Distributions

class quantflow.utils.distributions.Distribution1D

Base class for 1D distributions to be used as Jump distributions in Compound Poisson processes

Methods:

asymmetry	Asymmetry of the distribution, 0 for symmetric
from_variance_and_asymmetry	Create a distribution from variance and asymmetry
sample	Sample from the distribution
set_asymmetry	Set the asymmetry of the distribution
set_variance	Set the variance of the distribution

Attributes:

model_config

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

asymmetry() → float

Asymmetry of the distribution, 0 for symmetric

Implemented by distributions that have asymmetry

classmethod from_variance_and_asymmetry(variance: float, asymmetry: float) → Self

Create a distribution from variance and asymmetry

abstract sample(n: int) → ndarray

Sample from the distribution

set_asymmetry(asymmetry: float) → None

Set the asymmetry of the distribution

Implemented by distributions that have asymmetry

set_variance(variance: float) → None

Set the variance of the distribution

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

class quantflow.utils.distributions.Exponential(*, decay: Annotated[float, Gt(gt=0)] = 1)

A Distribution1D for the Exponential distribution

The exponential distribution is a continuous probability distribution with PDF given by

\[f(x) = \lambda e^{-\lambda x}\ \ \forall x \geq 0\]

Methods:

cdf	The analytical CDF of the exponential distribution
characteristic	Compute the characteristic function on support points n.
mean	Expected value
pdf	The analytical PDF of the exponential distribution as defined above
sample	Sample from the distribution
support	Compute the x axis.
variance	Variance

Attributes:

decay	The exponential decay rate \(\lambda\)
model_config	Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].
scale	The scale parameter, it is the inverse of the decay rate

cdf(x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

The analytical CDF of the exponential distribution

\[F(x) = 1 - e^{-\lambda x}\ \ \forall x \geq 0\]

characteristic(u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Compute the characteristic function on support points n.

mean() → float

Expected value

This should be overloaded if a more efficient way of computing the mean

pdf(x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

The analytical PDF of the exponential distribution as defined above

sample(n: int) → ndarray

Sample from the distribution

support(points: int = 100, *, std_mult: float = 4) → ndarray[tuple[int, ...], dtype[floating[Any]]]

Compute the x axis.

variance() → float

Variance

This should be overloaded if a more efficient way of computing the

decay: float

The exponential decay rate \(\lambda\)

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

property scale: float

The scale parameter, it is the inverse of the decay rate

class quantflow.utils.distributions.DoubleExponential(*, decay: Annotated[float, Gt(gt=0)] = 1, loc: float = 0, kappa: Annotated[float, Gt(gt=0)] = 1)

A Marginal1D for the generalized double exponential distribution

This is also know as the Asymmetric Laplace distribution (ALD) which is a continuous probability distribution with PDF

\[\begin{split}\begin{align} f(x) &= \frac{\lambda}{\kappa + \frac{1}{\kappa}} e^{-\left(x - m\right) \lambda s(x) \kappa^{s(x)}}\\ s(x) &= {\tt sgn}\left({x - m}\right) \end{align}\end{split}\]

where m is the loc parameter, \(\lambda\) is the decay parameter, and \(\kappa\) is the asymmetric kappa parameter.

The Asymmetric Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak, it has fatter tails and allow for skewness. It represents the difference between two independent, exponential random variables.

Methods:

asymmetry	Asymmetry of the distribution
characteristic	Characteristic function of the double exponential distribution
from_moments	Create a double exponential distribution from the mean, variance and asymmetry
from_variance_and_asymmetry	Create a distribution from variance and asymmetry
mean	The mean of the double exponential distribution
pdf	The analytical PDF as defined above
sample	Sample from the double exponential distribution
set_asymmetry	Set the asymmetry of the distribution
set_variance	Set the variance of the distribution
support	Compute the x axis.
variance	Variance

Attributes:

decay	The exponential decay rate \(\lambda\)
kappa	Asymmetric parameter - when k=1, the distribution is symmetric
loc	The location parameter m
log_kappa	The log of the kappa parameter
model_config	Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

asymmetry() → float

Asymmetry of the distribution

characteristic(u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Characteristic function of the double exponential distribution

\[\phi(u) = \frac{e^{i u m}}{\left(1 + \frac{i u \kappa}{\lambda}\right) \left(1 - \frac{i u}{\lambda \kappa}\right)}\]

classmethod from_moments(*, mean: float = 0, variance: float = 1, kappa: float = 1) → Self

Create a double exponential distribution from the mean, variance and asymmetry

Parameters:

• mean – The mean of the distribution

• variance – The variance of the distribution

• kappa – The asymmetry parameter of the distribution, 1 for symmetric

classmethod from_variance_and_asymmetry(variance: float, asymmetry: float) → Self

Create a distribution from variance and asymmetry

mean() → float

The mean of the double exponential distribution

pdf(x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

The analytical PDF as defined above

sample(n: int) → ndarray

Sample from the double exponential distribution

set_asymmetry(asymmetry: float) → None

Set the asymmetry of the distribution

Implemented by distributions that have asymmetry

set_variance(variance: float) → None

Set the variance of the distribution

support(points: int = 100, *, std_mult: float = 4) → ndarray[tuple[int, ...], dtype[floating[Any]]]

Compute the x axis.

variance() → float

Variance

This should be overloaded if a more efficient way of computing the

decay: float

The exponential decay rate \(\lambda\)

kappa: float

Asymmetric parameter - when k=1, the distribution is symmetric

loc: float

The location parameter m

property log_kappa: float

The log of the kappa parameter

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

class quantflow.utils.distributions.Normal(*, mu: float = 0, sigma: Annotated[float, Gt(gt=0)] = 1)

A Distribution1D for the Normal distribution

The normal distribution is a continuous probability distribution with PDF given by

\[f(x) = \frac{e^{-\frac{\left(x - \mu\right)^2}{2\sigma^2}}}{\sqrt{2\pi\sigma^2}}\]

Methods:

characteristic	Compute the characteristic function on support points n.
from_variance_and_asymmetry	The normal distribution is symmetric, so the asymmetry is ignored
mean	Expected value
sample	Sample from the distribution
set_variance	Set the variance of the distribution
support	Compute the x axis.
variance	Variance

Attributes:

model_config	Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].
mu	The mean \(\mu\) of the normal distribution
sigma	The standard deviation \(\sigma\) of the normal distribution

characteristic(u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Compute the characteristic function on support points n.

classmethod from_variance_and_asymmetry(variance: float, asymmetry: float) → Self

The normal distribution is symmetric, so the asymmetry is ignored

mean() → float

Expected value

This should be overloaded if a more efficient way of computing the mean

sample(n: int) → ndarray

Sample from the distribution

set_variance(variance: float) → None

Set the variance of the distribution

support(points: int = 100, *, std_mult: float = 4) → ndarray[tuple[int, ...], dtype[floating[Any]]]

Compute the x axis.

variance() → float

Variance

This should be overloaded if a more efficient way of computing the

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

mu: float

The mean \(\mu\) of the normal distribution

sigma: float

The standard deviation \(\sigma\) of the normal distribution