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2𝛑 Interval

Von Mises Distribution

von Mises distribution.

The von Mises distribution is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution.

Examples:

Python Console Session
>>> # PDF Example
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from umf.functions.distributions.continuous_2pi_interval import (
... VonMisesDistribution
... )
>>> x = np.linspace(-np.pi, np.pi, 1000)
>>> y_05 = VonMisesDistribution(x, mu=0, kappa=0.5).__eval__
>>> y_07 = VonMisesDistribution(x, mu=0, kappa=0.7).__eval__
>>> y_09 = VonMisesDistribution(x, mu=0, kappa=0.9).__eval__
>>> fig = plt.figure()
>>> ax = plt.subplot()
>>> _  = ax.plot(x, y_05, label=r"$\kappa=0.5$")
>>> _  = ax.plot(x, y_07, label=r"$\kappa=0.7$")
>>> _  = ax.plot(x, y_09, label=r"$\kappa=0.9$")
>>> _  = ax.set_xlabel("x")
>>> _  = ax.set_ylabel("f(x)")
>>> _  = ax.legend()
>>> plt.savefig("VonMisesDistribution.png", dpi=300, transparent=True)
Notes

The von Mises distribution is defined as follows for probability density:

f(x;μ,κ)=eκcos⁡(x−μ)2πI0(κ)

where x∈[−π,π] and μ∈[−π,π]. The parameter κ∈[0,∞) controls the concentration of the distribution around μ. The function I0 is the modified Bessel function of the first kind and order zero.

Parameters:

Name Type Description Default
*x UniversalArray

The points at which to evaluate the distribution.

 ()
mu float

The mean of the distribution. Defaults to 0.

 0
kappa float

The concentration of the distribution around mu. Defaults to 1.

 1
Source code in umf/functions/distributions/continuous_2pi_interval.py
Python
class VonMisesDistribution(Continuous2PiInterval):
    r"""von Mises distribution.

    The von Mises distribution is a continuous probability distribution on the
    circle. It is a close approximation to the wrapped normal distribution,
    which is the circular analogue of the normal distribution.

    Examples:
        >>> # PDF Example
        >>> import matplotlib.pyplot as plt
        >>> import numpy as np
        >>> from umf.functions.distributions.continuous_2pi_interval import (
        ... VonMisesDistribution
        ... )
        >>> x = np.linspace(-np.pi, np.pi, 1000)
        >>> y_05 = VonMisesDistribution(x, mu=0, kappa=0.5).__eval__
        >>> y_07 = VonMisesDistribution(x, mu=0, kappa=0.7).__eval__
        >>> y_09 = VonMisesDistribution(x, mu=0, kappa=0.9).__eval__
        >>> fig = plt.figure()
        >>> ax = plt.subplot()
        >>> _  = ax.plot(x, y_05, label=r"$\kappa=0.5$")
        >>> _  = ax.plot(x, y_07, label=r"$\kappa=0.7$")
        >>> _  = ax.plot(x, y_09, label=r"$\kappa=0.9$")
        >>> _  = ax.set_xlabel("x")
        >>> _  = ax.set_ylabel("f(x)")
        >>> _  = ax.legend()
        >>> plt.savefig("VonMisesDistribution.png", dpi=300, transparent=True)


    Notes:
        The von Mises distribution is defined as follows for probability density:

        $$
        f(x;\mu,\kappa) = \frac{e^{\kappa \cos(x-\mu)}}{2\pi I_0(\kappa)}
        $$

        where $x \in [-\pi, \pi]$ and $\mu \in [-\pi, \pi]$. The parameter
        $\kappa \in [0, \infty)$ controls the concentration of the distribution
        around $\mu$. The function $I_0$ is the modified Bessel function of the
        first kind and order zero.


    Args:
        *x (UniversalArray): The points at which to evaluate the distribution.
        mu (float): The mean of the distribution. Defaults to 0.
        kappa (float): The concentration of the distribution around mu.
            Defaults to 1.
    """

    def probability_density_function(self) -> UniversalArray:
        """Probability density function of the von Mises distribution."""
        return np.exp(self.kappa * np.cos(self._x - self.mu)) / (
            2 * np.pi * special.i0(self.kappa)
        )

    @property
    def __summary__(self) -> SummaryStatisticsAPI:
        """Summary statistics of the von Mises distribution."""

        def _mode() -> float | tuple[float, float]:
            """Mode of the von Mises distribution."""
            return self.mu

        return SummaryStatisticsAPI(
            mean=self.mu,
            variance=1 - special.i1(self.kappa) / special.i0(self.kappa),
            mode=_mode(),
            doc=self.__doc__,
        )
Probability Density Function
VonMisesDistribution

Wrapped Asymmetric Laplace Distribution

Wrapped asymmetric Laplace distribution.

The wrapped (asymmetric) Laplace distribution is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution.

Examples:

Python Console Session
>>> # PDF Example
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from umf.functions.distributions.continuous_2pi_interval import (
... WrappedAsymLaplaceDistribution
... )
>>> x = np.linspace(-np.pi, np.pi, 1000)
>>> y_05 = WrappedAsymLaplaceDistribution(
... x,
... mu=0,
... lambda_=0.5,
... kappa=0.5,
... ).__eval__
>>> y_07 = WrappedAsymLaplaceDistribution(
... x,
... mu=0,
... lambda_=0.7,
... kappa=0.7,
... ).__eval__
>>> y_09 = WrappedAsymLaplaceDistribution(
... x,
... mu=0,
... lambda_=0.9,
... kappa=0.9,
... ).__eval__
>>> fig = plt.figure()
>>> ax = plt.subplot()
>>> _  = ax.plot(x, y_05, label=r"$\lambda=0.5$")
>>> _  = ax.plot(x, y_07, label=r"$\lambda=0.7$")
>>> _  = ax.plot(x, y_09, label=r"$\lambda=0.9$")
>>> _  = ax.set_xlabel("x")
>>> _  = ax.set_ylabel("f(x)")
>>> _  = ax.legend()
>>> plt.savefig("WrappedAsymLaplaceDistribution.png", dpi=300, transparent=True)
Notes

The wrapped Laplace distribution is defined as follows for probability density:

fWAL(θ;m,λ,κ)=∑k=−∞∞fAL(θ+2πk,m,λ,κ)=κλκ2+1{e−(θ−m)λκ1−e−2πλκ−e(θ−m)λ/κ1−e2πλ/κif θ≥me−(θ−m)λκe2πλκ−1−e(θ−m)λ/κe−2πλ/κ−1if θ<m

where x∈[−π,π] and μ∈[−π,π]. The parameter κ∈[0,∞) controls the concentration of the distribution around μ. The function I0 is the modified Bessel function of the first kind and order zero.1

  1. Wrapped asymmetric Laplace distribution. (2022, January 24). In Wikipedia. en.wikipedia.org/wiki/Wrapped_asymmetric_Laplace_distribution ↩

Parameters:

Name Type Description Default
x UniversalArray

The points at which to evaluate the distribution.

 ()
mu float

The mean of the distribution.

 0
lambda_ float

The scale parameter of the distribution.

 1
kappa float

The concentration of the distribution around mu.

 1
Source code in umf/functions/distributions/continuous_2pi_interval.py
Python
class WrappedAsymLaplaceDistribution(Continuous2PiInterval):
    r"""Wrapped asymmetric Laplace distribution.

    The wrapped (asymmetric) Laplace distribution is a continuous probability
    distribution on the circle. It is a close approximation to the wrapped normal
    distribution, which is the circular analogue of the normal distribution.

    Examples:
        >>> # PDF Example
        >>> import matplotlib.pyplot as plt
        >>> import numpy as np
        >>> from umf.functions.distributions.continuous_2pi_interval import (
        ... WrappedAsymLaplaceDistribution
        ... )
        >>> x = np.linspace(-np.pi, np.pi, 1000)
        >>> y_05 = WrappedAsymLaplaceDistribution(
        ... x,
        ... mu=0,
        ... lambda_=0.5,
        ... kappa=0.5,
        ... ).__eval__
        >>> y_07 = WrappedAsymLaplaceDistribution(
        ... x,
        ... mu=0,
        ... lambda_=0.7,
        ... kappa=0.7,
        ... ).__eval__
        >>> y_09 = WrappedAsymLaplaceDistribution(
        ... x,
        ... mu=0,
        ... lambda_=0.9,
        ... kappa=0.9,
        ... ).__eval__
        >>> fig = plt.figure()
        >>> ax = plt.subplot()
        >>> _  = ax.plot(x, y_05, label=r"$\lambda=0.5$")
        >>> _  = ax.plot(x, y_07, label=r"$\lambda=0.7$")
        >>> _  = ax.plot(x, y_09, label=r"$\lambda=0.9$")
        >>> _  = ax.set_xlabel("x")
        >>> _  = ax.set_ylabel("f(x)")
        >>> _  = ax.legend()
        >>> plt.savefig("WrappedAsymLaplaceDistribution.png", dpi=300, transparent=True)

    Notes:
        The wrapped Laplace distribution is defined as follows for probability
        density:

        $$
        \begin{aligned}f_{WAL}(\theta ;m,\lambda ,\kappa )&=
        \sum _{k=-\infty }^{\infty }f_{AL}(\theta +2\pi k,m,\lambda ,\kappa )\\[10pt]
        &={\dfrac {\kappa \lambda }{\kappa ^{2}+1}}{\begin{cases}
        {\dfrac {e^{-(\theta -m)\lambda \kappa }}
        {1-e^{-2\pi \lambda \kappa }}}-{\dfrac {e^{(\theta -m)\lambda /\kappa }}
        {1-e^{2\pi \lambda /\kappa }}}&{\text{if }}
        \theta \geq m\\[12pt]{\dfrac {e^{-(\theta -m)\lambda \kappa }}
        {e^{2\pi \lambda \kappa }-1}}-{\dfrac {e^{(\theta -m)\lambda /\kappa }}
        {e^{-2\pi \lambda /\kappa }-1}}&{\text{if }}\theta <m\end{cases}}\end{aligned}
        $$

        where $x \in [-\pi, \pi]$ and $\mu \in [-\pi, \pi]$. The parameter
        $\kappa \in [0, \infty)$ controls the concentration of the distribution
        around $\mu$. The function $I_0$ is the modified Bessel function of the
        first kind and order zero.[^1]

        [^1]: Wrapped asymmetric Laplace distribution. (2022, January 24).
            _In Wikipedia._
            https://en.wikipedia.org/wiki/Wrapped_asymmetric_Laplace_distribution

    Args:
        x: The points at which to evaluate the distribution.
        mu: The mean of the distribution.
        lambda_: The scale parameter of the distribution.
        kappa: The concentration of the distribution around mu.
    """

    def __init__(
        self,
        *x: UniversalArray,
        mu: float = 0,
        lambda_: float = 1,
        kappa: float = 1,
    ) -> None:
        r"""Initialize a wrapped Laplace distribution.

        Args:
            *x (UniversalArray): The points at which to evaluate the distribution.
            mu (float): The mean of the distribution. Defaults to 0.
            lambda_ (float): The scale parameter of the distribution. Defaults to 1.
            kappa (float): The concentration of the distribution around mu.
                Defaults to 1.
        """
        super().__init__(*x, mu=mu, kappa=kappa)
        self.lambda_ = lambda_

    def probability_density_function(self) -> UniversalArray:
        """Probability density function of the wrapped Laplace distribution."""
        part_1 = (
            self.kappa
            * self.lambda_
            / (self.kappa**2 + 1)
            * (
                np.exp(-(self._x - self.mu) * self.lambda_ * self.kappa)
                / (1 - np.exp(-2 * np.pi * self.lambda_ * self.kappa))
                - np.exp((self._x - self.mu) * self.lambda_ / self.kappa)
                / (1 - np.exp(2 * np.pi * self.lambda_ / self.kappa))
            )
        )
        part_2 = (
            self.kappa
            * self.lambda_
            / (self.kappa**2 + 1)
            * (
                np.exp(-(self._x - self.mu) * self.lambda_ * self.kappa)
                / (np.exp(2 * np.pi * self.lambda_ * self.kappa) - 1)
                - np.exp((self._x - self.mu) * self.lambda_ / self.kappa)
                / (np.exp(-2 * np.pi * self.lambda_ / self.kappa) - 1)
            )
        )

        # Combine the two parts
        return np.where(self._x >= self.mu, part_1, part_2)

    @property
    def __summary__(self) -> SummaryStatisticsAPI:
        """Summary statistics of the wrapped Laplace distribution."""
        return SummaryStatisticsAPI(
            mean=self.mu,
            variance=1
            - self.lambda_**2
            / np.sqrt(
                (1 / self.kappa**2 + self.lambda_**2)
                * (self.kappa**2 + self.lambda_**2),
            ),
            mode=None,
            doc=self.__doc__,
        )
Probability Density Function
WrappedAsymmetricLaplaceDistribution