amep.functions.SkewGaussian#

class amep.functions.SkewGaussian#

Bases: BaseFunction

Skewed Gaussian distribution.

__init__() → None#

Initializes a function object of a skewed normal distribution.

Notes

The skewed normal distribution has three parameters \(\mu\) (location), \(\sigma\) (width), and \(\alpha\) (skewness). It is given by [1]

\[g_{\rm skeq}(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left\lbrace-\frac{(x-\mu)^2}{2\sigma^2}\right\rbrace \left[1 + {\rm erf}\left(\frac{\alpha (x-\mu)}{\sqrt{2}\sigma}\right)\right].\]

The parameters are ordered as follows:

p[0] : \(\mu\)

p[1] : \(\sigma\)

p[2] : \(\alpha\)

References

Return type:: None

Examples

>>> import amep
>>> import numpy as np
>>> g = amep.functions.SkewGaussian()
>>> x = np.linspace(-10, 10, 500)
>>> y = g.generate(
...     x, p=[-1.0, 2.0, 4.0]
... ) + 0.01*np.random.normal(size=x.shape)
>>> g.fit(x, y)
>>> print(g.results)
{'mu': (-0.997718763927072, 0.009639684113005365),
 'sigma': (1.9920368667592616, 0.016343709582216908),
 'alpha': (3.8547382025213865, 0.12910248961985227)}
>>> print(g.mean)
0.5407700079557437
>>> print(g.std)
1.2654102802326845

>>> fig, axs = amep.plot.new()
>>> axs.plot(x, y, label='raw data', ls="")
>>> axs.plot(
...     x, g.generate(x), marker="", lw=2,
...     label='skewed Gaussian fit'
... )
>>> axs.set_xlabel(r'$x$')
>>> axs.set_ylabel(r'$g(x)$')
>>> axs.legend()
>>> fig.savefig('./figures/functions/functions-SkewGaussian.png')

Methods

`__init__`()	Initializes a function object of a skewed normal distribution.
`f`(p, x)	Skewed Gaussian function of the form
`fit`(xdata, ydata[, p0, sigma, maxit, verbose])	Fits the function self.f to the given data by using ODR (orthogonal distance regression).
`generate`(x[, p])	Returns the y values for given x values.

Attributes

`errors`	Returns the fit errors for each parameter as an array.
`keys`
`mean`
`name`
`nparams`
`output`
`params`	Returns an array of the optimal fit parameters.
`results`	Returns the dictionary of fit results including parameter names, parameter values, and fit errors.
`std`

property errors: ndarray#

Returns the fit errors for each parameter as an array.

Returns:: Fit errors.
Return type:: np.ndarray

f(p: list | ndarray, x: float | ndarray) → float | ndarray#

Skewed Gaussian function of the form

\[g_{\rm skeq}(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left\lbrace-\frac{(x-\mu)^2}{2\sigma^2}\right\rbrace \left[1 + {\rm erf}\left(\frac{\alpha (x-\mu)}{\sqrt{2}\sigma}\right)\right].\]

Parameters:

p (list | np.ndarray) – Parameters \(\mu, \sigma\), and \(\alpha\) of the skewed Gaussian function.
x (float | np.ndarray) – Value(s) at which the function is evaluated.

Returns:

Function evaluated at the given x value(s).

Return type:

float or np.ndarray

fit(xdata: ndarray, ydata: ndarray, p0: list | None = None, sigma: ndarray | None = None, maxit: int | None = None, verbose: bool = False) → None#

Fits the function self.f to the given data by using ODR (orthogonal distance regression).

Parameters:

xdata (np.ndarray) – x values.
ydata (np.ndarray) – y values.
p0 (list or None, optional) – List of initial values. The default is None.
sigma (np.ndarray or None, optional) – Absolute error for each data point. The default is None.
maxit (int, optional) – Maximum number of iterations. The default is 200.
verbose (bool, optional) – If True, the main results are printed. The default is False.

Return type:

None.

generate(x: ndarray, p: list | None = None) → ndarray#

Returns the y values for given x values.

Parameters:

x (np.ndarray) – x values.
p (list, optional) – List of parameters. The default is None.

Returns:

y – f(x)

Return type:

np.ndarray

property params: ndarray#

Returns an array of the optimal fit parameters.

Returns:: Fit parameter values.
Return type:: np.ndarray

property results: dict#

Returns the dictionary of fit results including parameter names, parameter values, and fit errors.

Returns:: Fit results.
Return type:: dict