# Parameter and Parameters¶

This chapter describes the Parameter object, which is a key concept of lmfit.

A Parameter is the quantity to be optimized in all minimization problems, replacing the plain floating point number used in the optimization routines from scipy.optimize. A Parameter has a value that can either be varied in the fit or held at a fixed value, and can have upper and/or lower bounds placed on the value. It can even have a value that is constrained by an algebraic expression of other Parameter values. Since Parameter objects live outside the core optimization routines, they can be used in all optimization routines from scipy.optimize. By using Parameter objects instead of plain variables, the objective function does not have to be modified to reflect every change of what is varied in the fit, or whether bounds can be applied. This simplifies the writing of models, allowing general models that describe the phenomenon and gives the user more flexibility in using and testing variations of that model.

Whereas a Parameter expands on an individual floating point variable, the optimization methods actually still need an ordered group of floating point variables. In the scipy.optimize routines this is required to be a one-dimensional numpy.ndarray. In lmfit, this one-dimensional array is replaced by a Parameters object, which works as an ordered dictionary of Parameter objects with a few additional features and methods. That is, while the concept of a Parameter is central to lmfit, one normally creates and interacts with a Parameters instance that contains many Parameter objects. For example, the objective functions you write for lmfit will take an instance of Parameters as its first argument. A table of parameter values, bounds and other attributes can be printed using Parameters.pretty_print().

## The Parameter class¶

class Parameter(name=None, value=None, vary=True, min=-inf, max=inf, expr=None, brute_step=None, user_data=None)

A Parameter is an object that can be varied in a fit, or one of the controlling variables in a model. It is a central component of lmfit, and all minimization and modeling methods use Parameter objects.

A Parameter has a name attribute, and a scalar floating point value. It also has a vary attribute that describes whether the value should be varied during the minimization. Finite bounds can be placed on the Parameter’s value by setting its min and/or max attributes. A Parameter can also have its value determined by a mathematical expression of other Parameter values held in the expr attrribute. Additional attributes include brute_step used as the step size in a brute-force minimization, and user_data reserved exclusively for user’s need.

After a minimization, a Parameter may also gain other attributes, including stderr holding the estimated standard error in the Parameter’s value, and correl, a dictionary of correlation values with other Parameters used in the minimization.

Parameters: name (str, optional) – Name of the Parameter. value (float, optional) – Numerical Parameter value. vary (bool, optional) – Whether the Parameter is varied during a fit (default is True). min (float, optional) – Lower bound for value (default is -numpy.inf, no lower bound). max (float, optional) – Upper bound for value (default is numpy.inf, no upper bound). expr (str, optional) – Mathematical expression used to constrain the value during the fit. brute_step (float, optional) – Step size for grid points in the brute method. user_data (optional) – User-definable extra attribute used for a Parameter.
stderr

float – The estimated standard error for the best-fit value.

correl

dict – A dictionary of the correlation with the other fitted Parameters of the form:

{'decay': 0.404, 'phase': -0.020, 'frequency': 0.102}


See Bounds Implementation for details on the math used to implement the bounds with min and max.

The expr attribute can contain a mathematical expression that will be used to compute the value for the Parameter at each step in the fit. See Using Mathematical Constraints for more details and examples of this feature.

set(value=None, vary=None, min=None, max=None, expr=None, brute_step=None)

Set or update Parameter attributes.

Parameters: value (float, optional) – Numerical Parameter value. vary (bool, optional) – Whether the Parameter is varied during a fit. min (float, optional) – Lower bound for value. To remove a lower bound you must use -numpy.inf. max (float, optional) – Upper bound for value. To remove an upper bound you must use numpy.inf. expr (str, optional) – Mathematical expression used to constrain the value during the fit. To remove a constraint you must supply an empty string. brute_step (float, optional) – Step size for grid points in the brute method. To remove the step size you must use 0.

Notes

Each argument to set() has a default value of None, which will leave the current value for the attribute unchanged. Thus, to lift a lower or upper bound, passing in None will not work. Instead, you must set these to -numpy.inf or numpy.inf, as with:

par.set(min=None)        # leaves lower bound unchanged
par.set(min=-numpy.inf)  # removes lower bound


Similarly, to clear an expression, pass a blank string, (not None!) as with:

par.set(expr=None)  # leaves expression unchanged
par.set(expr='')    # removes expression


Explicitly setting a value or setting vary=True will also clear the expression.

Finally, to clear the brute_step size, pass 0, not None:

par.set(brute_step=None)  # leaves brute_step unchanged
par.set(brute_step=0)     # removes brute_step


## The Parameters class¶

class Parameters(asteval=None, *args, **kwds)

An ordered dictionary of all the Parameter objects required to specify a fit model. All minimization and Model fitting routines in lmfit will use exactly one Parameters object, typically given as the first argument to the objective function.

All keys of a Parameters() instance must be strings and valid Python symbol names, so that the name must match [a-z_][a-z0-9_]* and cannot be a Python reserved word.

All values of a Parameters() instance must be Parameter objects.

A Parameters() instance includes an asteval interpreter used for evaluation of constrained Parameters.

Parameters() support copying and pickling, and have methods to convert to and from serializations using json strings.

Parameters: asteval (asteval.Interpreter, optional) – Instance of the asteval Interpreter to use for constraint expressions. If None, a new interpreter will be created. *args (optional) – Arguments. **kwds (optional) – Keyword arguments.
add(name, value=None, vary=True, min=-inf, max=inf, expr=None, brute_step=None)

Parameters: name (str) – Name of parameter. Must match [a-z_][a-z0-9_]* and cannot be a Python reserved word. value (float, optional) – Numerical Parameter value, typically the initial value. vary (bool, optional) – Whether the Parameter is varied during a fit (default is True). min (float, optional) – Lower bound for value (default is -numpy.inf, no lower bound). max (float, optional) – Upper bound for value (default is numpy.inf, no upper bound). expr (str, optional) – Mathematical expression used to constrain the value during the fit. brute_step (float, optional) – Step size for grid points in the brute method.

Examples

>>> params = Parameters()


which is equivalent to:

>>> params = Parameters()
>>> params['xvar'] = Parameter(name='xvar', value=0.50, min=0, max=1)
>>> params['yvar'] = Parameter(name='yvar', expr='1.0 - xvar')

add_many(*parlist)

Add many parameters, using a sequence of tuples.

Parameters: parlist (sequence of tuple or Parameter) – A sequence of tuples, or a sequence of Parameter instances. If it is a sequence of tuples, then each tuple must contain at least the name. The order in each tuple must be (name, value, vary, min, max, expr, brute_step).

Examples

>>>  params = Parameters()
# add with tuples: (NAME VALUE VARY MIN  MAX  EXPR  BRUTE_STEP)
>>> params.add_many(('amp', 10, True, None, None, None, None),
...                 ('cen', 4, True, 0.0, None, None, None),
...                 ('wid', 1, False, None, None, None, None),
...                 ('frac', 0.5))
# add a sequence of Parameters
>>> f = Parameter('par_f', 100)
>>> g = Parameter('par_g', 2.)

pretty_print(oneline=False, colwidth=8, precision=4, fmt='g', columns=['value', 'min', 'max', 'stderr', 'vary', 'expr', 'brute_step'])

Pretty-print of parameters data.

Parameters: oneline (bool, optional) – If True prints a one-line parameters representation (default is False). colwidth (int, optional) – Column width for all columns specified in columns. precision (int, optional) – Number of digits to be printed after floating point. fmt ({'g', 'e', 'f'}, optional) – Single-character numeric formatter. Valid values are: ‘f’ floating point, ‘g’ floating point and exponential, or ‘e’ exponential. columns (list of str, optional) – List of Parameter attribute names to print.
valuesdict()

Return an ordered dictionary of parameter values.

Returns: An ordered dictionary of name:value pairs for each Parameter. OrderedDict
dumps(**kws)

Represent Parameters as a JSON string.

Parameters: **kws (optional) – Keyword arguments that are passed to json.dumps(). JSON string representation of Parameters. str

dump(), loads(), load(), json.dumps()

dump(fp, **kws)

Write JSON representation of Parameters to a file-like object.

Parameters: fp (file-like object) – An open and .write()-supporting file-like object. **kws (optional) – Keyword arguments that are passed to dumps(). Return value from fp.write(). None for Python 2.7 and the number of characters written in Python 3. None or int

dump(), load(), json.dump()

loads(s, **kws)

Load Parameters from a JSON string.

Parameters: **kws (optional) – Keyword arguments that are passed to json.loads(). Updated Parameters from the JSON string. Parameters

Notes

Current Parameters will be cleared before loading the data from the JSON string.

dump(), dumps(), load(), json.loads()

load(fp, **kws)

Load JSON representation of Parameters from a file-like object.

Parameters: fp (file-like object) – An open and .read()-supporting file-like object. **kws (optional) – Keyword arguments that are passed to loads(). Updated Parameters loaded from fp. Parameters

dump(), loads(), json.load()

## Simple Example¶

A basic example making use of Parameters and the minimize() function (discussed in the next chapter) might look like this:

#!/usr/bin/env python

# <examples/doc_parameters_basic.py>
import numpy as np

from lmfit import Minimizer, Parameters, report_fit

# create data to be fitted
x = np.linspace(0, 15, 301)
data = (5. * np.sin(2*x - 0.1) * np.exp(-x*x*0.025) +
np.random.normal(size=len(x), scale=0.2))

# define objective function: returns the array to be minimized
def fcn2min(params, x, data):
"""Model a decaying sine wave and subtract data."""
amp = params['amp']
shift = params['shift']
omega = params['omega']
decay = params['decay']
model = amp * np.sin(x*omega + shift) * np.exp(-x*x*decay)
return model - data

# create a set of Parameters
params = Parameters()

# do fit, here with leastsq model
minner = Minimizer(fcn2min, params, fcn_args=(x, data))
result = minner.minimize()

# calculate final result
final = data + result.residual

# write error report
report_fit(result)

# try to plot results
try:
import matplotlib.pyplot as plt
plt.plot(x, data, 'k+')
plt.plot(x, final, 'r')
plt.show()
except ImportError:
pass
# <end of examples/doc_parameters_basic.py>


Here, the objective function explicitly unpacks each Parameter value. This can be simplified using the Parameters valuesdict() method, which would make the objective function fcn2min above look like:

def fcn2min(params, x, data):
"""Model a decaying sine wave and subtract data."""
v = params.valuesdict()

model = v['amp'] * np.sin(x*v['omega'] + v['shift']) * np.exp(-x*x*v['decay'])
return model - data


The results are identical, and the difference is a stylistic choice.