Module Documentation

This page contains documentation to everything symfit has to offer.

Fit

class symfit.core.fit.BaseFit(model, *ordered_data, absolute_sigma=None, **named_data)

Bases: symfit.core.fit.TakesData

Abstract base class for all fitting objects.

error_func(*args, **kwargs)

Every fit object has to define an error_func method, giving the function to be minimized.

eval_jacobian(*args, **kwargs)

Every fit object has to define an eval_jacobian method, giving the jacobian of the function to be minimized.

execute(*args, **kwargs)

Every fit object has to define an execute method. Any * and ** arguments will be passed to the fitting module that is being wrapped, e.g. leastsq.

Args kwargs: Returns:Instance of FitResults

class symfit.core.fit.BaseModel(model)

Bases: collections.abc.Mapping

ABC for Model’s. Makes sure models are iterable. Models can be initiated from Mappings or Iterables of Expressions, or from an expression directly. Expressions are not enforced for ducktyping purposes.

__eq__(other)

Model’s are considered equal when they have the same dependent variables, and the same expressions for those dependent variables. The same is defined here as passing sympy == for the vars themselves, and as expr1 - expr2 == 0 for the expressions. For more info check the sympy docs.

Parameters:other – Instance of Model. Returns:bool

__getitem__(dependent_var)

Returns the expression belonging to a given dependent variable.

Parameters:dependent_var (Variable) – Instance of Variable Returns:The expression belonging to dependent_var

__init__(model)

Initiate a Model from a dict:

a = Model({y: x**2})

Preferred way of initiating Model, since now you know what the dependent variable is called.

Parameters:model – dict of Expr, where dependent variables are the keys.

__iter__()

Returns:iterable over self.model_dict

__len__()

Returns:the number of dependent variables for this model.

__neg__()

Returns:new model with opposite sign. Does not change the model in-place, but returns a new copy.

__str__()

A representation upon printing has to provided.

bounds

Returns:List of tuples of all bounds on parameters.

shared_parameters

Returns:bool, indicating if parameters are shared between the vector components of this model.

vars

Returns:Returns a list of dependent, independent and sigma variables, in that order.

class symfit.core.fit.CallableModel(model)

Bases: symfit.core.fit.BaseModel

Defines a callable model. The usual rules apply to the ordering of the arguments:

• first independent variables, then dependent variables, then parameters.
• within each of these groups they are ordered alphabetically.

__call__(*args, **kwargs)

Evaluate the model for a certain value of the independent vars and parameters. Signature for this function contains independent vars and parameters, NOT dependent and sigma vars.

Can be called with both ordered and named parameters. Order is independent vars first, then parameters. Alphabetical order within each group.

Parameters:

• args –
• kwargs –

Returns:

A namedtuple of all the dependent vars evaluated at the desired point. Will always return a tuple, even for scalar valued functions. This is done for consistency.

eval_components(*args, **kwargs)

Evaluate the components of the model with the given data. Used for numerical evaluation.

eval_jacobian(*args, **kwargs)

Returns:The jacobian matrix of the function.

finite_difference(*args, dx=1e-08, **kwargs)

Calculates a numerical approximation of the Jacobian of the model using the sixth order central finite difference method. Accepts a dx keyword to tune the relative stepsize used. Makes 6*n_params calls to the model.

Returns:A numerical approximation of the Jacobian of the model as a list with length n_components containing numpy arrays of shape (n_params, n_datapoints)

numerical_components

Returns:lambda functions of each of the components in model_dict, to be used in numerical calculation.

class symfit.core.fit.Constraint(constraint, model)

Bases: symfit.core.fit.Model

Constraints are a special type of model in that they have a type: >=, == etc. They are made to have lhs - rhs == 0 of the original expression.

For example, Eq(y + x, 4) -> Eq(y + x - 4, 0)

Since a constraint belongs to a certain model, it has to be initiated with knowledge of it’s parent model. This is important because all numerical_ methods are done w.r.t. the parameters and variables of the parent model, not the constraint! This is because the constraint might not have all the parameter or variables that the model has, but in order to compute for example the Jacobian we still want to derive w.r.t. all the parameters, not just those present in the constraint.

__init__(constraint, model)

Parameters:

• constraint – constraint that model should be subjected to.
• model – A constraint is always tied to a model.

__neg__()

Returns:new model with opposite sign. Does not change the model in-place, but returns a new copy.

jacobian

Returns:Jacobian ‘Matrix’ filled with the symbolic expressions for all the partial derivatives. Partial derivatives are of the components of the function with respect to the Parameter’s, not the independent Variable’s.

numerical_components

Returns:lambda functions of each of the components in model_dict, to be used in numerical calculation.

numerical_jacobian

Returns:lambda functions of the jacobian matrix of the function, which can be used in numerical optimization.

class symfit.core.fit.Fit(model, *ordered_data, objective=None, constraints=None, minimizer=None, **named_data)

Bases: symfit.core.fit.HasCovarianceMatrix

Your one stop fitting solution! Based on the nature of the input, this object will attempt to select the right fitting type for your problem.

If you need very specific control over how the problem is solved, you can pass it the minimizer or objective function you would like to use.

Example usage:

a, b = parameters('a, b')
x, y = variables('x, y')

model = {y: a * x + b}

# Fit will use its default settings
fit = Fit(model, x=xdata, y=ydata)
fit_result = fit.execute()

# Use Nelder-Mead instead
fit = Fit(model, x=xdata, y=ydata, minimizer=NelderMead)
fit_result = fit.execute()

# Use Nelder-Mead to get close, and BFGS to polish it off
fit = Fit(model, x=xdata, y=ydata, minimizer=[NelderMead, BFGS])
fit_result = fit.execute(minimizer_kwargs=[dict(xatol=0.1), {}])

__init__(model, *ordered_data, objective=None, constraints=None, minimizer=None, **named_data)

Parameters:

• model – (dict of) sympy expression(s) or Model object.
• constraints – iterable of Relation objects to be used as constraints.
• absolute_sigma (bool) – True by default. If the sigma is only used for relative weights in your problem, you could consider setting it to False, but if your sigma are measurement errors, keep it at True. Note that curve_fit has this set to False by default, which is wrong in experimental science.
• objective – Have Fit use your specified objective. Can be one of the predefined symfit objectives or any callable which accepts fit parameters and returns a scalar.
• minimizer – Have Fit use your specified symfit.core.minimizers.BaseMinimizer. Can be a Sequence of symfit.core.minimizers.BaseMinimizer.
• ordered_data – data for dependent, independent and sigma variables. Assigned in the following order: independent vars are assigned first, then dependent vars, then sigma’s in dependent vars. Within each group they are assigned in alphabetical order.
• named_data – assign dependent, independent and sigma variables data by name.

execute(**minimize_options)

Execute the fit.

Parameters:minimize_options – keyword arguments to be passed to the specified minimizer. Returns:FitResults instance

class symfit.core.fit.HasCovarianceMatrix(model, *ordered_data, absolute_sigma=None, **named_data)

Bases: symfit.core.fit.TakesData

Mixin class for calculating the covariance matrix for any model that has a well-defined Jacobian \(J\). The covariance is then approximated as \(J^T W J\), where W contains the weights of each data point.

Supports vector valued models, but is unable to estimate covariances for those, just variances. Therefore, take the result with a grain of salt for vector models.

covariance_matrix(best_fit_params)

Given best fit parameters, this function finds the covariance matrix. This matrix gives the (co)variance in the parameters.

Parameters:best_fit_params – dict of best fit parameters as given by .best_fit_params() Returns:covariance matrix.

class symfit.core.fit.LinearLeastSquares(*args, **kwargs)

Bases: symfit.core.fit.BaseFit

Experimental. Solves the linear least squares problem analytically. Involves no iterations or approximations, and therefore gives the best possible fit to the data.

The Model provided has to be linear.

Currently, since this object still has to mature, it suffers from the following limitations:

• It does not check if the model can be linearized by a simple substitution. For example, exp(a * x) -> b * exp(x). You will have to do this manually.
• Does not use bounds or guesses on the Parameter’s. Then again, it doesn’t have to, since you have an exact solution. No guesses required.
• It only works with scalar functions. This is strictly enforced.

__init__(*args, **kwargs)

Raises:ModelError in case of a non-linear model or when a vector valued function is provided.

best_fit_params()

Fits to the data and returns the best fit parameters.

Returns:dict containing parameters and their best-fit values.

covariance_matrix(best_fit_params)

Given best fit parameters, this function finds the covariance matrix. This matrix gives the (co)variance in the parameters.

Parameters:best_fit_params – dict of best fit parameters as given by .best_fit_params() Returns:covariance matrix.

execute()

Execute an analytical (Linear) Least Squares Fit. This object works by symbolically solving when \(\nabla \chi^2 = 0\).

To perform this task the expression of \(\nabla \chi^2\) is determined, ignoring that \(\chi^2\) involves summing over all terms. Then the sum is performed by substituting the variables by their respective data and summing all terms, while leaving the parameters symbolic.

The resulting system of equations is then easily solved with sympy.solve.

Returns:FitResult

static is_linear(model)

Test whether model is of linear form in it’s parameters.

Currently this function does not recognize if a model can be considered linear by a simple substitution, such as exp(k x) = k’ exp(x).

Parameters:model – Model instance Returns:True or False

class symfit.core.fit.Model(model)

Bases: symfit.core.fit.CallableModel

Model represents a symbolic function and all it’s derived properties such as sum of squares, jacobian etc. Models can be initiated from several objects:

a = Model({y: x**2})
b = Model(y=x**2)

Models are callable. The usual rules apply to the ordering of the arguments:

• first independent variables, then dependent variables, then parameters.
• within each of these groups they are ordered alphabetically.

Models are also iterable, behaving as their internal model_dict. In the example above, a[y] returns x**2, len(a) == 1, y in a == True, etc.

__str__()

Printable representation of this model.

Returns:str

chi

Returns:Symbolic Square root of \(\chi^2\). Required for MINPACK optimization only. Denoted as \(\sqrt(\chi^2)\)

chi_jacobian

Return a symbolic jacobian of the \(\sqrt(\chi^2)\) function. Vector of derivatives w.r.t. each parameter. Not a Matrix but a vector! This is because that’s what leastsq needs.

chi_squared

Returns:Symbolic \(\chi^2\)

chi_squared_jacobian

Return a symbolic jacobian of the \(\chi^2\) function. Vector of derivatives w.r.t. each parameter. Not a Matrix but a vector!

eval_components(*args, **kwargs)

Returns:lambda functions of each of the components in model_dict, to be used in numerical calculation.

eval_jacobian(*args, **kwargs)

Returns:Jacobian evaluated at the specified point.

jacobian

Returns:Jacobian ‘Matrix’ filled with the symbolic expressions for all the partial derivatives. Partial derivatives are of the components of the function with respect to the Parameter’s, not the independent Variable’s.

numerical_chi

Returns:lambda function of the .chi method, to be used in MINPACK optimisation.

numerical_chi_jacobian

Returns:lambda functions of the jacobian of the .chi method, which can be used in numerical optimization.

numerical_chi_squared

Returns:lambda function of the .chi_squared method, to be used in numerical optimisation.

numerical_chi_squared_jacobian

Returns:lambda functions of the jacobian of the .chi_squared method.

numerical_jacobian

Returns:lambda functions of the jacobian matrix of the function, which can be used in numerical optimization.

exception symfit.core.fit.ModelError

Bases: Exception

Raised when a problem occurs with a model.

class symfit.core.fit.NonLinearLeastSquares(*args, **kwargs)

Bases: symfit.core.fit.BaseFit

Experimental. Implements non-linear least squares [wiki_nllsq]. Works by a two step process: First the model is linearised by doing a first order taylor expansion around the guesses for the parameters. Then a LinearLeastSquares fit is performed. This is iterated until a fit of sufficient quality is obtained.

Sensitive to good initial guesses. Providing good initial guesses is a must.

[wiki_nllsq]

https://en.wikipedia.org/wiki/Non-linear_least_squares

__init__(*args, **kwargs)

Parameters:

• model – (dict of) sympy expression or Model object.
• absolute_sigma (bool) – True by default. If the sigma is only used for relative weights in your problem, you could consider setting it to False, but if your sigma are measurement errors, keep it at True. Note that curve_fit has this set to False by default, which is wrong in experimental science.
• ordered_data – data for dependent, independent and sigma variables. Assigned in the following order: independent vars are assigned first, then dependent vars, then sigma’s in dependent vars. Within each group they are assigned in alphabetical order.
• named_data – assign dependent, independent and sigma variables data by name.

Standard deviation can be provided to any variable. They have to be prefixed with sigma_. For example, let x be a Variable. Then sigma_x will give the stdev in x.

execute(relative_error=1e-08, max_iter=500)

Perform a non-linear least squares fit.

Parameters:

• relative_error – Relative error between the sum of squares of subsequent itterations. Once smaller than the value specified, the fit is considered complete.
• max_iter – Maximum number of iterations before giving up.

Returns:

Instance of FitResults.

class symfit.core.fit.ODEModel(model_dict, initial, *lsoda_args, **lsoda_kwargs)

Bases: symfit.core.fit.CallableModel

Model build from a system of ODEs. When the model is called, the ODE is integrated using the LSODA package.

__call__(*args, **kwargs)

Evaluate the model for a certain value of the independent vars and parameters. Signature for this function contains independent vars and parameters, NOT dependent and sigma vars.

Can be called with both ordered and named parameters. Order is independent vars first, then parameters. Alphabetical order within each group.

Parameters:

• args – Ordered arguments for the parameters and independent variables
• kwargs – Keyword arguments for the parameters and independent variables

Returns:

A namedtuple of all the dependent vars evaluated at the desired point. Will always return a tuple, even for scalar valued functions. This is done for consistency.

__getitem__(dependent_var)

Gives the function defined for the derivative of dependent_var. e.g. \(y' = f(y, t)\), model[y] -> f(y, t)

Parameters:dependent_var – Returns:

__init__(model_dict, initial, *lsoda_args, **lsoda_kwargs)

Parameters:

• model_dict – Dictionary specifying ODEs. e.g. model_dict = {D(y, x): a * x**2}
• initial – dict of initial conditions for the ODE. Must be provided! e.g. initial = {y: 1.0, x: 0.0}
• lsoda_args – args to pass to the lsoda solver. See scipy’s odeint for more info.
• lsoda_kwargs – kwargs to pass to the lsoda solver.

__iter__()

Returns:iterable over self.model_dict

__neg__()

Returns:new model with opposite sign. Does not change the model in-place, but returns a new copy.

__str__()

Printable representation of this model.

Returns:str

eval_components(*args, **kwargs)

Numerically integrate the system of ODEs.

Parameters:

• args – Ordered arguments for the parameters and independent variables
• kwargs – Keyword arguments for the parameters and independent variables

Returns:

class symfit.core.fit.TakesData(model, *ordered_data, absolute_sigma=None, **named_data)

Bases: object

An base class for everything that takes data. Most importantly, it takes care of linking the provided data to variables. The allowed variables are extracted from the model.

__init__(model, *ordered_data, absolute_sigma=None, **named_data)

Parameters:

• model – (dict of) sympy expression or Model object.
• absolute_sigma (bool) – True by default. If the sigma is only used for relative weights in your problem, you could consider setting it to False, but if your sigma are measurement errors, keep it at True. Note that curve_fit has this set to False by default, which is wrong in experimental science.
• ordered_data – data for dependent, independent and sigma variables. Assigned in the following order: independent vars are assigned first, then dependent vars, then sigma’s in dependent vars. Within each group they are assigned in alphabetical order.
• named_data – assign dependent, independent and sigma variables data by name.

Standard deviation can be provided to any variable. They have to be prefixed with sigma_. For example, let x be a Variable. Then sigma_x will give the stdev in x.

data_shapes

Returns the shape of the data. In most cases this will be the same for all variables of the same type, if not this raises an Exception.

Ignores variables which are set to None by design so we know that those None variables can be assumed to have the same shape as the other in calculations where this is needed, such as the covariance matrix.

Returns:Tuple of all independent var shapes, dependent var shapes.

dependent_data

Read-only Property

Returns:Data belonging to each dependent variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

independent_data

Read-only Property

Returns:Data belonging to each independent variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

initial_guesses

Returns:Initial guesses for every parameter.

sigma_data

Read-only Property

Returns:Data belonging to each sigma variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

class symfit.core.fit.TaylorModel(model)

Bases: symfit.core.fit.Model

A first-order Taylor expansion of a model around given parameter values (\(p_0\)). Is used by NonLinearLeastSquares. Currently only a first order expansion is implemented.

__init__(model)

Make a first order Taylor expansion of model.

Parameters:model – Instance of Model

__str__()

When printing a TaylorModel, the point around which the expansion took place is included.

For example, a Taylor expansion of {y: sin(w * x)} at w = 0 would be printed as:

@{w: 0.0} -> y(x; w) = w*x

p0

Property of the \(p_0\) around which to expand. Should be set by the names of the parameters themselves.

Example:

a = Parameter()
x, y = variables('x, y')
model = TaylorModel({y: sin(a * x)})

model.p0 = {a: 0.0}

params

params returns only the free parameters. Strictly speaking, the expression for a TaylorModel contains both the parameters \(\vec{p}\) and \(\vec{p_0}\) around which to expand, but params should only give \(\vec{p}\). To get a mapping to the \(\vec{p_0}\), use .params_0.

symfit.core.fit.r_squared(model, fit_result, data)

Calculates the coefficient of determination, R^2, for the fit.

(Is not defined properly for vector valued functions.)

Parameters:

• model – Model instance
• fit_result – FitResults instance
• data – data with which the fit was performed.

Argument

class symfit.core.argument.Argument(name=None, *args, **assumptions)

Bases: sympy.core.symbol.Symbol

Base class for symfit symbols. This helps make symfit symbols distinguishable from sympy symbols.

If no name is explicitly provided a name will be generated.

For example:

y = Variable()
print(y.name)
>> 'x_0'

y = Variable('y')
print(y.name)
>> 'y'

__init__(name=None, *args, **assumptions)

Initialize self. See help(type(self)) for accurate signature.

static __new__(cls, name=None, *args, **assumptions)

Symbols are identified by name and assumptions:

>>> from sympy import Symbol
>>> Symbol("x") == Symbol("x")

True >>> Symbol(“x”, real=True) == Symbol(“x”, real=False) False

class symfit.core.argument.Parameter(name=None, value=1.0, min=None, max=None, fixed=False, **assumptions)

Bases: symfit.core.argument.Argument

Parameter objects are used to facilitate bounds on function parameters. Important change from symfit>0.4.1: the name needs to be the first keyword, followed by the guess value. If no name is provided, the initial value can be passed as a keyword argument, e.g.: value=0.1. A generic name will then be generated.

__call__(*values, **named_values)

Call an expression to evaluate it at the given point.

Future improvements: I would like if func and signature could be buffered after the first call so they don’t have to be recalculated for every call. However, nothing can be stored on self as sympy uses __slots__ for efficiency. This means there is no instance dict to put stuff in! And I’m pretty sure it’s ill advised to hack into the __slots__ of Expr.

However, for the moment I don’t really notice a performance penalty in running tests.

p.s. In the current setup signature is not even needed since no introspection is possible on the Expr before calling it anyway, which makes calculating the signature absolutely useless. However, I hope that someday some monkey patching expert in shining armour comes by and finds a way to store it in __signature__ upon __init__ of any symfit expr such that calling inspect_sig.signature on a symbolic expression will tell you which arguments to provide.

Parameters:

• self – Any subclass of sympy.Expr
• values – Values for the Parameters and Variables of the Expr.
• named_values – Values for the vars and params by name. named_values is allowed to contain too many values, as this sometimes happens when using **fit_result.params on a submodel. The irrelevant params are simply ignored.

Returns:

The function evaluated at values. The type depends entirely on the input. Typically an array or a float but nothing is enforced.

__init__(name=None, value=1.0, min=None, max=None, fixed=False, **assumptions)

Parameters:

• name – Name of the Parameter.
• value – Initial guess value.
• min – Lower bound on the parameter value.
• max – Upper bound on the parameter value.
• fixed (bool) – Fix the parameter to value during fitting.
• assumptions – assumptions to pass to sympy.

static __new__(cls, name=None, *args, **kwargs)

Symbols are identified by name and assumptions:

>>> from sympy import Symbol
>>> Symbol("x") == Symbol("x")

True >>> Symbol(“x”, real=True) == Symbol(“x”, real=False) False

class symfit.core.argument.Variable(name=None, *args, **assumptions)

Bases: symfit.core.argument.Argument

Variable type.

Operators

Monkey Patching module.

This module makes sympy Expressions callable, which makes the whole project feel more consistent.

symfit.core.operators.call(self, *values, **named_values)

Call an expression to evaluate it at the given point.

Future improvements: I would like if func and signature could be buffered after the first call so they don’t have to be recalculated for every call. However, nothing can be stored on self as sympy uses __slots__ for efficiency. This means there is no instance dict to put stuff in! And I’m pretty sure it’s ill advised to hack into the __slots__ of Expr.

However, for the moment I don’t really notice a performance penalty in running tests.

p.s. In the current setup signature is not even needed since no introspection is possible on the Expr before calling it anyway, which makes calculating the signature absolutely useless. However, I hope that someday some monkey patching expert in shining armour comes by and finds a way to store it in __signature__ upon __init__ of any symfit expr such that calling inspect_sig.signature on a symbolic expression will tell you which arguments to provide.

Parameters:

• self – Any subclass of sympy.Expr
• values – Values for the Parameters and Variables of the Expr.
• named_values – Values for the vars and params by name. named_values is allowed to contain too many values, as this sometimes happens when using **fit_result.params on a submodel. The irrelevant params are simply ignored.

Returns:

The function evaluated at values. The type depends entirely on the input. Typically an array or a float but nothing is enforced.

Fit Results

class symfit.core.fit_results.FitResults(model, popt, covariance_matrix, infodic, mesg, ier, **gof_qualifiers)

Bases: object

Class to display the results of a fit in a nice and unambiguous way. All things related to the fit are available on this class, e.g. - parameter values + stdev - R squared (Regression coefficient.) or other fit quality qualifiers. - fitting status message - covariance matrix

Contains the attribute params, which is an OrderedDict containing all the parameter names and their optimized values. Can be ** unpacked when evaluating Model’s.

__getattr__(item)

Return the requested item if it can be found in the gof_qualifiers dict.

Parameters:item – Name of Goodness of Fit qualifier. Returns:Goodness of Fit qualifier if present.

__init__(model, popt, covariance_matrix, infodic, mesg, ier, **gof_qualifiers)

Excuse the ugly names of most of these variables, they are inherited from scipy. Will be changed.

Parameters:

• model – Model that was fit to.
• popt – best fit parameters, same ordering as in model.params.
• pcov – covariance matrix.
• infodic – dict with fitting info.
• mesg – Status message.
• ier – Number of iterations.
• gof_qualifiers – Any remaining keyword arguments should be Goodness of fit (g.o.f.) qualifiers.

__str__()

Pretty print the results as a table.

covariance(param_1, param_2)

Return the covariance between param_1 and param_2.

Parameters:

• param_1 – Parameter Instance.
• param_2 – Parameter Instance.

Returns:

Covariance of the two params.

stdev(param)

Return the standard deviation in a given parameter as found by the fit.

Parameters:param – Parameter Instance. Returns:Standard deviation of param.

value(param)

Return the value in a given parameter as found by the fit.

Parameters:param – Parameter Instance. Returns:Value of param.

variance(param)

Return the variance in a given parameter as found by the fit.

Parameters:param – Parameter Instance. Returns:Variance of param.

Minimizers

class symfit.core.minimizers.BFGS(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyGradientMinimize

Wrapper around scipy.optimize.minimize()’s BFGS algorithm.

class symfit.core.minimizers.BaseMinimizer(objective, parameters)

Bases: object

ABC for all Minimizers.

__init__(objective, parameters)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

execute(**options)

The execute method should implement the actual minimization procedure, and should return a FitResults instance.

Parameters:options – options to be used by the minimization procedure. Returns:an instance of FitResults.

class symfit.core.minimizers.BasinHopping(*args, local_minimizer=<class 'symfit.core.minimizers.BFGS'>, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.BaseMinimizer

Wrapper around scipy.optimize.basinhopping()’s basin-hopping algorithm.

As always, the best way to use this algorithm is through Fit, as this will automatically select a local minimizer for you depending on whether you provided bounds, constraints, etc.

However, BasinHopping can also be used directly. Example (with jacobian):

import numpy as np
from symfit.core.minimizers import BFGS, BasinHopping
from symfit import parameters

def func2d(x1, x2):
    f = np.cos(14.5 * x1 - 0.3) + (x2 + 0.2) * x2 + (x1 + 0.2) * x1
    return f

def jac2d(x1, x2):
    df = np.zeros(2)
    df[0] = -14.5 * np.sin(14.5 * x1 - 0.3) + 2. * x1 + 0.2
    df[1] = 2. * x2 + 0.2
    return df

x0 = [1.0, 1.0]
np.random.seed(555)
x1, x2 = parameters('x1, x2', value=x0)
fit = BasinHopping(func2d, [x1, x2], local_minimizer=BFGS)
minimizer_kwargs = {'jac': fit.list2kwargs(jac2d)}
fit_result = fit.execute(niter=200, minimizer_kwargs=minimizer_kwargs)

See scipy.optimize.basinhopping() for more options.

__init__(*args, local_minimizer=<class 'symfit.core.minimizers.BFGS'>, **kwargs)

Parameters:

• local_minimizer – minimizer to be used for local minimization steps. Can be any subclass of symfit.core.minimizers.ScipyMinimize.
• args – positional arguments to be passed on to super.
• kwargs – keyword arguments to be passed on to super.

execute(**minimize_options)

Execute the basin-hopping minimization.

Parameters:minimize_options – options to be passed on to scipy.optimize.basinhopping(). Returns:symfit.core.fit_results.FitResults

class symfit.core.minimizers.BoundedMinimizer(objective, parameters)

Bases: symfit.core.minimizers.BaseMinimizer

ABC for Minimizers that support bounds.

class symfit.core.minimizers.COBYLA(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyConstrainedMinimize

Wrapper around scipy.optimize.minimize()’s COBYLA algorithm.

class symfit.core.minimizers.ChainedMinimizer(*args, minimizers=None, **kwargs)

Bases: symfit.core.minimizers.BaseMinimizer

A minimizer that consists of multiple other minimizers, each executed in order. This is valuable if you have minimizers that are not good at finding the exact minimum such as NelderMead or DifferentialEvolution.

__init__(*args, minimizers=None, **kwargs)

Parameters:

• minimizers – a Sequence of BaseMinimizer objects, which need to be run in order.
• *args – passed to symfit.core.minimizers.BaseMinimizer.__init__().
• **kwargs – passed to symfit.core.minimizers.BaseMinimizer.__init__().

execute(**minimizer_kwargs)

Execute the chained-minimization. In order to pass options to the seperate minimizers, they can be passed by using the names of the minimizers as keywords. For example:

fit = Fit(self.model, self.xx, self.yy, self.ydata,
          minimizer=[DifferentialEvolution, BFGS])
fit_result = fit.execute(
    DifferentialEvolution={'seed': 0, 'tol': 1e-4, 'maxiter': 10},
    BFGS={'tol': 1e-4}
)

In case of multiple identical minimizers an index is added to each keyword argument to make them identifiable. For example, if:

minimizer=[BFGS, DifferentialEvolution, BFGS])

then the keyword arguments will be ‘BFGS’, ‘DifferentialEvolution’, and ‘BFGS_2’.

Parameters:minimizer_kwargs – Minimizer options to be passed to the minimzers by name Returns:an instance of FitResults.

class symfit.core.minimizers.ConstrainedMinimizer(*args, constraints=None, **kwargs)

Bases: symfit.core.minimizers.BaseMinimizer

ABC for Minimizers that support constraints

__init__(*args, constraints=None, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

class symfit.core.minimizers.DifferentialEvolution(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.GlobalMinimizer, symfit.core.minimizers.BoundedMinimizer

A wrapper around scipy.optimize.differential_evolution().

execute(*, strategy='rand1bin', mutation=(0.423, 1.053), recombination=0.95, polish=False, popsize=40, init='latinhypercube', **de_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

class symfit.core.minimizers.DummyModel(params)

Bases: tuple

__getnewargs__()

Return self as a plain tuple. Used by copy and pickle.

static __new__(_cls, params)

Create new instance of DummyModel(params,)

__repr__()

Return a nicely formatted representation string

params

Alias for field number 0

class symfit.core.minimizers.GlobalMinimizer(*args, **kwargs)

Bases: symfit.core.minimizers.BaseMinimizer

A minimizer that looks for a global minimum, instead of a local one.

__init__(*args, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

class symfit.core.minimizers.GradientMinimizer(*args, jacobian=None, **kwargs)

Bases: symfit.core.minimizers.BaseMinimizer

ABC for Minizers that support the use of a jacobian

__init__(*args, jacobian=None, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

resize_jac(func)

Removes values with identical indices to fixed parameters from the output of func. func has to return the jacobian of a scalar function.

Parameters:func – Jacobian function to be wrapped. Is assumed to be the jacobian of a scalar function. Returns:Jacobian corresponding to non-fixed parameters only.

class symfit.core.minimizers.LBFGSB(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyGradientMinimize, symfit.core.minimizers.BoundedMinimizer

Wrapper around scipy.optimize.minimize()’s LBFGSB algorithm.

execute(**minimize_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

classmethod method_name()

Returns the name of the minimize method this object represents. This is needed because the name of the object is not always exactly what needs to be passed on to scipy as a string. :return:

class symfit.core.minimizers.MINPACK(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.GradientMinimizer, symfit.core.minimizers.BoundedMinimizer

Wrapper to scipy’s implementation of MINPACK, since it is the industry standard.

__init__(*args, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

execute(**minpack_options)

Parameters:**minpack_options – Any named arguments to be passed to leastsqbound

class symfit.core.minimizers.NelderMead(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.BaseMinimizer

Wrapper around scipy.optimize.minimize()’s NelderMead algorithm.

classmethod method_name()

Returns the name of the minimize method this object represents. This is needed because the name of the object is not always exactly what needs to be passed on to scipy as a string. :return:

class symfit.core.minimizers.SLSQP(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyConstrainedMinimize, symfit.core.minimizers.GradientMinimizer, symfit.core.minimizers.BoundedMinimizer

Wrapper around scipy.optimize.minimize()’s SLSQP algorithm.

__init__(*args, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

execute(**minimize_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

scipy_constraints(constraints)

Returns all constraints in a scipy compatible format.

Returns:dict of scipy compatible constraints, including jacobian term.

class symfit.core.minimizers.ScipyConstrainedMinimize(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.ConstrainedMinimizer

Base class for scipy.optimize.minimize()’s constrained-minimizers.

__init__(*args, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

execute(**minimize_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

scipy_constraints(constraints)

Returns all constraints in a scipy compatible format.

Returns:dict of scipy compatible statements.

class symfit.core.minimizers.ScipyGradientMinimize(*args, **kwargs)

Bases: symfit.core.minimizers.ScipyMinimize, symfit.core.minimizers.GradientMinimizer

Base class for scipy.optimize.minimize()’s gradient-minimizers.

__init__(*args, **kwargs)

Parameters:

• objective – Objective function to be used.
• parameters – List of Parameter instances

execute(**minimize_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

class symfit.core.minimizers.ScipyMinimize(*args, **kwargs)

Bases: object

Mix-in class that handles the execute calls to scipy.optimize.minimize().

__init__(*args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

execute(bounds=None, jacobian=None, constraints=None, *, tol=1e-09, **minimize_options)

Calls the wrapped algorithm.

Parameters:

• bounds – The bounds for the parameters. Usually filled by BoundedMinimizer.
• jacobian – The Jacobian. Usually filled by ScipyGradientMinimize.
• **minimize_options – Further keywords to pass to scipy.optimize.minimize(). Note that your method will usually be filled by a specific subclass.

list2kwargs(func)

Given an objective function func, make sure it is always called via keyword arguments with the relevant parameter names.

Parameters:func – Function to be wrapped to keyword only calls. Returns:wrapped function

classmethod method_name()

Returns the name of the minimize method this object represents. This is needed because the name of the object is not always exactly what needs to be passed on to scipy as a string. :return:

Objectives

class symfit.core.objectives.BaseObjective(model, data)

Bases: object

ABC for objective functions. Implements basic data handling.

__call__(**parameters)

Evaluate the objective function for given parameter values.

Parameters:parameters – Returns:float

__init__(model, data)

Parameters:

• model – symfit style model.
• data – data for all the variables of the model.

dependent_data

Read-only Property

Returns:Data belonging to each dependent variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

independent_data

Read-only Property

Returns:Data belonging to each independent variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

sigma_data

Read-only Property

Returns:Data belonging to each sigma variable as a dict with variable names as key, data as value. Return type:collections.OrderedDict

class symfit.core.objectives.GradientObjective(model, data)

Bases: symfit.core.objectives.BaseObjective

ABC for objectives that support gradient methods.

eval_jacobian(**parameters)

Evaluate the jacobian for given parameter values.

Parameters:parameters – Returns:float

class symfit.core.objectives.LeastSquares(model, data)

Bases: symfit.core.objectives.GradientObjective

Objective representing the \(\chi^2\) of a model.

__call__(*, flatten_components=True, **parameters)

Parameters:

• parameters – values of the Parameter’s to evaluate \(\chi^2\) at.
• flatten_components – if True, return the total \(\chi^2\). If False, return the \(\chi^2\) per component of the BaseModel.

Returns:

scalar or list of scalars depending on the value of flatten_components.

eval_jacobian(**parameters)

Jacobian of \(\chi^2\) in the Parameter’s (\(\nabla_\vec{p} \chi^2\)).

Parameters:parameters – values of the Parameter’s to evaluate \(\nabla_\vec{p} \chi^2\) at. Returns:np.array of length equal to the number of parameters..

class symfit.core.objectives.LogLikelihood(model, data)

Bases: symfit.core.objectives.GradientObjective

Error function to be minimized by a minimizer in order to maximize the log-likelihood.

__call__(**parameters)

Parameters:parameters – values for the fit parameters. Returns:scalar value of log-likelihood

eval_jacobian(*, apply_func=<function nansum>, **parameters)

Jacobian for log-likelihood is defined as \(\nabla_{\vec{p}}( \log( L(\vec{p} | \vec{x})))\).

Parameters:

• parameters – values for the fit parameters.
• apply_func – Function to apply to each component before returning it. The default is to sum away along the datapoint dimension using np.nansum.

Returns:

array of length number of Parameter’s in the model, with all partial derivatives evaluated at p, data.

class symfit.core.objectives.MinimizeModel(model, *args, **kwargs)

Bases: symfit.core.objectives.BaseObjective

Objective to use when the model itself is the quantity that should be minimized. This is only supported for scalar models.

__call__(**parameters)

Evaluate the objective function for given parameter values.

Parameters:parameters – Returns:float

__init__(model, *args, **kwargs)

Parameters:

• model – symfit style model.
• data – data for all the variables of the model.

class symfit.core.objectives.VectorLeastSquares(model, data)

Bases: symfit.core.objectives.GradientObjective

Implemented for MINPACK only. Returns the residuals/sigma before squaring and summing, rather then chi2 itself.

__call__(*, flatten_components=True, **parameters)

Returns the value of the square root of \(\chi^2\), summing over the components.

This function now supports setting variables to None.

Parameters:

• p – array of parameter values.
• flatten_components – If True, summing is performed over the data indices (default).

Returns:

\(\sqrt(\chi^2)\)

eval_jacobian(**parameters)

Evaluate the jacobian for given parameter values.

Parameters:parameters – Returns:float

Support

This module contains support functions and convenience methods used throughout symfit. Some are used predominantly internally, others are designed for users.

class symfit.core.support.D

Bases: sympy.core.function.Derivative

Convenience wrapper for sympy.Derivative. Used most notably in defining ODEModel’s.

class symfit.core.support.RequiredKeyword

Bases: object

Flag variable to indicate that this is a required keyword.

exception symfit.core.support.RequiredKeywordError

Bases: Exception

Error raised in case a keyword-only argument is not treated as such.

class symfit.core.support.cached_property

Bases: property

A property which cashes the output of the first ever call and always returns that value from then on, unless delete is called on the attribute.

This is typically used in converting sympy code into scipy compatible code, which is computationally a very expensive step we would like to perform only once.

Does not allow setting of the attribute.

__delete__(obj)

Calling delete on the attribute will delete the cache. :param obj: parent object.

__get__(obj, objtype=None)

In case of a first call, this will call the decorated function and return it’s output. On every subsequent call, the same output will be returned.

Parameters:

• obj – the parent object this property is attached to.
• objtype –

Returns:

Output of the first call to the decorated function.

class symfit.core.support.deprecated(replacement=None)

Bases: object

Decorator to raise a DeprecationWarning.

__call__(func)

Call self as a function.

__init__(replacement=None)

Parameters:replacement – The function which should now be used instead.

symfit.core.support.jacobian(expr, symbols)

Derive a symbolic expr w.r.t. each symbol in symbols. This returns a symbolic jacobian vector.

Parameters:

• expr – A sympy Expr.
• symbols – The symbols w.r.t. which to derive.

symfit.core.support.key2str(target)

In symfit there are many dicts with symbol: value pairs. These can not be used immediately as **kwargs, even though this would make a lot of sense from the context. This function wraps such dict to make them usable as **kwargs immediately.

Parameters:target – Mapping to be made save Returns:Mapping of str(symbol): value pairs.

class symfit.core.support.keywordonly(**kwonly_arguments)

Bases: object

Decorator class which wraps a python 2 function into one with keyword-only arguments.

Example:

@keywordonly(floor=True)
def f(x, **kwargs):
    floor = kwargs.pop('floor')
    return np.floor(x**2) if floor else x**2

This decorator is not much more than:

floor = kwargs.pop('floor') if 'floor' in kwargs else True

However, I prefer it’s usage because:

• it’s clear from reading the function declaration there is an option to provide this argument. The information on possible keywords is where you’d expect it to be.
• you’re guaranteed that the pop works.
• It is fully inspect compatible such that sphynx is able to index these properly as keyword only arguments just like it would for native py3 keyword only arguments.

Please note that this decorator needs a ** argument on the wrapped function in order to work.

__call__(func)

Returns a decorated version of func, who’s signature now includes the keyword-only arguments.

Parameters:func – the function to be decorated Returns:the decorated function

__init__(**kwonly_arguments)

Initialize self. See help(type(self)) for accurate signature.

symfit.core.support.parameters(names, **kwargs)

Convenience function for the creation of multiple parameters. For more control, consider using symbols(names, cls=Parameter, **kwargs) directly.

The Parameter attributes value, min, max and fixed can also be provided directly. If given as a single value, the same value will be set for all Parameter’s. When a sequence, it must be of the same length as the number of parameters created.

Example::

x1, x2 = parameters(‘x1, x2’, value=[2.0, 1.3], min=0.0)

Parameters:

• names – string of parameter names. Example: a, b = parameters(‘a, b’)
• kwargs – kwargs to be passed onto sympy.core.symbol.symbols(). value, min and max will be handled separately if they are sequences.

Returns:

iterable of symfit.core.argument.Parameter objects

symfit.core.support.seperate_symbols(func)

Seperate the symbols in symbolic function func. Return them in alphabetical order.

Parameters:func – scipy symbolic function. Returns:(vars, params), a tuple of all variables and parameters, each sorted in alphabetical order. Raises:TypeError – only symfit Variable and Parameter are allowed, not sympy Symbols.

symfit.core.support.sympy_to_py(func, vars, params)

Turn a symbolic expression into a Python lambda function, which has the names of the variables and parameters as it’s argument names.

Parameters:

• func – sympy expression
• vars – variables in this model
• params – parameters in this model

Returns:

lambda function to be used for numerical evaluation of the model. Ordering of the arguments will be vars first, then params.

symfit.core.support.sympy_to_scipy(func, vars, params)

Convert a symbolic expression to one scipy digs. Not used by symfit any more.

Parameters:

• func – sympy expression
• vars – variables
• params – parameters

Returns:

Scipy-style function to be used for numerical evaluation of the model.

symfit.core.support.variables(names, **kwargs)

Convenience function for the creation of multiple variables. For more control, consider using symbols(names, cls=Variable, **kwargs) directly.

Parameters:

• names – string of variable names. Example: x, y = variables(‘x, y’)
• kwargs – kwargs to be passed onto sympy.core.symbol.symbols()

Returns:

iterable of symfit.core.argument.Variable objects

Distributions

Some common distributions are defined in this module. That way, users can easily build more complicated expressions without making them look hard.

I have deliberately chosen to start these function with a capital, e.g. Gaussian instead of gaussian, because this makes the resulting expressions more readable.

symfit.distributions.Exp(x, l)

Exponential Distribution pdf. :param x: free variable. :param l: rate parameter. :return: sympy.Expr for an Exponential Distribution pdf.

symfit.distributions.Gaussian(x, mu, sig)

Gaussian pdf. :param x: free variable. :param mu: mean of the distribution. :param sig: standard deviation of the distribution. :return: sympy.Expr for a Gaussian pdf.

Contrib

Contrib modules are modules and extensions to symfit provided by other people. This usually means the code is of slightly less quality, and may not survive future versions.

class symfit.contrib.interactive_guess.interactive_guess.InteractiveGuess2D(*args, n_points=100, **kwargs)

Bases: symfit.core.fit.TakesData

A class that provides an graphical, interactive way of guessing initial fitting parameters.

__init__(*args, n_points=100, **kwargs)

Create a matplotlib window with sliders for all parameters in this model, so that you may graphically guess initial fitting parameters. n_points is the number of points drawn for the plot. Data points are plotted as blue points, the proposed model as a red line.

Slider extremes are taken from the parameters where possible. If these are not provided, the minimum is 0; and the maximum is value*2. If no initial value is provided, it defaults to 1.

This will modify the values of the parameters present in model.

Parameters:n_points (int) – The number of points used for drawing the fitted function.

__str__()

Represent the guesses in a human readable way.

Returns:string with the guessed values.

execute(*, show=True, block=True, **kwargs)

Execute the interactive guessing procedure.

Parameters:

• show (bool) – Whether or not to show the figure. Useful for testing.
• block – Blocking call to matplotlib