`, `fun`, `funl`, `nfev`, `nit`, `nlfev`, `nlhev`, and `nljev`) are stored as `shgo_<parname>` attributes.

  Return type

    MinimizerResult

  New in version 0.9.14.

Minimizer.dual_annealing(params=None, max_nfev=None, **kws)

  Use the dual_annealing algorithm to find the global minimum.

  This method calls scipy.optimize.dual_annealing using its default arguments.

  Parameters

    - params (Parameters, optional) – Contains the Parameters for the model. If None, then the Parameters used to initialize the Minimizer object are used.

    - max_nfev (int or None, optional) – Maximum number of function evaluations. Defaults to `200000*(nvars+1)`, where `nvars` is the number of variables.

    - **kws (dict, optional) – Minimizer options to pass to the dual_annealing algorithm.

  Returns

    Object containing the parameters from the dual_annealing method. The return values specific to scipy.optimize.dual_annealing (`x`, `fun`, `nfev`, `nhev`, `njev`, and `nit`) are stored as `da_<parname>` attributes.

  Return type

    MinimizerResult

  New in version 0.9.14.

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, progress=True, run_mcmc_kwargs={})

  Bayesian sampling of the posterior distribution.

  The method uses the `emcee` Markov Chain Monte Carlo package and assumes that the prior is Uniform. You need to have `emcee` version 3 or newer 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, deprecated) – ntemps has no effect.

    - pos (numpy.ndarray, optional) – Specify the initial positions for the sampler, an ndarray of shape `(nwalkers, nvarys)`. You can also initialise using a previous chain of the same 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=False`.

    - reuse_sampler (bool, optional) – Set to True if you have already run emcee with the Minimizer instance and want to continue to draw from its `sampler` (and so retain the chain history). If False, a new sampler is created. The keywords nwalkers, pos, and params will be ignored when this is set, as they will be set by the existing sampler. 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 (`nwalkers * nvarys`).

    - float_behavior (str, optional) – Meaning of float (scalar) output of objective function. Use ‘posterior’ if it returns a log-posterior probability or ‘chi2’ if it returns \(\chi^2\). See Notes for further details.

    - is_weighted (bool, optional) – Has your objective function been weighted by measurement uncertainties? If `is_weighted=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=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.

    - progress (bool, optional) – Print a progress bar to the console while running.

    - run_mcmc_kwargs (dict, optional) – Additional (optional) keyword arguments that are passed to `emcee.EnsembleSampler.run_mcmc`.

  Returns

    MinimizerResult object containing updated params, statistics, etc. The updated params represent the median of the samples, while the uncertainties are half the difference of the 15.87 and 84.13 percentiles. The MinimizerResult contains a few additional attributes: chain contain the samples and has shape `((steps - burn) // thin, nwalkers, nvarys)`. flatchain is a pandas.DataFrame of the flattened chain, that can be accessed with result.flatchain[parname]. lnprob contains the log probability for each sample in chain. The sample with the highest probability corresponds to the maximum likelihood estimate. acor is an array containing the autocorrelation time for each parameter if the autocorrelation time can be computed from the chain. Finally, acceptance_fraction (an array of the fraction of steps accepted for each walker).

  Return type

    MinimizerResult

  Notes

  This method samples the posterior distribution of the parameters using Markov Chain Monte Carlo. It calculates the log-posterior probability of the model parameters, F, given the data, D, \(\ln p(F_{true} | D)\). This ‘posterior probability’ is given by:

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

  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 known about the model that the log-prior probability is `-numpy.inf` (impossible) if any of the parameters is outside its limits, and is zero if all the parameters are inside their bounds (uniform prior). The log-likelihood function is 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 term represents the residual (\(g\) being the generative model, \(D_n\) the data and \(s_n\) the measurement uncertainty). This gives \(\chi^2\) when summed over all data points. The objective function may also return the log-posterior probability, \(\ln p(F_{true} | D)\). Since the default 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 the objective function returns a float value, this is assumed by default to be the log-posterior probability, (float_behavior default is ‘posterior’). If your objective function returns \(\chi^2\), then you should use `float_behavior='chi2'` instead.

  By default objective functions may return an ndarray of (possibly weighted) residuals. In this case, use is_weighted to select whether these are correctly weighted by measurement uncertainty. Note that this ignores the second term above, so that to calculate a correct log-posterior probability value your objective function should return a float value. With `is_weighted=False` the data uncertainty, s_n, will be treated as a nuisance parameter to be marginalized out. This uses strictly positive uncertainty (homoscedasticity) for each data point, \(s_n = \exp(\rm{\_\_lnsigma})\). `__lnsigma` will be present in MinimizerResult.params, as well as Minimizer.chain and `nvarys` will be increased by one.

  References

  1

    https://emcee.readthedocs.io

## `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. Note that this method does not actually perform a fit at all. Instead, it explores parameter space to determine the probability distributions for the parameters, but without an explicit goal of attempting to refine the solution. It should not be used for fitting, but it is a useful method to to more thoroughly explore the parameter space around the solution after a fit has been done and thereby get an improved understanding of the probability distribution for the parameters. It may be able to refine your estimate of the most likely values for a set of parameters, but it will not iteratively find a good solution to the minimization problem. To use this method effectively, you should first use another minimization method and then use this method to explore the parameter space around thosee best-fit values.

To illustrate this, we’ll use an example problem of fitting data to function of a double exponential decay, including a modest amount of Gaussian noise to the data. Note that this example is the same problem used in An advanced example for evaluating confidence intervals for evaluating confidence intervals in the parameters, which is a similar goal to the one here.

Create a Parameter set for the initial guesses:

Solving with `minimize()` gives the Maximum Likelihood solution. Note that we use the robust Nelder-Mead method here. The default Levenberg-Marquardt method seems to have difficulty with exponential decays, though it can refine the solution if starting near the solution:

and plotting the fit using the Maximum Likelihood solution gives the graph below:

Note that the fit here (for which the `numdifftools` package is installed) does estimate and report uncertainties in the parameters and correlations for the parameters, and reports the correlation of parameters `a2` and `t2` to be very high. As we’ll see, these estimates are pretty good, but when faced with such high correlation, it can be helpful to get the full probability distribution for the parameters. MCMC methods are very good for this.

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.

If the objective function returns an array of unweighted residuals (i.e., `data-model`) as is the case here, you can use `is_weighted=False` as an argument for `emcee`. In that case, `emcee` will automatically add/use the `__lnsigma` parameter to estimate the true uncertainty in the data. To place boundaries on this parameter one can do:

Now we have to set up the minimizer and do the sampling (again, just to be clear, this is not doing a fit):

As mentioned in the Notes for `Minimizer.emcee()`, the `is_weighted` argument will be ignored if your objective function returns a float instead of an array. For the documentation we set `progress=False`; the default is to print a progress bar to the Terminal if the `tqdm` package is installed.

The success of the method (i.e., whether or not the sampling went well) can be assessed by checking the integrated autocorrelation time and/or the acceptance fraction of the walkers. For this specific example the autocorrelation time could not be estimated because the “chain is too short”. Instead, we plot the acceptance fraction per walker and its mean value suggests that the sampling worked as intended (as a rule of thumb the value should be between 0.2 and 0.5).

With the results from `emcee`, we can visualize the posterior distributions for the parameters using the `corner` package:

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. Printing these values:

You can see that this recovered the right uncertainty level on the data. Note that these values agree pretty well with the results, uncertainties and correlations found by the fit and using `numdifftools` to estimate the covariance matrix. That is, even though the parameters `a2`, `t1`, and `t2` are all highly correlated and do not display perfectly Gaussian probability distributions, the probability distributions found by explicitly sampling the parameter space are not so far from elliptical as to make the simple (and much faster) estimates from inverting the covariance matrix completely invalid.

As mentioned above, the result from `emcee` reports the median values, which are not necessarily the same as the Maximum Likelihood Estimate. To obtain the values for the Maximum Likelihood Estimation (MLE) we find the location in the chain with the highest probability:

Here the difference between MLE and median value are seen to be below 0.5%, and well within the estimated 1-\(\sigma\) uncertainty.

Finally, we can use the samples from `emcee` to work out the 1- and 2-\(\sigma\) error estimates.

And we see that the initial estimates for the 1-\(\sigma\) standard error using `numdifftools` was not too bad. We’ll return to this example problem in An advanced example for evaluating confidence intervals and use a different method to calculate the 1- and 2-\(\sigma\) error bars.