Exponential¶

class plasmapy.analysis.fit_functions.Exponential(params: Optional[Tuple[float, …]] = None, param_errors: Optional[Tuple[float, …]] = None)¶

Bases: plasmapy.analysis.fit_functions.AbstractFitFunction

A sub-class of AbstractFitFunction to represent an exponential with an offset.

\[ \begin{align}\begin{aligned}y &= f(x) = a \, e^{\alpha \, x}\\\left( \frac{\delta y}{|y|} \right)^2 &= \left( \frac{\delta a}{a} \right)^2 + (x \, \delta \alpha)^2 + (\alpha \, \delta x)^2\end{aligned}\end{align} \]

where \(a\) and \(\alpha\) are the real constants to be fitted and \(x\) is the independent variable. \(\delta a\), \(\delta \alpha\), and \(\delta x\) are the respective uncertainties for \(a\), \(\alpha\), and \(x\).

Parameters

params (Tuple[float, ..], optional) – Tuple of values for the function parameters. Equal in size to param_names.
param_errors (Tuple[float, ..], optional) – Tuple of values for the errors associated with the function parameters. Equal in size to param_names.

Attributes Summary

`curve_fit_results`	The results returned by the curve fitting routine used by `curve_fit`.
`latex_str`	LaTeX friendly representation of the fit function.
`param_errors`	The associated errors of the fitted `params`.
`param_names`	Names of the fitted parameters.
`params`	The fitted parameters for the fit function.
`rsq`	Coefficient of determination (r-squared) value of the fit.

Methods Summary

`__call__`(x[, x_err, reterr])	Direct call of the fit function \(f(x)\).
`curve_fit`(xdata, ydata, **kwargs)	Use a non-linear least squares method to fit the fit function to (`xdata`, `ydata`), using `scipy.optimize.curve_fit`.
`func`(x, a, alpha)	The fit function, a exponential function.
`func_err`(x[, x_err, rety])	Calculate dependent variable uncertainties \(\delta y\) for dependent variables \(y=f(x)\).
`root_solve`(args, *kwargs)	The root \(f(x_r) = 0\) for the fit function.

Attributes Documentation

curve_fit_results¶: The results returned by the curve fitting routine used by curve_fit. This is typically from scipy.stats.linregress or scipy.optimize.curve_fit.

latex_str¶

param_errors¶: The associated errors of the fitted params.

param_names¶: Names of the fitted parameters.

params¶: The fitted parameters for the fit function.

rsq¶

Coefficient of determination (r-squared) value of the fit.

\[ \begin{align}\begin{aligned}r^2 &= 1 - \frac{SS_{res}}{SS_{tot}}\\SS_{res} &= \sum\limits_{i} (y_i - f(x_i))^2\\SS_{tot} &= \sum\limits_{i} (y_i - \bar{y})^2\end{aligned}\end{align} \]

where \((x_i, y_i)\) are the sample data pairs, \(f(x_i)\) is the fitted dependent variable corresponding to \(x_i\), and \(\bar{y}\) is the average of the \(y_i\) values.

The \(r^2\) value is an indicator of how close the points \((x_i, y_i)\) lie to the model \(f(x)\). \(r^2\) values range between 0 and 1. Values close to 0 indicate that the points are uncorrelated and have little tendency to lie close to the model, whereas, values close to 1 indicate a high correlation to the model.

Methods Documentation

__call__(x, x_err=None, reterr=False)¶

Direct call of the fit function \(f(x)\).

Parameters

x (array_like) – Dependent variables.
x_err (array_like, optional) – Errors associated with the independent variables x. Must be of size one or equal to the size of x.
reterr (bool, optional) – (Default: False) If True, return an array of uncertainties associated with the calculated independent variables

Returns

y (numpy.ndarray) – Corresponding dependent variables \(y=f(x)\) of the independent variables x.
y_err (numpy.ndarray) – Uncertainties associated with the calculated dependent variables \(\delta y\)

curve_fit(xdata, ydata, **kwargs) → None ¶

Use a non-linear least squares method to fit the fit function to (xdata, ydata), using scipy.optimize.curve_fit. This will set the attributes parameters, parameters_err, and rsq.

The results of scipy.optimize.curve_fit can be obtained via curve_fit_results.

Parameters

xdata (array_like) – The independent variable where data is measured. Should be 1D of length M.
ydata (array_like) – The dependent data associated with xdata.
**kwargs – Any keywords accepted by scipy.optimize.curve_fit.

Raises

ValueError – if either ydata or xdata contain numpy.nan’s, or if incompatible options are used.
RuntimeError – if the least-squares minimization fails.
OptimizeWarning – if covariance of the parameters can not be estimated.

func(x, a, alpha)¶

The fit function, a exponential function.

\[f(x) = a \, e^{\alpha \, x}\]

where \(a\) and \(\alpha\) are real constants and \(x\) is the independent variable.

Parameters

x (array_like) – Independent variable.
a (float) – value for the exponential “normalization” constant, \(a\)
alpha (float) – value for the growth constant, \(\alpha\)

Returns

y – dependent variables corresponding to x

Return type

array_like

func_err(x, x_err=None, rety=False)¶

Calculate dependent variable uncertainties \(\delta y\) for dependent variables \(y=f(x)\).

\[\left( \frac{\delta y}{|y|} \right)^2 = \left( \frac{\delta a}{a} \right)^2 + (x \, \delta \alpha)^2 + (\alpha \, \delta x)^2\]

Parameters

x (array_like) – Independent variables to be passed to the fit function.
x_err (array_like, optional) – Errors associated with the independent variables x. Must be of size one or equal to the size of x.
rety (bool) – Set True to also return the associated dependent variables \(y = f(x)\).

Returns

err (numpy.ndarray) – The calculated uncertainties \(\delta y\) of the dependent variables (\(y = f(x)\)) of the independent variables x.
y (numpy.ndarray, optional) – (if rety == True) The associated dependent variables \(y = f(x)\).

Notes

A good reference for formulating propagation of uncertainty expressions is:

J. R. Taylor. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, second edition, August 1996 (ISBN: 093570275X)

root_solve(*args, **kwargs)¶

The root \(f(x_r) = 0\) for the fit function. An exponential has no real roots.

Parameters

*args – Not needed. This is to ensure signature compatibility with AbstractFitFunction.
*kwargs – Not needed. This is to ensure signature compatibility with AbstractFitFunction.

Returns

root (float) – The root value for the given fit parameters.
err (float) – The uncertainty in the calculated root for the given fit parameters and parameters_err.