chaospy.Cauchy

class chaospy.Cauchy(scale=1, shift=0)

Cauchy distribution.

Also known as Lorentz distribution, Cachy-Lorentz distribution, and Breit-Wigner distribution.

Args:

shift (float, Distribution):

Location parameter

scale (float, Distribution):

Scaling parameter

Examples:

>>> distribution = chaospy.Cauchy()
>>> distribution
Cauchy()
>>> uloc = numpy.linspace(0.1, 0.9, 5)
>>> uloc
array([0.1, 0.3, 0.5, 0.7, 0.9])
>>> xloc = distribution.inv(uloc)
>>> xloc.round(3)
array([-3.078, -0.727,  0.   ,  0.727,  3.078])
>>> numpy.allclose(distribution.fwd(xloc), uloc)
True
>>> distribution.pdf(xloc).round(3)
array([0.03 , 0.208, 0.318, 0.208, 0.03 ])
>>> distribution.sample(4).round(3)
array([ 0.524, -2.646,  6.35 , -0.056])

Notes:

The Cauchy distribution is what is known as a “pathological” distribution. It is not only infinitely bound, but heavy tailed enough that approximate bounds is also infinite for any reasonable approximation. This makes both bounds and moments results in non-sensibel results. E.g.:

>>> distribution.lower < -1e10
array([ True])
>>> distribution.upper > 1e10
array([ True])

__init__(scale=1, shift=0)

Distribution initializer.

In addition to assigning some object variables, also checks for some consistency issues.

Args:

parameters (Optional[Distribution[str, Union[ndarray, Distribution]]]):

Collection of model parameters.

dependencies (Optional[Sequence[Set[int]]]):

Dependency identifiers. One collection for each dimension.

rotation (Optional[Sequence[int]]):

The order of which to resolve dependencies.

exclusion (Optional[Sequence[int]]):

Distributions that has been “taken out of play” and therefore can not be reused other places in the dependency hierarchy.

repr_args (Optional[Sequence[str]]):

Positional arguments to place in the object string representation. The repr output will then be: <class name>(<arg1>, <arg2>, …).

Raises:

StochasticallyDependentError:

For dependency structures that can not later be rectified. This include under-defined distributions, and inclusion of distributions that should be exclusion.

Methods

pdf(x_data[, decompose, allow_approx, step_size])	Probability density function.
cdf(x_data)	Cumulative distribution function.
fwd(x_data)	Forward Rosenblatt transformation.
inv(q_data[, max_iterations, tollerance])	Inverse Rosenblatt transformation.
sample([size, rule, antithetic, ...])	Create pseudo-random generated samples.
mom(K[, allow_approx])	Raw statistical moments.
ttr(kloc)	Three terms relation's coefficient generator.

Attributes

interpret_as_integer	Flag indicating that return value from the methods sample, and inv should be interpreted as integers instead of floating point.
lower	Lower bound for the distribution.
stochastic_dependent	True if distribution contains stochastically dependent components.
upper	Upper bound for the distribution.