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# `Parameter` and `Parameters`

This chapter describes `Parameter` objects which is the 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 be varied in the fit or have a fixed value, have upper and/or lower bounds. It can even have a value that is constrained by an algebraic expression of other Parameter values. Since `Parameters` 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. This simplifies the writing of models, allowing general models that describe the phenomenon to be written, 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 need an ordered group of floating point variables. In the `scipy.optimize` routines this is required to be a 1-dimensional numpy ndarray. For lmfit, where each `Parameter` has a name, this is replaced by a `Parameters` class, 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. The objective functions you write for lmfit will take an instance of `Parameters` as its first argument.

## The `Parameter` class

class `Parameter`(name=None[, value=None[, vary=True[, min=None[, max=None[, expr=None]]]]])

  create a Parameter object.

  Parameters:

  - name (`None` or string – will be overwritten during fit if `None`.) – parameter name
  - value – the numerical value for the parameter
  - vary (boolean (`True`/`False`) [default `True`]) – whether to vary the parameter or not.
  - min – lower bound for value (`None` = no lower bound).
  - max – upper bound for value (`None` = no upper bound).
  - expr (`None` or string) – mathematical expression to use to evaluate value during fit.

Each of these inputs is turned into an attribute of the same name.

After a fit, a Parameter for a fitted variable (that is with `vary = True`) may have its `value` attribute to hold the best-fit value. Depending on the success of the fit and fitting algorithm used, it may also have attributes `stderr` and `correl`.

`stderr`

  the estimated standard error for the best-fit value.

`correl`

  a dictionary of the correlation with the other fitted variables in the fit, 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]]]])

  set or update a Parameters value or other attributes.

  Parameters:

  - name – parameter name
  - value – the numerical value for the parameter
  - vary – whether to vary the parameter or not.
  - min – lower bound for value
  - max – upper bound for value
  - expr – mathematical expression to use to evaluate value during fit.

Each argument of `set()` has a default value of `None`, and will be set only if the provided value is not `None`. You can use this to update some Parameter attribute without affecting others, for example:

        p1 = Parameter('a', value=2.0)
        p2 = Parameter('b', value=0.0)
        p1.set(min=0)
        p2.set(vary=False)

    to set a lower bound, or to set a Parameter as have a fixed value.

    Note that to use this approach to lift a lower or upper bound, doing::

        p1.set(min=0)
        .....
        # now lift the lower bound
        p1.set(min=None)   # won't work!  lower bound NOT changed

    won't work -- this will not change the current lower bound.  Instead
    you'll have to use ``np.inf`` to remove a lower or upper bound::

        # now lift the lower bound
        p1.set(min=-np.inf)   # will work!

    Similarly, to clear an expression of a parameter, you need to pass an
    empty string, not ``None``.  You also need to give a value and
    explicitly tell it to vary::

        p3 = Parameter('c', expr='(a+b)/2')
        p3.set(expr=None)     # won't work!  expression NOT changed

        # remove constraint expression
        p3.set(value=1.0, vary=True, expr='')  # will work!  parameter now unconstrained

## The `Parameters` class

class `Parameters`

  create a Parameters object. This is little more than a fancy ordered dictionary, with the restrictions that:

  1. keys must be valid Python symbol names, so that they can be used in expressions of mathematical constraints. This means the names must match `[a-z_][a-z0-9_]*` and cannot be a Python reserved word.
  2. values must be valid `Parameter` objects.

  Two methods are for provided for convenient initialization of a `Parameters`, and one for extracting `Parameter` values into a plain dictionary.

`add`(name[, value=None[, vary=True[, min=None[, max=None[, expr=None]]]]])

  add a named parameter. This creates a `Parameter` object associated with the key name, with optional arguments passed to `Parameter`:

      p = Parameters()
      p.add('myvar', value=1, vary=True)

`add_many`(self, paramlist)

  add a list of named parameters. Each entry must be a tuple with the following entries:

      name, value, vary, min, max, expr

  This method is somewhat rigid and verbose (no default values), but can be useful when initially defining a parameter list so that it looks table-like:

      p = Parameters()
      #           (Name,  Value,  Vary,   Min,  Max,  Expr)
      p.add_many(('amp1',    10,  True, None, None,  None),
                 ('cen1',   1.2,  True,  0.5,  2.0,  None),
                 ('wid1',   0.8,  True,  0.1, None,  None),
                 ('amp2',   7.5,  True, None, None,  None),
                 ('cen2',   1.9,  True,  1.0,  3.0,  None),
                 ('wid2',  None, False, None, None, '2*wid1/3'))

`pretty_print`(oneline=False)

  prints a clean representation on the Parameters. If oneline is True, the result will be printed to a single (long) line.

`valuesdict`()

  return an ordered dictionary of name:value pairs with the Paramater name as the key and Parameter value as value.

  This is distinct from the `Parameters` itself, as the dictionary values are not `Parameter` objects, just the `value`. Using :method:`valuesdict` can be a very convenient way to get updated values in a objective function.

`dumps(**kws):`

  return a JSON string representation of the `Parameter` object. This can be saved or used to re-create or re-set parameters, using the `loads()` method.

  Optional keywords are sent `json.dumps()`.

`dump(file, **kws):`

  write a JSON representation of the `Parameter` object to a file or file-like object in file – really any object with a `write()` method. Optional keywords are sent `json.dumps()`.

`loads(sval, **kws):`

  use a JSON string representation of the `Parameter` object in sval to set all parameter settins. Optional keywords are sent `json.loads()`.

`load(file, **kws):`

  read and use a JSON string representation of the `Parameter` object from a file or file-like object in file – really any object with a `read()` method. Optional keywords are sent `json.loads()`.

## Simple Example

Using `Parameters`` and `minimize()` function (discussed in the next chapter) might look like this:

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 decaying sine wave, 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.