pystra.distributions.normal.Normal#

class Normal(name, mean, stdv, input_type=None, startpoint=None)[source]#

Bases: Distribution

Normal distribution

Attributes:

name (str): Name of the random variable
mean (float): Mean
stdv (float): Standard deviation
input_type (any): Change meaning of mean and stdv
startpoint (float): Start point for seach

Note: while we could use SciPy norm distribution here, there is a substantial perfromance hit, so use local implementation.

Leave initialization to the base class

Methods

`cdf`	cumulative distribution function
`dF_dtheta`	Analytical derivatives of the Normal CDF w.r.t.
`getMean`
`getName`
`getStartPoint`
`getStdv`
`jacobian`	Compute the Jacobian (e.g. Lemaire, eq.
`pdf`	probability density function
`plot`	Plot the probability density function.
`ppf`	inverse cumulative distribution function
`sample`	Override sample from base class due to bespoke implementation
`setStartPoint`
`set_location`	Updating the distribution location parameter.
`set_scale`	Updating the distribution scale parameter.
`u_to_x`	Transformation from u to x
`x_to_u`	Transformation from x to u

pdf(x)[source]#: probability density function

cdf(x)[source]#: cumulative distribution function

ppf(p)[source]#: inverse cumulative distribution function

sample(n=1000)[source]#: Override sample from base class due to bespoke implementation

u_to_x(u)[source]#: Transformation from u to x

x_to_u(x)[source]#: Transformation from x to u

jacobian(u, x)[source]#: Compute the Jacobian (e.g. Lemaire, eq. 4.9) For the Normal distribution, the more usual general function can be specialized as follows.

dF_dtheta(x)[source]#: Analytical derivatives of the Normal CDF w.r.t. μ and σ.

\[ \begin{align}\begin{aligned}\frac{\partial F}{\partial \mu} = -\frac{1}{\sigma}\, \varphi\!\left(\frac{x - \mu}{\sigma}\right)\\\frac{\partial F}{\partial \sigma} = -\frac{x - \mu}{\sigma^2}\, \varphi\!\left(\frac{x - \mu}{\sigma}\right)\end{aligned}\end{align} \]

set_location(loc=0)[source]#: Updating the distribution location parameter. For Normal, there is no need to update other properties as a result of this change.

set_scale(scale=1)[source]#: Updating the distribution scale parameter. For Normal, there is no need to update other properties as a result of this change.

plot(ax=None, **kwargs)#

Plot the probability density function.

Parameters:

ax (matplotlib.axes.Axes, optional) – Axes to plot on. A new figure is created if None.
**kwargs – Additional keyword arguments forwarded to ax.plot().

Returns:

The axes containing the plot.

Return type:

matplotlib.axes.Axes

property sensitivity_params#

Distribution parameters for which sensitivities are computed.

Returns a dict {param_name: current_value} listing every parameter with respect to which \(\partial\beta/\partial\theta\) should be evaluated.

The default implementation returns {"mean": μ, "std": σ}, which is appropriate for most distributions. Distributions with additional parameters of interest (e.g. the GEV shape parameter) should override this property to include them.

Parameters listed in _ctor_kwargs but not in sensitivity_params are held fixed during sensitivity analysis — they are only used by _make_copy() to faithfully reconstruct the distribution.

Returns:: {param_name: current_value}
Return type:: dict