Performing Fits, 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.
Warning
Upgrading scripts from version 0.8.3 to 0.9.0? See Version 0.9.0 Release Notes
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
(function, params[, args=None[, kws=None[, method='leastsq'[, scale_covar=True[, iter_cb=None[, **fit_kws]]]]]])¶ find values for the
params
so that the sum-of-squares of the array returned fromfunction
is minimized.Parameters: - function (callable.) – function to return fit residual. See Writing a Fitting Function for details.
- params (
Parameters
.) – aParameters
dictionary. Keywords must be strings that match[a-z_][a-z0-9_]*
and cannot be a python reserved word. Each value must beParameter
. - args (tuple) – arguments tuple to pass to the residual function as positional arguments.
- kws (dict) – dictionary to pass to the residual function as keyword arguments.
- method (string (default
leastsq
)) – name of fitting method to use. See Choosing Different Fitting Methods for details - scale_covar (bool (default
True
)) – whether to automatically scale covariance matrix (leastsq
only) - iter_cb (callable or
None
) – function to be called at each fit iteration. See Using a Iteration Callback Function for details. - fit_kws (dict) – dictionary to pass to
scipy.optimize.leastsq()
orscipy.optimize.minimize()
.
Returns: MinimizerResult
instance, which will contain the optimized parameter, and several goodness-of-fit statistics.
Changed in version 0.9.0: return value changed to MinimizerResult
On output, the 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.
For clarity, it should be emphasized that this function is simply a
wrapper around Minimizer
that runs a single fit, implemented as:
fitter = Minimizer(fcn, params, fcn_args=args, fcn_kws=kws,
iter_cb=iter_cb, scale_covar=scale_covar, **fit_kws)
return 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 tominimize()
- kws – keyword arguments. Must match
kws
argument tominimize()
Returns: residual array (generally data-model) to be minimized in the least-squares sense.
Return type: numpy array. The length of this array cannot change between calls.
- params (
A common use for the positional and keyword arguments would be to pass in other data needed to calculate the residual, including such things as the data array, dependent variable, uncertainties in the data, and other data structures for the model calculation.
The objective function should return the value to be minimized. For the
Levenberg-Marquardt algorithm from leastsq()
, this returned value must be an
array, with a length greater than or equal to the number of fitting variables in the
model. For the other methods, the return value can either be a scalar or an array. If an
array is returned, the sum of squares of the array will be sent to the underlying fitting
method, effectively doing a least-squares optimization of the return values.
Since the function will be passed in a dictionary of Parameters
, it is advisable
to unpack these to get numerical values at the top of the function. A
simple way to do this is with Parameters.valuesdict()
, as with:
def residual(pars, x, data=None, eps=None):
# unpack parameters:
# extract .value attribute for each parameter
parvals = pars.valuesdict()
period = parvals['period']
shift = parvals['shift']
decay = parvals['decay']
if abs(shift) > pi/2:
shift = shift - sign(shift)*pi
if abs(period) < 1.e-10:
period = sign(period)*1.e-10
model = parvals['amp'] * sin(shift + x/period) * exp(-x*x*decay*decay)
if data is None:
return model
if eps is None:
return (model - data)
return (model - data)/eps
In this example, x
is a positional (required) argument, while the
data
array is actually optional (so that the function returns the model
calculation if the data is neglected). Also note that the model
calculation will divide x
by the value of the ‘period’ Parameter. It
might be wise to ensure this parameter cannot be 0. It would be possible
to use the bounds on the Parameter
to do this:
params['period'] = Parameter(value=2, min=1.e-10)
but putting this directly in the function with:
if abs(period) < 1.e-10:
period = sign(period)*1.e-10
is also a reasonable approach. Similarly, one could place bounds on the
decay
parameter to take values only between -pi/2
and pi/2
.
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 Method, eithers:
Fitting Method method
arg tominimize()
orMinimizer.minimize()
Levenberg-Marquardt leastsq
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
Note
The objective function for the Levenberg-Marquardt method must return an array, with more elements than variables. All other methods can return either a scalar value or an array.
Warning
Much of this documentation assumes that the Levenberg-Marquardt method is the method used. Many of the fit statistics and estimates for uncertainties in parameters discussed in MinimizerResult – the optimization result are done only for this method.
MinimizerResult
– the optimization result¶
-
class
MinimizerResult
(**kws)¶
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.
-
params
¶ the
Parameters
actually used in the fit, with updated values,stderr
andcorrel
.
-
var_names
¶ ordered list of variable parameter names used in optimization, and useful for understanding the the values in
init_vals
andcovar
.
-
nfev
¶ number of function evaluations
-
success
¶ boolean (
True
/False
) for whether fit succeeded.
-
errorbars
¶ boolean (
True
/False
) for whether uncertainties were estimated.
-
message
¶ message about fit success.
-
ier
¶ integer error value from
scipy.optimize.leastsq()
(leastsq only).
-
lmdif_message
¶ message from
scipy.optimize.leastsq()
(leastsq only).
-
nvarys
¶ number of variables in fit \(N_{\rm varys}\)
-
ndata
¶ number of data points: \(N\)
-
nfree `
degrees of freedom in fit: \(N - N_{\rm varys}\)
-
residual
¶ residual array, return value of
func()
: \({\rm Resid}\)
-
chisqr
¶ chi-square: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\)
-
redchi
¶ reduced chi-square: \(\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}\)
-
aic
¶ Akaike Information Criterion statistic (see below)
-
bic
¶ Bayesian Information Criterion statistic (see below).
Goodness-of-Fit Statistics¶
Table of Fit Results: These values, including the standard Goodness-of-Fit statistics, are all attributes of theMinimizerResult
object returned byminimize()
orMinimizer.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, return value of func() : \({\rm Resid}\) |
chisqr | chi-square: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\) |
redchi | reduced chi-square: \(\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}\) |
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) go into the stderr
attribute of the Parameter. The
correlations with all other variables will be put into the
correl
attribute of the Parameter – a dictionary with keys for
all other Parameters and values of the corresponding correlation.
In some cases, it may not be possible to estimate the errors and
correlations. For example, if a variable actually has no practical effect
on the fit, it will likely cause the covariance matrix to be singular,
making standard errors impossible to estimate. Placing bounds on varied
Parameters makes it more likely that errors cannot be estimated, as being
near the maximum or minimum value makes the covariance matrix singular. In
these cases, the errorbars
attribute of the fit result
(Minimizer
object) will be False
.
Akaike and Bayesian Information Criteria¶
The MinimizerResult
includes the tradtional chi-square and reduced chi-square statistics:
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
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 (integer) – iteration number
- resid (ndarray) – residual array.
- args – positional arguments. Must match
args
argument tominimize()
- kws – keyword arguments. Must match
kws
argument tominimize()
Returns: residual array (generally data-model) to be minimized in the least-squares sense.
Return type: None
for normal behavior, any value likeTrue
to abort fit.- params (
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’ll want to create a
Minimizer
object.
-
class
Minimizer
(function, params, fcn_args=None, fcn_kws=None, iter_cb=None, scale_covar=True, **kws)¶ creates a Minimizer, for more detailed access to fitting methods and attributes.
Parameters: - function (callable.) – objective function to return fit residual. See Writing a Fitting Function for details.
- params (dict) – a dictionary of Parameters. Keywords must be strings
that match
[a-z_][a-z0-9_]*
and is not a python reserved word. Each value must beParameter
. - fcn_args (tuple) – arguments tuple to pass to the residual function as positional arguments.
- fcn_kws (dict) – dictionary to pass to the residual function as keyword arguments.
- iter_cb (callable or
None
) – function to be called at each fit iteration. See Using a Iteration Callback Function for details. - scale_covar – flag for automatically scaling covariance matrix and uncertainties to reduced chi-square (
leastsq
only) - kws (dict) – dictionary to pass as keywords to the underlying
scipy.optimize
method.
The Minimizer object has a few public methods:
-
minimize
(method='leastsq', params=None, **kws)¶ perform fit using either
leastsq()
orscalar_minimize()
.Parameters: - method (str.) – name of fitting method. Must be one of the naemes in Table of Supported Fitting Methods
- params (
Parameters
or None) – aParameters
dictionary for starting values
Returns: MinimizerResult
object, containing updated parameters, fitting statistics, and information.
Changed in version 0.9.0: return value changed to MinimizerResult
Additonal keywords are passed on to the correspond leastsq()
or scalar_minimize()
method.
-
leastsq
(params=None, scale_covar=True, **kws)¶ perform fit with Levenberg-Marquardt algorithm. Keywords will be passed directly to
scipy.optimize.leastsq()
. By default, numerical derivatives are used, and the following arguments are set:leastsq()
argDefault 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
Changed in version 0.9.0: return value changed to MinimizerResult
-
scalar_minimize
(method='Nelder-Mead', params=None, hess=None, tol=None, **kws)¶ perform fit with any of the scalar minimization algorithms supported by
scipy.optimize.minimize()
.scalar_minimize()
argDefault Value Description method Nelder-Mead
fitting method tol 1.e-7 fitting and parameter tolerance hess None Hessian of objective function
Changed in version 0.9.0: return value changed to MinimizerResult
-
prepare_fit
(**kws)¶ 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 aMinimizerResult
object that contains the copy of the Parameters that will actually be varied in the fit.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
Getting and Printing Fit Reports¶
-
fit_report
(result, modelpars=None, show_correl=True, min_correl=0.1)¶ generate and return text of report of best-fit values, uncertainties, and correlations from fit.
Parameters: - result –
MinimizerResult
object as returned byminimize()
. - modelpars – Parameters with “Known Values” (optional, default None)
- show_correl – whether to show list of sorted correlations [
True
] - min_correl – smallest correlation absolute value to show [0.1]
If the first argument is a
Parameters
object, goodness-of-fit statistics will not be included.- result –
-
report_fit
(result, modelpars=None, show_correl=True, min_correl=0.1)¶ print text of report from
fit_report()
.
An example fit with report would be
which would write out:
[[Fit Statistics]]
# function evals = 85
# data points = 1001
# variables = 4
chi-square = 498.812
reduced chi-square = 0.500
[[Variables]]
amp: 13.9121944 +/- 0.141202 (1.01%) (init= 13)
period: 5.48507044 +/- 0.026664 (0.49%) (init= 2)
shift: 0.16203677 +/- 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