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Circular Distributions

circularuniform_gen

Bases: rv_continuous

Continuous Circular Uniform Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Source code in pycircstat2/distributions.py 
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class circularuniform_gen(rv_continuous):
    """Continuous Circular Uniform Distribution

    Methods
    -------
    pdf(x)
        Probability density function.

    cdf(x)
        Cumulative distribution function.
    """

    def _pdf(self, x):
        return 1 / np.pi

    def pdf(self, x, *args, **kwargs):
        r"""
        Probability density function of the Circular Uniform distribution.

        $$
        f(\theta) = \frac{1}{\pi}
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, *args, **kwargs)

    def _cdf(self, x):
        return x / (2 * np.pi)

    def cdf(self, x, *args, **kwargs):
        r"""
        Cumulative distribution function of the Circular Uniform distribution.

        $$
        F(\theta) = \frac{\theta}{2\pi}
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the cumulative distribution function.

        Returns
        -------
        cdf_values : array_like
            Cumulative distribution function evaluated at `x`.
        """
        return super().cdf(x, *args, **kwargs)

    def _ppf(self, q):
        return 2 * np.pi * q

    def ppf(self, q, *args, **kwargs):
        r"""
        Percent-point function (inverse of the CDF) of the Circular Uniform distribution.

        $$
        Q(q) = F^{-1}(q) = 2\pi q, \space 0 \leq q \leq 1
        $$

        Parameters
        ----------
        q : array_like
            Quantiles to evaluate.

        Returns
        -------
        ppf_values : array_like
            Values at the given quantiles.
        """
        return super().ppf(q, *args, **kwargs)

pdf(x, *args, **kwargs)

Probability density function of the Circular Uniform distribution.

\[ f(\theta) = \frac{1}{\pi} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, *args, **kwargs):
    r"""
    Probability density function of the Circular Uniform distribution.

    $$
    f(\theta) = \frac{1}{\pi}
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, *args, **kwargs)

cdf(x, *args, **kwargs)

Cumulative distribution function of the Circular Uniform distribution.

\[ F(\theta) = \frac{\theta}{2\pi} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the cumulative distribution function.

 required

Returns:

Name Type Description
cdf_values array_like

Cumulative distribution function evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, *args, **kwargs):
    r"""
    Cumulative distribution function of the Circular Uniform distribution.

    $$
    F(\theta) = \frac{\theta}{2\pi}
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the cumulative distribution function.

    Returns
    -------
    cdf_values : array_like
        Cumulative distribution function evaluated at `x`.
    """
    return super().cdf(x, *args, **kwargs)

ppf(q, *args, **kwargs)

Percent-point function (inverse of the CDF) of the Circular Uniform distribution.

\[ Q(q) = F^{-1}(q) = 2\pi q, \space 0 \leq q \leq 1 \]

Parameters:

Name Type Description Default
q array_like

Quantiles to evaluate.

 required

Returns:

Name Type Description
ppf_values array_like

Values at the given quantiles.
Source code in pycircstat2/distributions.py 
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def ppf(self, q, *args, **kwargs):
    r"""
    Percent-point function (inverse of the CDF) of the Circular Uniform distribution.

    $$
    Q(q) = F^{-1}(q) = 2\pi q, \space 0 \leq q \leq 1
    $$

    Parameters
    ----------
    q : array_like
        Quantiles to evaluate.

    Returns
    -------
    ppf_values : array_like
        Values at the given quantiles.
    """
    return super().ppf(q, *args, **kwargs)

cardioid_gen

Bases: rv_continuous

Cardioid (cosine) Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Notes

Implementation based on Section 4.3.4 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class cardioid_gen(rv_continuous):
    """Cardioid (cosine) Distribution

    Methods
    -------
    pdf(x, mu, rho)
        Probability density function.

    cdf(x, mu, rho)
        Cumulative distribution function.

    Notes
    -----
    Implementation based on Section 4.3.4 of Pewsey et al. (2014)
    """

    def _argcheck(self, mu, rho):
        return 0 <= mu <= np.pi * 2 and 0 <= rho <= 0.5

    def _pdf(self, x, mu, rho):
        return (1 + 2 * rho * np.cos(x - mu)) / 2.0 / np.pi

    def pdf(self, x, mu, rho, *args, **kwargs):
        r"""
        Probability density function of the Cardioid distribution.

        $$
        f(\theta) = \frac{1}{2\pi} \left(1 + 2\rho \cos(\theta - \mu)\right), \space \rho \in [0, 1/2]
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Mean resultant length, 0 <= rho <= 0.5.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, mu, rho, *args, **kwargs)

    def _cdf(self, x, mu, rho):
        return (x + 2 * rho * (np.sin(x - mu) + np.sin(mu))) / (2 * np.pi)

    def cdf(self, x, mu, rho, *args, **kwargs):
        r"""
        Cumulative distribution function of the Cardioid distribution.

        $$
        F(\theta) = \frac{\theta + 2\rho (\sin(\mu) + \sin(\theta - \mu))}{2\pi}
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the cumulative distribution function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Mean resultant length, 0 <= rho <= 0.5.

        Returns
        -------
        cdf_values : array_like
            Cumulative distribution function evaluated at `x`.
        """
        return super().cdf(x, mu, rho, *args, **kwargs)

pdf(x, mu, rho, *args, **kwargs)

Probability density function of the Cardioid distribution.

\[ f(\theta) = \frac{1}{2\pi} \left(1 + 2\rho \cos(\theta - \mu)\right), \space \rho \in [0, 1/2] \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Mean resultant length, 0 <= rho <= 0.5.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, rho, *args, **kwargs):
    r"""
    Probability density function of the Cardioid distribution.

    $$
    f(\theta) = \frac{1}{2\pi} \left(1 + 2\rho \cos(\theta - \mu)\right), \space \rho \in [0, 1/2]
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Mean resultant length, 0 <= rho <= 0.5.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, mu, rho, *args, **kwargs)

cdf(x, mu, rho, *args, **kwargs)

Cumulative distribution function of the Cardioid distribution.

\[ F(\theta) = \frac{\theta + 2\rho (\sin(\mu) + \sin(\theta - \mu))}{2\pi} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the cumulative distribution function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Mean resultant length, 0 <= rho <= 0.5.

 required

Returns:

Name Type Description
cdf_values array_like

Cumulative distribution function evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, mu, rho, *args, **kwargs):
    r"""
    Cumulative distribution function of the Cardioid distribution.

    $$
    F(\theta) = \frac{\theta + 2\rho (\sin(\mu) + \sin(\theta - \mu))}{2\pi}
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the cumulative distribution function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Mean resultant length, 0 <= rho <= 0.5.

    Returns
    -------
    cdf_values : array_like
        Cumulative distribution function evaluated at `x`.
    """
    return super().cdf(x, mu, rho, *args, **kwargs)

cartwright_gen

Bases: rv_continuous

Cartwright's Power-of-Cosine Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation based on Section 4.3.5 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class cartwright_gen(rv_continuous):
    """Cartwright's Power-of-Cosine Distribution

    Methods
    -------
    pdf(x, mu, zeta)
        Probability density function.

    cdf(x, mu, zeta)
        Cumulative distribution function.

    Note
    ----
    Implementation based on Section 4.3.5 of Pewsey et al. (2014)
    """

    def _argcheck(self, mu, zeta):
        return 0 <= mu <= 2 * np.pi and zeta > 0

    def _pdf(self, x, mu, zeta):
        return (
            (2 ** (-1 + 1 / zeta) * (gamma(1 + 1 / zeta)) ** 2)
            * (1 + np.cos(x - mu)) ** (1 / zeta)
            / (np.pi * gamma(1 + 2 / zeta))
        )

    def pdf(self, x, mu, zeta, *args, **kwargs):
        r"""
        Probability density function of the Cartwright distribution.

        $$
        f(\theta) = \frac{2^{- 1+1/\zeta} \Gamma^2(1 + 1/\zeta)}{\pi \Gamma(1 + 2/\zeta)} (1 + \cos(\theta - \mu))^{1/\zeta}
        $$

        , where $\Gamma$ is the gamma function.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        zeta : float
            Shape parameter, zeta > 0.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """

        return super().pdf(x, mu, zeta, *args, **kwargs)

    def _cdf(self, x, mu, zeta):
        @np.vectorize
        def _cdf_single(x, mu, zeta):
            return quad(self._pdf, a=0, b=x, args=(mu, zeta))

        return _cdf_single(x, mu, zeta)

    def cdf(self, x, mu, zeta, *args, **kwargs):
        r"""
        Cumulative distribution function of the Cartwright distribution.

        No closed-form solution is available, so the CDF is computed numerically.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the cumulative distribution function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        zeta : float
            Shape parameter, zeta > 0.

        Returns
        -------
        cdf_values : array_like
            Cumulative distribution function evaluated at `x`.
        """
        return super().cdf(x, mu, zeta, *args, **kwargs)

pdf(x, mu, zeta, *args, **kwargs)

Probability density function of the Cartwright distribution.

\[ f(\theta) = \frac{2^{- 1+1/\zeta} \Gamma^2(1 + 1/\zeta)}{\pi \Gamma(1 + 2/\zeta)} (1 + \cos(\theta - \mu))^{1/\zeta} \]

, where \(\Gamma\) is the gamma function.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
zeta float

Shape parameter, zeta > 0.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, zeta, *args, **kwargs):
    r"""
    Probability density function of the Cartwright distribution.

    $$
    f(\theta) = \frac{2^{- 1+1/\zeta} \Gamma^2(1 + 1/\zeta)}{\pi \Gamma(1 + 2/\zeta)} (1 + \cos(\theta - \mu))^{1/\zeta}
    $$

    , where $\Gamma$ is the gamma function.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    zeta : float
        Shape parameter, zeta > 0.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """

    return super().pdf(x, mu, zeta, *args, **kwargs)

cdf(x, mu, zeta, *args, **kwargs)

Cumulative distribution function of the Cartwright distribution.

No closed-form solution is available, so the CDF is computed numerically.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the cumulative distribution function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
zeta float

Shape parameter, zeta > 0.

 required

Returns:

Name Type Description
cdf_values array_like

Cumulative distribution function evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, mu, zeta, *args, **kwargs):
    r"""
    Cumulative distribution function of the Cartwright distribution.

    No closed-form solution is available, so the CDF is computed numerically.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the cumulative distribution function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    zeta : float
        Shape parameter, zeta > 0.

    Returns
    -------
    cdf_values : array_like
        Cumulative distribution function evaluated at `x`.
    """
    return super().cdf(x, mu, zeta, *args, **kwargs)

wrapnorm_gen

Bases: rv_continuous

Wrapped Normal Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.

Examples:

from pycircstat2.distributions import wrapnorm
Notes

Implementation based on Section 4.3.7 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class wrapnorm_gen(rv_continuous):
    """Wrapped Normal Distribution

    Methods
    -------
    pdf(x, mu, rho)
        Probability density function.

    cdf(x, mu, rho)
        Cumulative distribution function.

    Examples
    --------
    ```
    from pycircstat2.distributions import wrapnorm
    ```

    Notes
    -----
    Implementation based on Section 4.3.7 of Pewsey et al. (2014)
    """

    def _argcheck(self, mu, rho):
        return 0 <= mu <= np.pi * 2 and 0 < rho <= 1

    def _pdf(self, x, mu, rho):
        return (
            1
            + 2
            * np.sum([rho ** (p**2) * np.cos(p * (x - mu)) for p in range(1, 30)], 0)
        ) / (2 * np.pi)

    def pdf(self, x, mu, rho, *args, **kwargs):
        r"""
        Probability density function of the Wrapped Normal distribution.

        $$
        f(\theta) = \frac{1}{2\pi} \left(1 + 2\sum_{p=1}^{\infty} \rho^{p^2} \cos(p(\theta - \mu))\right)
        $$

        , here we approximate the infinite sum by summing the first 30 terms.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Shape parameter, 0 < rho <= 1.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, mu, rho, *args, **kwargs)

    def _cdf(self, x, mu, rho):
        @np.vectorize
        def _cdf_single(x, mu, rho):
            return quad(self._pdf, a=0, b=x, args=(mu, rho))

        return _cdf_single(x, mu, rho)

    def cdf(self, x, mu, rho, *args, **kwargs):
        """
        Cumulative distribution function of the Wrapped Normal distribution.

        No closed-form solution is available, so the CDF is computed numerically.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the cumulative distribution function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Shape parameter, 0 < rho <= 1.

        Returns
        -------
        cdf_values : array_like
            Cumulative distribution function evaluated at `x`.
        """
        return super().cdf(x, mu, rho, *args, **kwargs)

pdf(x, mu, rho, *args, **kwargs)

Probability density function of the Wrapped Normal distribution.

\[ f(\theta) = \frac{1}{2\pi} \left(1 + 2\sum_{p=1}^{\infty} \rho^{p^2} \cos(p(\theta - \mu))\right) \]

, here we approximate the infinite sum by summing the first 30 terms.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Shape parameter, 0 < rho <= 1.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, rho, *args, **kwargs):
    r"""
    Probability density function of the Wrapped Normal distribution.

    $$
    f(\theta) = \frac{1}{2\pi} \left(1 + 2\sum_{p=1}^{\infty} \rho^{p^2} \cos(p(\theta - \mu))\right)
    $$

    , here we approximate the infinite sum by summing the first 30 terms.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Shape parameter, 0 < rho <= 1.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, mu, rho, *args, **kwargs)

cdf(x, mu, rho, *args, **kwargs)

Cumulative distribution function of the Wrapped Normal distribution.

No closed-form solution is available, so the CDF is computed numerically.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the cumulative distribution function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Shape parameter, 0 < rho <= 1.

 required

Returns:

Name Type Description
cdf_values array_like

Cumulative distribution function evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, mu, rho, *args, **kwargs):
    """
    Cumulative distribution function of the Wrapped Normal distribution.

    No closed-form solution is available, so the CDF is computed numerically.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the cumulative distribution function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Shape parameter, 0 < rho <= 1.

    Returns
    -------
    cdf_values : array_like
        Cumulative distribution function evaluated at `x`.
    """
    return super().cdf(x, mu, rho, *args, **kwargs)

wrapcauchy_gen

Bases: rv_continuous

Wrapped Cauchy Distribution.

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
rvs

Random variates.
fit

Fit the distribution to the data and return the parameters (mu, rho).
Notes

Implementation based on Section 4.3.6 of Pewsey et al. (2014).

Source code in pycircstat2/distributions.py 
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class wrapcauchy_gen(rv_continuous):
    """Wrapped Cauchy Distribution.

    Methods
    -------
    pdf(x, mu, rho)
        Probability density function.

    cdf(x, mu, rho)
        Cumulative distribution function.

    rvs(mu, rho, size=None, random_state=None)
        Random variates.

    fit(data, method="analytical", *args, **kwargs)
        Fit the distribution to the data and return the parameters (mu, rho).

    Notes
    -----
    Implementation based on Section 4.3.6 of Pewsey et al. (2014).
    """

    def _argcheck(self, mu, rho):
        return 0 <= mu <= np.pi * 2 and 0 < rho <= 1

    def _pdf(self, x, mu, rho):
        return (1 - rho**2) / (2 * np.pi * (1 + rho**2 - 2 * rho * np.cos(x - mu)))

    def pdf(self, x, mu, rho, *args, **kwargs):
        r"""
        Probability density function of the Wrapped Cauchy distribution.

        $$
        f(\theta) = \frac{1 - \rho^2}{2\pi(1 + \rho^2 - 2\rho \cos(\theta - \mu))}
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Shape parameter, 0 < rho <= 1.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, mu, rho, *args, **kwargs)

    def _logpdf(self, x, mu, rho):
        return np.log(np.clip(self._pdf(x, mu, rho), 1e-16, None))

    def logpdf(self, x, mu, rho, *args, **kwargs):
        """
        Logarithm of the probability density function.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the log-PDF.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Mean resultant length, 0 < rho <= 1.

        Returns
        -------
        logpdf_values : array_like
            Logarithm of the probability density function evaluated at `x`.
        """
        return super().logpdf(x, mu, rho, *args, **kwargs)

    def _cdf(self, x, mu, rho):
        @np.vectorize
        def _cdf_single(x, mu, rho):
            integral, _ = quad(self._pdf, a=0, b=x, args=(mu, rho))
            return integral

        return _cdf_single(x, mu, rho)

    def cdf(self, x, mu, rho, *args, **kwargs):
        """
        Cumulative distribution function of the Wrapped Cauchy distribution.

        No closed-form solution is available, so the CDF is computed numerically.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the CDF.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Shape parameter, 0 < rho <= 1.

        Returns
        -------
        cdf_values : array_like
            CDF evaluated at `x`.
        """
        return super().cdf(x, mu, rho, *args, **kwargs)

    def _rvs(self, mu, rho, size=None, random_state=None):
        """
        Random variate generation for the Wrapped Cauchy distribution.

        Parameters
        ----------

        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        rho : float
            Mean resultant length, 0 <= rho <= 1.
        size : int or tuple, optional
            Number of samples to generate.
        random_state : RandomState, optional
            Random number generator instance.

        Returns
        -------
        samples : ndarray
            Random variates from the Wrapped Cauchy distribution.
        """
        rng = self._random_state if random_state is None else random_state

        if rho == 0:
            return rng.uniform(0, 2 * np.pi, size=size)
        elif rho == 1:
            return np.full(size, mu % (2 * np.pi))
        else:
            from scipy.stats import cauchy

            scale = -np.log(rho)
            samples = cauchy.rvs(loc=mu, scale=scale, size=size, random_state=rng)
            return np.mod(samples, 2 * np.pi)

    def fit(self, data, method="analytical", *args, **kwargs):
        """
        Fit the Wrapped Cauchy distribution to the data.

        Parameters
        ----------
        data : array_like
            Input data (angles in radians).
        method : str, optional
            The approach for fitting the distribution. Options are:
            - "analytical": Compute `rho` and `mu` using closed-form solutions.
            - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer.
            Default is "analytical".

        *args, **kwargs :
            Additional arguments passed to the optimizer (if used).

        Returns
        -------
        rho : float
            Estimated shape parameter.
        mu : float
            Estimated mean direction.
        """

        # Validate the fitting method
        valid_methods = ["analytical", "numerical"]
        if method not in valid_methods:
            raise ValueError(
                f"Invalid method '{method}'. Available methods are {valid_methods}."
            )

        # Validate the data
        if not np.all((0 <= data) & (data < 2 * np.pi)):
            raise ValueError("Data must be in the range [0, 2π).")

        # Analytical solution for the Von Mises distribution
        mu, rho = circ_mean_and_r(alpha=data)

        # Use analytical estimates for mu and rho
        if method == "analytical":
            return mu, rho
        elif method == "numerical":
            # Numerical optimization
            def nll(params):
                mu, rho = params
                if not self._argcheck(mu, rho):
                    return np.inf
                return -np.sum(self._logpdf(data, mu, rho))

            start_params = [mu, np.clip(rho, 1e-4, 1 - 1e-4)]
            bounds = [(0, 2 * np.pi), (1e-6, 1)]
            algo = kwargs.pop("algorithm", "L-BFGS-B")
            result = minimize(
                nll, start_params, bounds=bounds, method=algo, *args, **kwargs
            )
            if not result.success:
                raise RuntimeError(f"Optimization failed: {result.message}")
            mu, rho = result.x
            return mu, rho
        else:
            raise ValueError(
                "Invalid method. Supported methods are 'analytical' and " "'numerical'."
            )

pdf(x, mu, rho, *args, **kwargs)

Probability density function of the Wrapped Cauchy distribution.

\[ f(\theta) = \frac{1 - \rho^2}{2\pi(1 + \rho^2 - 2\rho \cos(\theta - \mu))} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Shape parameter, 0 < rho <= 1.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, rho, *args, **kwargs):
    r"""
    Probability density function of the Wrapped Cauchy distribution.

    $$
    f(\theta) = \frac{1 - \rho^2}{2\pi(1 + \rho^2 - 2\rho \cos(\theta - \mu))}
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Shape parameter, 0 < rho <= 1.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, mu, rho, *args, **kwargs)

logpdf(x, mu, rho, *args, **kwargs)

Logarithm of the probability density function.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the log-PDF.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Mean resultant length, 0 < rho <= 1.

 required

Returns:

Name Type Description
logpdf_values array_like

Logarithm of the probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def logpdf(self, x, mu, rho, *args, **kwargs):
    """
    Logarithm of the probability density function.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the log-PDF.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Mean resultant length, 0 < rho <= 1.

    Returns
    -------
    logpdf_values : array_like
        Logarithm of the probability density function evaluated at `x`.
    """
    return super().logpdf(x, mu, rho, *args, **kwargs)

cdf(x, mu, rho, *args, **kwargs)

Cumulative distribution function of the Wrapped Cauchy distribution.

No closed-form solution is available, so the CDF is computed numerically.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the CDF.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
rho float

Shape parameter, 0 < rho <= 1.

 required

Returns:

Name Type Description
cdf_values array_like

CDF evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, mu, rho, *args, **kwargs):
    """
    Cumulative distribution function of the Wrapped Cauchy distribution.

    No closed-form solution is available, so the CDF is computed numerically.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the CDF.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    rho : float
        Shape parameter, 0 < rho <= 1.

    Returns
    -------
    cdf_values : array_like
        CDF evaluated at `x`.
    """
    return super().cdf(x, mu, rho, *args, **kwargs)

fit(data, method='analytical', *args, **kwargs)

Fit the Wrapped Cauchy distribution to the data.

Parameters:

Name Type Description Default
data array_like

Input data (angles in radians).

 required
method str

The approach for fitting the distribution. Options are: - "analytical": Compute rho and mu using closed-form solutions. - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer. Default is "analytical".

 'analytical'
*args

Additional arguments passed to the optimizer (if used).

 ()
**kwargs

Additional arguments passed to the optimizer (if used).

 ()

Returns:

Name Type Description
rho float

Estimated shape parameter.
mu float

Estimated mean direction.
Source code in pycircstat2/distributions.py 
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def fit(self, data, method="analytical", *args, **kwargs):
    """
    Fit the Wrapped Cauchy distribution to the data.

    Parameters
    ----------
    data : array_like
        Input data (angles in radians).
    method : str, optional
        The approach for fitting the distribution. Options are:
        - "analytical": Compute `rho` and `mu` using closed-form solutions.
        - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer.
        Default is "analytical".

    *args, **kwargs :
        Additional arguments passed to the optimizer (if used).

    Returns
    -------
    rho : float
        Estimated shape parameter.
    mu : float
        Estimated mean direction.
    """

    # Validate the fitting method
    valid_methods = ["analytical", "numerical"]
    if method not in valid_methods:
        raise ValueError(
            f"Invalid method '{method}'. Available methods are {valid_methods}."
        )

    # Validate the data
    if not np.all((0 <= data) & (data < 2 * np.pi)):
        raise ValueError("Data must be in the range [0, 2π).")

    # Analytical solution for the Von Mises distribution
    mu, rho = circ_mean_and_r(alpha=data)

    # Use analytical estimates for mu and rho
    if method == "analytical":
        return mu, rho
    elif method == "numerical":
        # Numerical optimization
        def nll(params):
            mu, rho = params
            if not self._argcheck(mu, rho):
                return np.inf
            return -np.sum(self._logpdf(data, mu, rho))

        start_params = [mu, np.clip(rho, 1e-4, 1 - 1e-4)]
        bounds = [(0, 2 * np.pi), (1e-6, 1)]
        algo = kwargs.pop("algorithm", "L-BFGS-B")
        result = minimize(
            nll, start_params, bounds=bounds, method=algo, *args, **kwargs
        )
        if not result.success:
            raise RuntimeError(f"Optimization failed: {result.message}")
        mu, rho = result.x
        return mu, rho
    else:
        raise ValueError(
            "Invalid method. Supported methods are 'analytical' and " "'numerical'."
        )

vonmises_gen

Bases: rv_continuous

Von Mises Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
ppf

Percent-point function (inverse of CDF).
rvs

Random variates.
fit

Fit the distribution to the data and return the parameters (mu, kappa).

Examples:

from pycircstat2.distributions import vonmises
References
  • Section 4.3.8 of Pewsey et al. (2014)
Source code in pycircstat2/distributions.py 
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class vonmises_gen(rv_continuous):
    """Von Mises Distribution

    Methods
    -------
    pdf(x, mu, kappa)
        Probability density function.

    cdf(x, mu, kappa)
        Cumulative distribution function.

    ppf(q, mu, kappa)
        Percent-point function (inverse of CDF).

    rvs(mu, kappa, size=None, random_state=None)
        Random variates.

    fit(data, *args, **kwargs)
        Fit the distribution to the data and return the parameters (mu, kappa).

    Examples
    --------
    ```
    from pycircstat2.distributions import vonmises
    ```

    References
    ----------
    - Section 4.3.8 of Pewsey et al. (2014)

    """

    _freeze_doc = """
    Freeze the distribution with specific parameters.

    Parameters
    ----------
    mu : float
        The mean direction of the distribution (0 <= mu <= 2*pi).
    kappa : float
        The concentration parameter of the distribution (kappa > 0).

    Returns
    -------
    rv_frozen : rv_frozen instance
        The frozen distribution instance with fixed parameters.
    """

    def __call__(self, *args, **kwds):
        return self.freeze(*args, **kwds)

    __call__.__doc__ = _freeze_doc

    def _argcheck(self, mu, kappa):
        return 0 <= mu <= np.pi * 2 and kappa > 0

    def _pdf(self, x, mu, kappa):
        return np.exp(kappa * np.cos(x - mu)) / (2 * np.pi * i0(kappa))

    def pdf(self, x, mu, kappa, *args, **kwargs):
        r"""
        Probability density function of the Von Mises distribution.

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

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            The mean direction of the distribution (0 <= mu <= 2*pi).
        kappa : float
            The concentration parameter of the distribution (kappa > 0).

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, mu, kappa, *args, **kwargs)

    def _logpdf(self, x, mu, kappa):
        return kappa * np.cos(x - mu) - np.log(2 * np.pi * i0(kappa))

    def logpdf(self, x, mu, kappa, *args, **kwargs):
        """
        Logarithm of the probability density function of the Von Mises
        distribution.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the logarithm of the probability density function.
        mu : float
            The mean direction of the distribution (0 <= mu <= 2*pi).
        kappa : float
            The concentration parameter of the distribution (kappa > 0).

        Returns
        -------
        logpdf_values : array_like
            Logarithm of the probability density function evaluated at `x`.
        """
        return super().logpdf(x, mu, kappa, *args, **kwargs)

    def _cdf(self, x, mu, kappa):
        @np.vectorize
        def _cdf_single(x, mu, kappa):
            integral, _ = quad(self._pdf, a=0, b=x, args=(mu, kappa))
            return integral

        return _cdf_single(x, mu, kappa)

    def cdf(self, x, mu, kappa, *args, **kwargs):
        r"""
        Cumulative distribution function of the Von Mises distribution.

        $$
        F(\theta) = \frac{1}{2 \pi I_0(\kappa)}\int_{0}^{\theta} e^{\kappa \cos(\theta - \mu)} dx
        $$

        No closed-form solution is available, so the CDF is computed numerically.

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the cumulative distribution function.
        mu : float
            The mean direction of the distribution (0 <= mu <= 2*pi).
        kappa : float
            The concentration parameter of the distribution (kappa > 0).

        Returns
        -------
        cdf_values : array_like
            Cumulative distribution function evaluated at `x`.
        """
        return super().cdf(x, mu, kappa, *args, **kwargs)

    def ppf(self, q, mu, kappa, *args, **kwargs):
        """
        Percent-point function (inverse of the CDF) of the Von Mises distribution.

        Parameters
        ----------
        q : array_like
            Quantiles to evaluate.
        mu : float
            The mean direction of the distribution (0 <= mu <= 2*pi).
        kappa : float
            The concentration parameter of the distribution (kappa > 0).

        Returns
        -------
        ppf_values : array_like
            Values at the given quantiles.
        """
        return super().ppf(q, mu, kappa, *args, **kwargs)

    def _rvs(self, mu, kappa, size=None, random_state=None):
        # Use the random_state attribute or a new default random generator
        rng = self._random_state if random_state is None else random_state

        # Handle size being a tuple
        if size is None:
            size = 1
        num_samples = np.prod(size)  # Total number of samples

        # Best-Fisher algorithm
        a = 1 + np.sqrt(1 + 4 * kappa**2)
        b = (a - np.sqrt(2 * a)) / (2 * kappa)
        r = (1 + b**2) / (2 * b)

        def sample():
            while True:
                u1 = rng.uniform()
                z = np.cos(np.pi * u1)
                f = (1 + r * z) / (r + z)
                c = kappa * (r - f)
                u2 = rng.uniform()
                if u2 < c * (2 - c) or u2 <= c * np.exp(1 - c):
                    break
            u3 = rng.uniform()
            theta = mu + np.sign(u3 - 0.5) * np.arccos(f)
            return theta % (2 * np.pi)

        samples = np.array([sample() for _ in range(num_samples)])
        return samples

    def rvs(self, size=None, random_state=None, *args, **kwargs):
        """
        Draw random variates.

        Parameters
        ----------
        size : int or tuple, optional
            Number of samples to generate.
        random_state : RandomState, optional
            Random number generator instance.

        Returns
        -------
        samples : ndarray
            Random variates.
        """
        # Check if instance-level parameters are set
        mu = getattr(self, "mu", None)
        kappa = getattr(self, "kappa", None)

        # Override instance parameters if provided in args/kwargs
        mu = kwargs.pop("mu", mu)
        kappa = kwargs.pop("kappa", kappa)

        # Ensure required parameters are provided
        if mu is None or kappa is None:
            raise ValueError("Both 'mu' and 'kappa' must be provided.")

        # Call the private _rvs method
        return self._rvs(mu, kappa, size=size, random_state=random_state)

    def support(self, *args, **kwargs):
        return (0, 2 * np.pi)

    def mean(self, *args, **kwargs):
        """
        Circular mean of the Von Mises distribution.

        Returns
        -------
        mean : float
            The circular mean direction (in radians), equal to `mu`.
        """
        (mu, _) = self._parse_args(*args, **kwargs)[0]
        return mu

    def median(self, *args, **kwargs):
        """
        Circular median of the Von Mises distribution.

        Returns
        -------
        median : float
            The circular median direction (in radians), equal to `mu`.
        """
        return self.mean(*args, **kwargs)

    def var(self, *args, **kwargs):
        """
        Circular variance of the Von Mises distribution.

        Returns
        -------
        variance : float
            The circular variance, derived from `kappa`.
        """
        (_, kappa) = self._parse_args(*args, **kwargs)[0]
        return 1 - i1(kappa) / i0(kappa)

    def std(self, *args, **kwargs):
        """
        Circular standard deviation of the Von Mises distribution.

        Returns
        -------
        std : float
            The circular standard deviation, derived from `kappa`.
        """
        (_, kappa) = self._parse_args(*args, **kwargs)[0]
        r = i1(kappa) / i0(kappa)

        return np.sqrt(-2 * np.log(r))

    def entropy(self, *args, **kwargs):
        """
        Entropy of the Von Mises distribution.

        Returns
        -------
        entropy : float
            The entropy of the distribution.
        """
        (_, kappa) = self._parse_args(*args, **kwargs)[0]
        return -np.log(i0(kappa)) + (kappa * i1(kappa)) / i0(kappa)

    def _nnlf(self, theta, data):
        """
        Custom negative log-likelihood function for the Von Mises distribution.
        """
        mu, kappa = theta

        if not self._argcheck(mu, kappa):  # Validate parameter range
            return np.inf

        # Compute log-likelihood robustly
        log_likelihood = self._logpdf(data, mu, kappa)

        # Negative log-likelihood
        return -np.sum(log_likelihood)

    def fit(self, data, method="analytical", *args, **kwargs):
        """
        Fit the Von Mises distribution to the given data.

        Parameters
        ----------
        data : array_like
            The data to fit the distribution to. Assumes values are in radians.
        method : str, optional
            The approach for fitting the distribution. Options are:
            - "analytical": Compute `mu` and `kappa` using closed-form solutions.
            - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer.
            Default is "analytical".

            When `method="numerical"`, the optimization algorithm can be specified via `algorithm` in `kwargs`.
            Supported algorithms include any method from `scipy.optimize.minimize`, such as "L-BFGS-B" (default) or "Nelder-Mead".

        *args : tuple, optional
            Additional positional arguments passed to the optimizer (if used).
        **kwargs : dict, optional
            Additional keyword arguments passed to the optimizer (if used).

        Returns
        -------
        kappa : float
            The estimated concentration parameter of the Von Mises distribution.
        mu : float
            The estimated mean direction of the Von Mises distribution.

        Notes
        -----
        - The "analytical" method directly computes the parameters using the circular mean
        and resultant vector length (`r`) for `mu` and `kappa`, respectively.
        - For numerical methods, the negative log-likelihood (NLL) is minimized using `_nnlf` as the objective function.


        Examples
        --------
        ```python
        # MLE fitting using analytical solution
        mu, kappa = vonmises.fit(data, method="analytical")

        # MLE fitting with numerical method using L-BFGS-B
        mu, kappa = vonmises.fit(data, method="L-BFGS-B")
        ```
        """

        # Validate the fitting method
        valid_methods = ["analytical", "numerical"]
        if method not in valid_methods:
            raise ValueError(
                f"Invalid method '{method}'. Available methods are {valid_methods}."
            )

        # Validate the data
        if not np.all((0 <= data) & (data < 2 * np.pi)):
            raise ValueError("Data must be in the range [0, 2π).")

        # Analytical solution for the Von Mises distribution
        mu, r = circ_mean_and_r(alpha=data)
        kappa = circ_kappa(r=r, n=len(data))

        if method == "analytical":
            if np.isclose(r, 0):
                raise ValueError(
                    "Resultant vector length (r) is zero, e.g. uniform data or low directional bias."
                )
            return mu, kappa
        elif method == "numerical":
            # Use analytical solution as initial guess
            start_params = [mu, kappa]
            bounds = [(0, 2 * np.pi), (0, None)]  # 0 <= mu < 2*pi, kappa > 0,

            algo = kwargs.pop("algorithm", "L-BFGS-B")

            # Define the objective function (NLL) using `_nnlf`
            def nll(params):
                return self._nnlf(params, data)

            # Use the optimizer to minimize NLL
            result = minimize(
                nll, start_params, bounds=bounds, method=algo, *args, **kwargs
            )

            # Extract parameters from optimization result
            if not result.success:
                raise RuntimeError(f"Optimization failed: {result.message}")

            mu, kappa = result.x
            return mu, kappa
        else:
            raise ValueError(
                f"Invalid method '{method}'. Supported methods are 'analytical' and 'numerical'."
            )

pdf(x, mu, kappa, *args, **kwargs)

Probability density function of the Von Mises distribution.

\[ f(\theta) = \frac{e^{\kappa \cos(\theta - \mu)}}{2\pi I_0(\kappa)} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

The mean direction of the distribution (0 <= mu <= 2*pi).

 required
kappa float

The concentration parameter of the distribution (kappa > 0).

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, kappa, *args, **kwargs):
    r"""
    Probability density function of the Von Mises distribution.

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

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        The mean direction of the distribution (0 <= mu <= 2*pi).
    kappa : float
        The concentration parameter of the distribution (kappa > 0).

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, mu, kappa, *args, **kwargs)

logpdf(x, mu, kappa, *args, **kwargs)

Logarithm of the probability density function of the Von Mises distribution.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the logarithm of the probability density function.

 required
mu float

The mean direction of the distribution (0 <= mu <= 2*pi).

 required
kappa float

The concentration parameter of the distribution (kappa > 0).

 required

Returns:

Name Type Description
logpdf_values array_like

Logarithm of the probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def logpdf(self, x, mu, kappa, *args, **kwargs):
    """
    Logarithm of the probability density function of the Von Mises
    distribution.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the logarithm of the probability density function.
    mu : float
        The mean direction of the distribution (0 <= mu <= 2*pi).
    kappa : float
        The concentration parameter of the distribution (kappa > 0).

    Returns
    -------
    logpdf_values : array_like
        Logarithm of the probability density function evaluated at `x`.
    """
    return super().logpdf(x, mu, kappa, *args, **kwargs)

cdf(x, mu, kappa, *args, **kwargs)

Cumulative distribution function of the Von Mises distribution.

\[ F(\theta) = \frac{1}{2 \pi I_0(\kappa)}\int_{0}^{\theta} e^{\kappa \cos(\theta - \mu)} dx \]

No closed-form solution is available, so the CDF is computed numerically.

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the cumulative distribution function.

 required
mu float

The mean direction of the distribution (0 <= mu <= 2*pi).

 required
kappa float

The concentration parameter of the distribution (kappa > 0).

 required

Returns:

Name Type Description
cdf_values array_like

Cumulative distribution function evaluated at x.
Source code in pycircstat2/distributions.py 
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def cdf(self, x, mu, kappa, *args, **kwargs):
    r"""
    Cumulative distribution function of the Von Mises distribution.

    $$
    F(\theta) = \frac{1}{2 \pi I_0(\kappa)}\int_{0}^{\theta} e^{\kappa \cos(\theta - \mu)} dx
    $$

    No closed-form solution is available, so the CDF is computed numerically.

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the cumulative distribution function.
    mu : float
        The mean direction of the distribution (0 <= mu <= 2*pi).
    kappa : float
        The concentration parameter of the distribution (kappa > 0).

    Returns
    -------
    cdf_values : array_like
        Cumulative distribution function evaluated at `x`.
    """
    return super().cdf(x, mu, kappa, *args, **kwargs)

ppf(q, mu, kappa, *args, **kwargs)

Percent-point function (inverse of the CDF) of the Von Mises distribution.

Parameters:

Name Type Description Default
q array_like

Quantiles to evaluate.

 required
mu float

The mean direction of the distribution (0 <= mu <= 2*pi).

 required
kappa float

The concentration parameter of the distribution (kappa > 0).

 required

Returns:

Name Type Description
ppf_values array_like

Values at the given quantiles.
Source code in pycircstat2/distributions.py 
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def ppf(self, q, mu, kappa, *args, **kwargs):
    """
    Percent-point function (inverse of the CDF) of the Von Mises distribution.

    Parameters
    ----------
    q : array_like
        Quantiles to evaluate.
    mu : float
        The mean direction of the distribution (0 <= mu <= 2*pi).
    kappa : float
        The concentration parameter of the distribution (kappa > 0).

    Returns
    -------
    ppf_values : array_like
        Values at the given quantiles.
    """
    return super().ppf(q, mu, kappa, *args, **kwargs)

rvs(size=None, random_state=None, *args, **kwargs)

Draw random variates.

Parameters:

Name Type Description Default
size int or tuple

Number of samples to generate.

 None
random_state RandomState

Random number generator instance.

 None

Returns:

Name Type Description
samples ndarray

Random variates.
Source code in pycircstat2/distributions.py 
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def rvs(self, size=None, random_state=None, *args, **kwargs):
    """
    Draw random variates.

    Parameters
    ----------
    size : int or tuple, optional
        Number of samples to generate.
    random_state : RandomState, optional
        Random number generator instance.

    Returns
    -------
    samples : ndarray
        Random variates.
    """
    # Check if instance-level parameters are set
    mu = getattr(self, "mu", None)
    kappa = getattr(self, "kappa", None)

    # Override instance parameters if provided in args/kwargs
    mu = kwargs.pop("mu", mu)
    kappa = kwargs.pop("kappa", kappa)

    # Ensure required parameters are provided
    if mu is None or kappa is None:
        raise ValueError("Both 'mu' and 'kappa' must be provided.")

    # Call the private _rvs method
    return self._rvs(mu, kappa, size=size, random_state=random_state)

mean(*args, **kwargs)

Circular mean of the Von Mises distribution.

Returns:

Name Type Description
mean float

The circular mean direction (in radians), equal to mu.
Source code in pycircstat2/distributions.py 
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def mean(self, *args, **kwargs):
    """
    Circular mean of the Von Mises distribution.

    Returns
    -------
    mean : float
        The circular mean direction (in radians), equal to `mu`.
    """
    (mu, _) = self._parse_args(*args, **kwargs)[0]
    return mu

median(*args, **kwargs)

Circular median of the Von Mises distribution.

Returns:

Name Type Description
median float

The circular median direction (in radians), equal to mu.
Source code in pycircstat2/distributions.py 
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def median(self, *args, **kwargs):
    """
    Circular median of the Von Mises distribution.

    Returns
    -------
    median : float
        The circular median direction (in radians), equal to `mu`.
    """
    return self.mean(*args, **kwargs)

var(*args, **kwargs)

Circular variance of the Von Mises distribution.

Returns:

Name Type Description
variance float

The circular variance, derived from kappa.
Source code in pycircstat2/distributions.py 
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def var(self, *args, **kwargs):
    """
    Circular variance of the Von Mises distribution.

    Returns
    -------
    variance : float
        The circular variance, derived from `kappa`.
    """
    (_, kappa) = self._parse_args(*args, **kwargs)[0]
    return 1 - i1(kappa) / i0(kappa)

std(*args, **kwargs)

Circular standard deviation of the Von Mises distribution.

Returns:

Name Type Description
std float

The circular standard deviation, derived from kappa.
Source code in pycircstat2/distributions.py 
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def std(self, *args, **kwargs):
    """
    Circular standard deviation of the Von Mises distribution.

    Returns
    -------
    std : float
        The circular standard deviation, derived from `kappa`.
    """
    (_, kappa) = self._parse_args(*args, **kwargs)[0]
    r = i1(kappa) / i0(kappa)

    return np.sqrt(-2 * np.log(r))

entropy(*args, **kwargs)

Entropy of the Von Mises distribution.

Returns:

Name Type Description
entropy float

The entropy of the distribution.
Source code in pycircstat2/distributions.py 
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def entropy(self, *args, **kwargs):
    """
    Entropy of the Von Mises distribution.

    Returns
    -------
    entropy : float
        The entropy of the distribution.
    """
    (_, kappa) = self._parse_args(*args, **kwargs)[0]
    return -np.log(i0(kappa)) + (kappa * i1(kappa)) / i0(kappa)

fit(data, method='analytical', *args, **kwargs)

Fit the Von Mises distribution to the given data.

Parameters:

Name Type Description Default
data array_like

The data to fit the distribution to. Assumes values are in radians.

 required
method str

The approach for fitting the distribution. Options are: - "analytical": Compute mu and kappa using closed-form solutions. - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer. Default is "analytical".

When method="numerical", the optimization algorithm can be specified via algorithm in kwargs. Supported algorithms include any method from scipy.optimize.minimize, such as "L-BFGS-B" (default) or "Nelder-Mead".

 'analytical'
*args tuple

Additional positional arguments passed to the optimizer (if used).

 ()
**kwargs dict

Additional keyword arguments passed to the optimizer (if used).

 {}

Returns:

Name Type Description
kappa float

The estimated concentration parameter of the Von Mises distribution.
mu float

The estimated mean direction of the Von Mises distribution.
Notes
  • The "analytical" method directly computes the parameters using the circular mean and resultant vector length (r) for mu and kappa, respectively.
  • For numerical methods, the negative log-likelihood (NLL) is minimized using _nnlf as the objective function.

Examples:

# MLE fitting using analytical solution
mu, kappa = vonmises.fit(data, method="analytical")

# MLE fitting with numerical method using L-BFGS-B
mu, kappa = vonmises.fit(data, method="L-BFGS-B")
Source code in pycircstat2/distributions.py 
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def fit(self, data, method="analytical", *args, **kwargs):
    """
    Fit the Von Mises distribution to the given data.

    Parameters
    ----------
    data : array_like
        The data to fit the distribution to. Assumes values are in radians.
    method : str, optional
        The approach for fitting the distribution. Options are:
        - "analytical": Compute `mu` and `kappa` using closed-form solutions.
        - "numerical": Fit the parameters by minimizing the negative log-likelihood using an optimizer.
        Default is "analytical".

        When `method="numerical"`, the optimization algorithm can be specified via `algorithm` in `kwargs`.
        Supported algorithms include any method from `scipy.optimize.minimize`, such as "L-BFGS-B" (default) or "Nelder-Mead".

    *args : tuple, optional
        Additional positional arguments passed to the optimizer (if used).
    **kwargs : dict, optional
        Additional keyword arguments passed to the optimizer (if used).

    Returns
    -------
    kappa : float
        The estimated concentration parameter of the Von Mises distribution.
    mu : float
        The estimated mean direction of the Von Mises distribution.

    Notes
    -----
    - The "analytical" method directly computes the parameters using the circular mean
    and resultant vector length (`r`) for `mu` and `kappa`, respectively.
    - For numerical methods, the negative log-likelihood (NLL) is minimized using `_nnlf` as the objective function.


    Examples
    --------
    ```python
    # MLE fitting using analytical solution
    mu, kappa = vonmises.fit(data, method="analytical")

    # MLE fitting with numerical method using L-BFGS-B
    mu, kappa = vonmises.fit(data, method="L-BFGS-B")
    ```
    """

    # Validate the fitting method
    valid_methods = ["analytical", "numerical"]
    if method not in valid_methods:
        raise ValueError(
            f"Invalid method '{method}'. Available methods are {valid_methods}."
        )

    # Validate the data
    if not np.all((0 <= data) & (data < 2 * np.pi)):
        raise ValueError("Data must be in the range [0, 2π).")

    # Analytical solution for the Von Mises distribution
    mu, r = circ_mean_and_r(alpha=data)
    kappa = circ_kappa(r=r, n=len(data))

    if method == "analytical":
        if np.isclose(r, 0):
            raise ValueError(
                "Resultant vector length (r) is zero, e.g. uniform data or low directional bias."
            )
        return mu, kappa
    elif method == "numerical":
        # Use analytical solution as initial guess
        start_params = [mu, kappa]
        bounds = [(0, 2 * np.pi), (0, None)]  # 0 <= mu < 2*pi, kappa > 0,

        algo = kwargs.pop("algorithm", "L-BFGS-B")

        # Define the objective function (NLL) using `_nnlf`
        def nll(params):
            return self._nnlf(params, data)

        # Use the optimizer to minimize NLL
        result = minimize(
            nll, start_params, bounds=bounds, method=algo, *args, **kwargs
        )

        # Extract parameters from optimization result
        if not result.success:
            raise RuntimeError(f"Optimization failed: {result.message}")

        mu, kappa = result.x
        return mu, kappa
    else:
        raise ValueError(
            f"Invalid method '{method}'. Supported methods are 'analytical' and 'numerical'."
        )

jonespewsey_gen

Bases: rv_continuous

Jones-Pewsey Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation based on Section 4.3.9 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class jonespewsey_gen(rv_continuous):
    """Jones-Pewsey Distribution

    Methods
    -------
    pdf(x, mu, kappa, psi)
        Probability density function.

    cdf(x, mu, kappa, psi)
        Cumulative distribution function.


    Note
    ----
    Implementation based on Section 4.3.9 of Pewsey et al. (2014)
    """

    def _validate_params(self, mu, kappa, psi):
        return (0 <= mu <= np.pi * 2) and (kappa >= 0) and (-np.inf <= psi <= np.inf)

    def _argcheck(self, mu, kappa, psi):
        if self._validate_params(mu, kappa, psi):
            self._c = _c_jonespewsey(
                mu, kappa, psi
            )  # Precompute the normalizing constant
            return True
        else:
            return False

    def _pdf(self, x, mu, kappa, psi):

        if np.all(kappa < 0.001):
            return 1 / (2 * np.pi)
        else:
            if np.isclose(np.abs(psi), 0).all():
                return 1 / (2 * np.pi * i0(kappa)) * np.exp(kappa * np.cos(x - mu))
            else:
                return _kernel_jonespewsey(x, mu, kappa, psi) / self._c

    def pdf(self, x, mu, kappa, psi, *args, **kwargs):
        r"""
        Probability density function of the Jones-Pewsey distribution.

        $$
        f(\theta) = \frac{(\cosh(\kappa \psi) + \sinh(\kappa \psi) \cos(\theta - \mu))^{1/\psi}}{2\pi \cosh(\kappa \pi)}
        $$

        Parameters
        ----------
        x : array_like
            Points at which to evaluate the probability density function.
        mu : float
            Mean direction, 0 <= mu <= 2*pi.
        kappa : float
            Concentration parameter, kappa >= 0.
        psi : float
            Skewness parameter, -∞ <= psi <= ∞.

        Returns
        -------
        pdf_values : array_like
            Probability density function evaluated at `x`.
        """
        return super().pdf(x, mu, kappa, psi, *args, **kwargs)

    def _cdf(self, x, mu, kappa, psi):
        def vonmises_pdf(x, mu, kappa, psi, c):
            return c * np.exp(kappa * np.cos(x - mu))

        if np.isclose(np.abs(psi), 0).all():
            c = self._c

            @np.vectorize
            def _cdf_single(x, mu, kappa, psi, c):
                return quad(vonmises_pdf, a=0, b=x, args=(mu, kappa, psi, c))

            return _cdf_single(x, mu, kappa, psi, c)
        else:

            @np.vectorize
            def _cdf_single(x, mu, kappa, psi):
                return quad(self._pdf, a=0, b=x, args=(mu, kappa, psi))

            return _cdf_single(x, mu, kappa, psi)

pdf(x, mu, kappa, psi, *args, **kwargs)

Probability density function of the Jones-Pewsey distribution.

\[ f(\theta) = \frac{(\cosh(\kappa \psi) + \sinh(\kappa \psi) \cos(\theta - \mu))^{1/\psi}}{2\pi \cosh(\kappa \pi)} \]

Parameters:

Name Type Description Default
x array_like

Points at which to evaluate the probability density function.

 required
mu float

Mean direction, 0 <= mu <= 2*pi.

 required
kappa float

Concentration parameter, kappa >= 0.

 required
psi float

Skewness parameter, -∞ <= psi <= ∞.

 required

Returns:

Name Type Description
pdf_values array_like

Probability density function evaluated at x.
Source code in pycircstat2/distributions.py 
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def pdf(self, x, mu, kappa, psi, *args, **kwargs):
    r"""
    Probability density function of the Jones-Pewsey distribution.

    $$
    f(\theta) = \frac{(\cosh(\kappa \psi) + \sinh(\kappa \psi) \cos(\theta - \mu))^{1/\psi}}{2\pi \cosh(\kappa \pi)}
    $$

    Parameters
    ----------
    x : array_like
        Points at which to evaluate the probability density function.
    mu : float
        Mean direction, 0 <= mu <= 2*pi.
    kappa : float
        Concentration parameter, kappa >= 0.
    psi : float
        Skewness parameter, -∞ <= psi <= ∞.

    Returns
    -------
    pdf_values : array_like
        Probability density function evaluated at `x`.
    """
    return super().pdf(x, mu, kappa, psi, *args, **kwargs)

vonmises_ext_gen

Bases: rv_continuous

Flat-topped von Mises Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation based on Section 4.3.10 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class vonmises_ext_gen(rv_continuous):
    """Flat-topped von Mises Distribution

    Methods
    -------
    pdf(x, mu, kappa, nu)
        Probability density function.

    cdf(x, mu, kappa, nu)
        Cumulative distribution function.

    Note
    ----
    Implementation based on Section 4.3.10 of Pewsey et al. (2014)
    """

    def _validate_params(self, mu, kappa, nu):
        return (0 <= mu <= np.pi * 2) and (kappa >= 0) and (-1 <= nu <= 1)

    def _argcheck(self, mu, kappa, nu):
        if self._validate_params(mu, kappa, nu):
            self._c = _c_vmext(mu, kappa, nu)
            return True
        else:
            return False

    def _pdf(self, x, mu, kappa, nu):
        return self._c * _kernel_vmext(x, mu, kappa, nu)

    def _cdf(self, x, mu, kappa, nu):
        @np.vectorize
        def _cdf_single(x, mu, kappa, nu):
            return quad(self._pdf, a=0, b=x, args=(mu, kappa, nu))

        return _cdf_single(x, mu, kappa, nu)

jonespewsey_sineskewed_gen

Bases: rv_continuous

Sine-Skewed Jones-Pewsey Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation based on Section 4.3.11 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class jonespewsey_sineskewed_gen(rv_continuous):
    """Sine-Skewed Jones-Pewsey Distribution

    Methods
    -------
    pdf(x, xi, kappa, psi, lmbd)
        Probability density function.

    cdf(x, xi, kappa, psi, lmbd)
        Cumulative distribution function.


    Note
    ----
    Implementation based on Section 4.3.11 of Pewsey et al. (2014)
    """

    def _validate_params(self, xi, kappa, psi, lmbd):
        return (
            (0 <= xi <= np.pi * 2)
            and (kappa >= 0)
            and (-np.inf <= psi <= np.inf)
            and (-1 <= lmbd <= 1)
        )

    def _argcheck(self, xi, kappa, psi, lmbd):
        if self._validate_params(xi, kappa, psi, lmbd):
            self._c = _c_jonespewsey(xi, kappa, psi)
            return True
        else:
            return False

    def _pdf(self, x, xi, kappa, psi, lmbd):

        if np.all(kappa < 0.001):
            return 1 / (2 * np.pi) * (1 + lmbd * np.sin(x - xi))
        else:
            if np.isclose(np.abs(psi), 0).all():
                return (
                    1
                    / (2 * np.pi * i0(kappa))
                    * np.exp(kappa * np.cos(x - xi))
                    * (1 + lmbd * np.sin(x - xi))
                )
            else:
                return (
                    (1 + lmbd * np.sin(x - xi))
                    * _kernel_jonespewsey(x, xi, kappa, psi)
                    / self._c
                )

    def _cdf(self, x, xi, kappa, psi, lmbd):
        @np.vectorize
        def _cdf_single(x, xi, kappa, psi, lmbd):
            return quad(self._pdf, a=0, b=x, args=(xi, kappa, psi, lmbd))

        return _cdf_single(x, xi, kappa, psi, lmbd)

jonespewsey_asymext_gen

Bases: rv_continuous

Asymmetric Extended Jones-Pewsey Distribution

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation from 4.3.12 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class jonespewsey_asymext_gen(rv_continuous):
    """Asymmetric Extended Jones-Pewsey Distribution

    Methods
    -------
    pdf(x, xi, kappa, psi, nu)
        Probability density function.

    cdf(x, xi, kappa, psi, nu)
        Cumulative distribution function.


    Note
    ----
    Implementation from 4.3.12 of Pewsey et al. (2014)
    """

    def _validate_params(self, xi, kappa, psi, nu):
        return (
            (0 <= xi <= np.pi * 2)
            and (kappa >= 0)
            and (-np.inf <= psi <= np.inf)
            and (0 <= nu < 1)
        )

    def _argcheck(self, xi, kappa, psi, nu):
        if self._validate_params(xi, kappa, psi, nu):
            self._c = _c_jonespewsey_asymext(xi, kappa, psi, nu)
            return True
        else:
            return False

    def _pdf(self, x, xi, kappa, psi, nu):
        return _kernel_jonespewsey_asymext(x, xi, kappa, psi, nu) / self._c

    def _cdf(self, x, xi, kappa, psi, nu):
        @np.vectorize
        def _cdf_single(x, xi, kappa, psi, nu):
            return quad(self._pdf, a=0, b=x, args=(xi, kappa, psi, nu))

        return _cdf_single(x, xi, kappa, psi, nu)

inverse_batschelet_gen

Bases: rv_continuous

Inverse Batschelet distribution.

Methods:

Name Description
pdf

Probability density function.
cdf

Cumulative distribution function.
Note

Implementation from 4.3.13 of Pewsey et al. (2014)

Source code in pycircstat2/distributions.py 
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class inverse_batschelet_gen(rv_continuous):
    """Inverse Batschelet distribution.

    Methods
    -------
    pdf(x, xi, kappa, psi, nu, lmbd)
        Probability density function.

    cdf(x, xi, kappa, psi, nu, lmbd)
        Cumulative distribution function.


    Note
    ----
    Implementation from 4.3.13 of Pewsey et al. (2014)
    """

    def _validate_params(self, xi, kappa, nu, lmbd):
        return (
            (0 <= xi <= np.pi * 2)
            and (kappa >= 0)
            and (-1 <= nu <= 1)
            and (-1 <= lmbd <= 1)
        )

    def _argcheck(self, xi, kappa, nu, lmbd):
        if self._validate_params(xi, kappa, nu, lmbd):
            self._c = _c_invbatschelet(kappa, lmbd)
            if np.isclose(lmbd, -1).all():
                self.con1, self.con2 = 0, 0
            else:
                self.con1 = (1 - lmbd) / (1 + lmbd)
                self.con2 = (2 * lmbd) / (1 + lmbd)
            return True
        else:
            return False

    def _pdf(self, x, xi, kappa, nu, lmbd):

        arg1 = _tnu(x, nu, xi)
        arg2 = _slmbdinv(arg1, lmbd)

        if np.isclose(lmbd, -1).all():
            return self._c * np.exp(kappa * np.cos(arg1 - np.sin(arg1)))
        else:
            return self._c * np.exp(kappa * np.cos(self.con1 * arg1 + self.con2 * arg2))

    def _cdf(self, x, xi, kappa, nu, lmbd):
        @np.vectorize
        def _cdf_single(x, xi, kappa, nu, lmbd):
            return quad(self._pdf, a=0, b=x, args=(xi, kappa, nu, lmbd))

        return _cdf_single(x, xi, kappa, nu, lmbd)