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Descriptive Statistics

circ_r(alpha=None, w=None, Cbar=None, Sbar=None)

Circular mean resultant vector length (r).

\[ r = \sqrt{\bar{C}^2 + \bar{S}^2} \]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
Cbar Optional[float]

Precomputed intermediate values

 None
Sbar Optional[float]

Precomputed intermediate values

 None

Returns:

Name Type Description
r float

Resultant vector length
References

Implementation of Example 26.5 (Zar, 2010)

Source code in pycircstat2/descriptive.py 
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def circ_r(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    Cbar: Optional[float] = None,
    Sbar: Optional[float] = None,
) -> float:
    r"""
    Circular mean resultant vector length (r).

    $$
    r = \sqrt{\bar{C}^2 + \bar{S}^2}
    $$

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,)
        Frequencies or weights
    Cbar, Sbar: float
        Precomputed intermediate values

    Returns
    -------
    r: float
        Resultant vector length

    References
    ----------
    Implementation of Example 26.5 (Zar, 2010)
    """
    if alpha is None and (Cbar is None or Sbar is None):
        raise ValueError("`alpha` is needed for computing the resultant vector length.")

    if w is None:
        w = np.ones_like(alpha)

    if Cbar is None or Sbar is None:
        Cbar, Sbar = compute_C_and_S(alpha, w)

    # mean resultant vecotr length
    r = np.sqrt(Cbar**2 + Sbar**2)

    return r

circ_mean(alpha, w=None)

Circular mean (m).

\[\cos\bar\theta = C/R,\space \sin\bar\theta = S/R\]

or

\[ \bar\theta = \begin{cases} \tan^{-1}\left(S/C\right), & \text{if } S > 0, C > 0 \\ \tan^{-1}\left(S/C\right) + \pi, & \text{if } C < 0 \\ \tan^{-1}\left(S/C\right) + 2\pi, & \text{S < 0, C > 0} \end{cases} \]

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None

Returns:

Name Type Description
m float or NaN

Circular mean
Note

Implementation of Example 26.5 (Zar, 2010)

Source code in pycircstat2/descriptive.py 
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def circ_mean(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
) -> Union[np.ndarray, float]:
    r"""
    Circular mean (m).

    $$\cos\bar\theta = C/R,\space \sin\bar\theta = S/R$$

    or 

    $$
    \bar\theta =
    \begin{cases} 
    \tan^{-1}\left(S/C\right), & \text{if } S > 0, C > 0 \\ 
    \tan^{-1}\left(S/C\right) + \pi, & \text{if } C < 0 \\ 
    \tan^{-1}\left(S/C\right) + 2\pi, & \text{S < 0, C > 0}
    \end{cases}
    $$

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,)
        Frequencies or weights

    Returns
    -------
    m: float or NaN
        Circular mean

    Note
    ----
    Implementation of Example 26.5 (Zar, 2010)
    """
    if w is None:
        w = np.ones_like(alpha)

    # mean resultant vecotr length
    Cbar, Sbar = compute_C_and_S(alpha, w)
    r = circ_r(alpha, w, Cbar, Sbar)

    # angular mean
    if np.isclose(r, 0):
        m = np.nan
    else:
        m = np.arctan2(Sbar, Cbar)

    return angmod(m)

circ_mean_and_r(alpha, w=None)

Circular mean (m) and resultant vector length (r).

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None

Returns:

Name Type Description
m float or NaN

Circular mean
r float

Resultant vector length
Note

Implementation of Example 26.5 (Zar, 2010)

Source code in pycircstat2/descriptive.py 
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def circ_mean_and_r(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
) -> Tuple[Union[float, np.ndarray], float]:
    """
    Circular mean (m) and resultant vector length (r).

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,)
        Frequencies or weights

    Returns
    -------
    m: float or NaN
        Circular mean
    r: float
        Resultant vector length

    Note
    ----
    Implementation of Example 26.5 (Zar, 2010)
    """
    if w is None:
        w = np.ones_like(alpha)

    # mean resultant vecotr length
    Cbar, Sbar = compute_C_and_S(alpha, w)
    r = circ_r(alpha, w, Cbar, Sbar)

    # angular mean
    if np.isclose(r, 0):
        m = np.nan
        return m, r
    else:
        m = np.arctan2(Sbar, Cbar)

        return angmod(m), r

circ_mean_and_r_of_means(circs=None, ms=None, rs=None)

The Mean of a set of Mean Angles

Parameters:

Name Type Description Default
circs Union[list, None]

a list of Circular Objects

 None
ms Optional[ndarray]

a set of mean angles in radian

 None
rs Optional[ndarray]

a set of mean resultant vecotr lengths

 None

Returns:

Name Type Description
m float

mean of means in radian
r float

mean of mean resultant vector lengths
Source code in pycircstat2/descriptive.py 
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def circ_mean_and_r_of_means(
    circs: Union[list, None] = None,
    ms: Optional[np.ndarray] = None,
    rs: Optional[np.ndarray] = None,
) -> Tuple[float, float]:
    """The Mean of a set of Mean Angles

    Parameters
    ----------
    circs: list
        a list of Circular Objects

    ms: np.array (n, )
        a set of mean angles in radian

    rs: np.array (n, )
        a set of mean resultant vecotr lengths

    Returns
    -------
    m: float
        mean of means in radian

    r: float
        mean of mean resultant vector lengths

    """

    if circs is None:
        assert isinstance(ms, np.ndarray) and isinstance(
            rs, np.ndarray
        ), "If `circs` is None, then `ms` and `rs` are needed."
    else:
        ms, rs = map(np.array, zip(*[(circ.mean, circ.r) for circ in circs]))

    X = np.mean(np.cos(ms) * rs)
    Y = np.mean(np.sin(ms) * rs)
    r = np.sqrt(X**2 + Y**2)
    C = X / r
    S = Y / r

    m = angmod(np.arctan2(S, C))

    return m, r

circ_moment(alpha, w=None, p=1, mean=None, centered=False)

Compute the p-th circular moment.

\[ m^{\prime}_{p} = \bar{C}_{p} + i\bar{S}_{p} \]

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights. If None, equal weights are used.

 None
p int

Order of the moment to compute.

 1
mean Union[float, ndarray, None]

Precomputed circular mean. If None, mean is computed internally.

 None
centered bool

If True, center alpha by subtracting the mean.

 False

Returns:

Name Type Description
mp complex

The p-th circular moment as a complex number.
Note

Implementation of Equation 2.24 (Fisher, 1993).

Source code in pycircstat2/descriptive.py 
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def circ_moment(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
    p: int = 1,
    mean: Union[float, np.ndarray, None] = None,
    centered: bool = False,
) -> complex:
    r"""
    Compute the p-th circular moment.

    $$
    m^{\prime}_{p} = \bar{C}_{p} + i\bar{S}_{p}
    $$

    Parameters
    ----------
    alpha: np.ndarray
        Angles in radian.
    w: np.ndarray, optional
        Frequencies or weights. If None, equal weights are used.
    p: int, optional
        Order of the moment to compute.
    mean: float, optional
        Precomputed circular mean. If None, mean is computed internally.
    centered: bool, optional
        If True, center alpha by subtracting the mean.

    Returns
    -------
    mp: complex
        The p-th circular moment as a complex number.

    Note
    ----
    Implementation of Equation 2.24 (Fisher, 1993).
    """
    if w is None:
        w = np.ones_like(alpha)

    if mean is None:
        mean = circ_mean(alpha, w) if centered else 0.0

    Cbar, Sbar = compute_C_and_S(alpha, w, p, mean)

    return Cbar + 1j * Sbar

circ_dispersion(alpha, w=None, mean=None)

Sample Circular Dispersion, defined by Equation 2.28 (Fisher, 1993):

\[ \hat\delta = (1 - \hat\rho_{2})/(2 \hat\rho_{1}^{2}) \]

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None
mean

Precomputed circular mean.

 None

Returns:

Name Type Description
dispersion float

Sample Circular Dispersion
Source code in pycircstat2/descriptive.py 
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def circ_dispersion(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
    mean=None,
) -> float:
    r"""
    Sample Circular Dispersion, defined by Equation 2.28 (Fisher, 1993):

    $$
    \hat\delta = (1 - \hat\rho_{2})/(2 \hat\rho_{1}^{2})
    $$

    Parameters
    ----------

    alpha: np.array, (n, )
        Angles in radian.
    w: None or np.array, (n)
        Frequencies or weights
    mean: None or float
        Precomputed circular mean.

    Returns
    -------
    dispersion: float
        Sample Circular Dispersion
    """

    if w is None:
        w = np.ones_like(alpha)

    mp1 = circ_moment(alpha=alpha, w=w, p=1, mean=mean, centered=False)  # eq(2.26)
    mp2 = circ_moment(alpha=alpha, w=w, p=2, mean=mean, centered=False)  # eq(2.27)

    r1 = np.abs(mp1)
    r2 = np.abs(mp2)

    dispersion = (1 - r2) / (2 * r1**2)  # eq(2.28)

    return dispersion

circ_skewness(alpha, w=None)

Circular skewness, as defined by Equation 2.29 (Fisher, 1993):

\[\hat s = [\hat\rho_2 \sin(\hat\mu_2 - 2 \hat\mu_1)] / (1 - \hat\rho_1)^{\frac{3}{2}}\]

But unlike the implementation of Fisher (1993), here we followed Pewsey et al. (2014) by NOT centering the second moment.

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None

Returns:

Name Type Description
skewness float

Circular Skewness
Source code in pycircstat2/descriptive.py 
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def circ_skewness(alpha: np.ndarray, w: Optional[np.ndarray] = None) -> float:
    r"""
    Circular skewness, as defined by Equation 2.29 (Fisher, 1993):

    $$\hat s = [\hat\rho_2 \sin(\hat\mu_2 - 2 \hat\mu_1)] / (1 - \hat\rho_1)^{\frac{3}{2}}$$

    But unlike the implementation of Fisher (1993), here we followed Pewsey et al. (2014) by NOT centering the second moment.

    Parameters
    ----------

    alpha: np.array, (n, )
        Angles in radian.
    w: None or np.array, (n)
        Frequencies or weights

    Returns
    -------
    skewness: float
        Circular Skewness
    """

    if w is None:
        w = np.ones_like(alpha)

    mp1 = circ_moment(alpha=alpha, w=w, p=1, mean=None, centered=False)
    mp2 = circ_moment(alpha=alpha, w=w, p=2, mean=None, centered=False)  # eq(2.27)

    u1, r1 = convert_moment(mp1)
    u2, r2 = convert_moment(mp2)

    skewness = (r2 * np.sin(u2 - 2 * u1)) / (1 - r1) ** 1.5

    return skewness

circ_kurtosis(alpha, w=None)

Circular kurtosis, as defined by Equation 2.30 (Fisher, 1993):

\[\hat k = [\hat\rho_2 \cos(\hat\mu_2 - 2 \hat\mu_1) - \hat\rho_1^4] / (1 - \hat\rho_1)^{2}\]

But unlike the implementation of Fisher (1993), here we followed Pewsey et al. (2014) by NOT centering the second moment.

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None

Returns:

Name Type Description
kurtosis float

Circular Kurtosis
Source code in pycircstat2/descriptive.py 
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def circ_kurtosis(alpha: np.ndarray, w: Optional[np.ndarray] = None) -> float:
    r"""
    Circular kurtosis, as defined by Equation 2.30 (Fisher, 1993):

    $$\hat k = [\hat\rho_2 \cos(\hat\mu_2 - 2 \hat\mu_1) - \hat\rho_1^4] / (1 - \hat\rho_1)^{2}$$

    But unlike the implementation of Fisher (1993), here we followed Pewsey et al. (2014) by **NOT** centering the second moment.

    Parameters
    ----------

    alpha: np.array, (n, )
        Angles in radian.
    w: None or np.array, (n)
        Frequencies or weights

    Returns
    -------
    kurtosis: float
        Circular Kurtosis
    """

    if w is None:
        w = np.ones_like(alpha)

    mp1 = circ_moment(alpha=alpha, w=w, p=1, mean=None, centered=False)
    mp2 = circ_moment(alpha=alpha, w=w, p=2, mean=None, centered=False)  # eq(2.27)

    u1, r1 = convert_moment(mp1)
    u2, r2 = convert_moment(mp2)

    kurtosis = (r2 * np.cos(u2 - 2 * u1) - r1**4) / (1 - r1) ** 2

    return kurtosis

angular_var(alpha=None, w=None, r=None, bin_size=None)

Angular variance

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
r Optional[float]

Resultant vector length

 None
bin_size Optional[float]

Interval size of grouped data. Needed for correcting biased r.

 None

Returns:

Name Type Description
angular_variance float

Angular variance, range from 0 to 2.
References
  • Batschlet (1965, 1981), from Section 26.5 of Zar (2010)
Source code in pycircstat2/descriptive.py 
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def angular_var(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    r: Optional[float] = None,
    bin_size: Optional[float] = None,
) -> float:
    r"""
    Angular variance

    Parameters
    ----------
    alpha: np.array (n, ) or None
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    r: float or None
        Resultant vector length
    bin_size: float
        Interval size of grouped data. Needed for correcting biased r.

    Returns
    -------
    angular_variance: float
        Angular variance, range from 0 to 2.

    References
    ----------
    - Batschlet (1965, 1981), from Section 26.5 of Zar (2010)
    """

    variance = circ_var(alpha=alpha, w=w, r=r, bin_size=bin_size)
    angular_variance = 2 * variance
    return angular_variance

angular_std(alpha=None, w=None, r=None, bin_size=None)

Angular (standard) deviation

\[ s = \sqrt{2V} = \sqrt{2(1 - r)} \]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
r Optional[float]

Resultant vector length

 None
bin_size Optional[float]

Interval size of grouped data. Needed for correcting biased r.

 None

Returns:

Name Type Description
angular_std float

Angular (standard) deviation, range from 0 to sqrt(2).
References
  • Equation 26.20 of Zar (2010)
Source code in pycircstat2/descriptive.py 
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def angular_std(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    r: Optional[float] = None,
    bin_size: Optional[float] = None,
) -> float:
    r"""
    Angular (standard) deviation

    $$
    s = \sqrt{2V} = \sqrt{2(1 - r)}
    $$

    Parameters
    ----------
    alpha: np.array (n, ) or None
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    r: float or None
        Resultant vector length
    bin_size: float
        Interval size of grouped data. Needed for correcting biased r.

    Returns
    -------
    angular_std: float
        Angular (standard) deviation, range from 0 to sqrt(2).

    References
    ----------
    - Equation 26.20 of Zar (2010)
    """

    angular_variance = angular_var(alpha=alpha, w=w, r=r, bin_size=bin_size)
    angular_std = np.sqrt(angular_variance)
    return angular_std

circ_var(alpha=None, w=None, r=None, bin_size=None)

Circular variance

\[ V = 1 - r \]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
r Optional[float]

Resultant vector length

 None
bin_size Optional[float]

Interval size of grouped data. Needed for correcting biased r.

 None

Returns:

Name Type Description
variance float

Circular variance, range from 0 to 1.
References
  • Equation 2.11 of Fisher (1993)
  • Equation 26.17 of Zar (2010)
Source code in pycircstat2/descriptive.py 
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def circ_var(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    r: Optional[float] = None,
    bin_size: Optional[float] = None,
) -> float:
    r"""
    Circular variance

    $$ V = 1 - r $$

    Parameters
    ----------
    alpha: np.array (n, ) or None
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    r: float or None
        Resultant vector length
    bin_size: float
        Interval size of grouped data. Needed for correcting biased r.

    Returns
    -------
    variance: float
        Circular variance, range from 0 to 1.

    References
    ----------
    - Equation 2.11 of Fisher (1993)
    - Equation 26.17 of Zar (2010)
    """

    if w is None:
        w = np.ones_like(alpha)

    if r is None:
        assert isinstance(alpha, np.ndarray) and isinstance(
            w, np.ndarray
        ), "If `r` is None, then `alpha` and `w` are needed."
        r = circ_r(alpha, w)

    if bin_size is None:
        assert isinstance(alpha, np.ndarray) and isinstance(
            w, np.ndarray
        ), "If `bin_size` is None, then `alpha` and `w` are needed."
        if (w == w[0]).all():  #
            bin_size = 0
        else:
            bin_size = np.diff(alpha).min()

    ## corrected r if data are grouped.
    if bin_size == 0:
        rc = r

    else:
        c = bin_size / 2 / np.sin(bin_size / 2)  # eq(26.16)
        rc = r * c  # eq(26.15)

    variance = 1 - rc

    return variance

circ_std(alpha=None, w=None, r=None, bin_size=None)

Circular standard deviation (s).

\[ s = \sqrt{-2 \ln(1 - V)} \]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
r Optional[float]

Resultant vector length

 None
bin_size Optional[float]

Interval size of grouped data. Needed for correcting biased r.

 None

Returns:

Name Type Description
s float

Circular standard deviation.
References

Implementation of Equation 26.15-16/20-21 (Zar, 2010)

Source code in pycircstat2/descriptive.py 
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def circ_std(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    r: Optional[float] = None,
    bin_size: Optional[float] = None,
) -> tuple:
    r"""
    Circular standard deviation (s).

    $$ s = \sqrt{-2 \ln(1 - V)} $$

    Parameters
    ----------
    alpha: np.array (n, ) or None
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    r: float or None
        Resultant vector length
    bin_size: float
        Interval size of grouped data.
        Needed for correcting biased r.

    Returns
    -------
    s: float
        Circular standard deviation.

    References
    ----------
    Implementation of Equation 26.15-16/20-21 (Zar, 2010)
    """
    var = circ_var(alpha=alpha, w=w, r=r, bin_size=bin_size)

    # circular standard deviation
    s = np.sqrt(-2 * np.log(1 - var))  # eq(26.21)

    return s

circ_median(alpha, w=None, method='deviation', return_average=True, average_method='all')

Circular median.

Two ways to compute the circular median for ungrouped data (Fisher, 1993):

  • deviation: find the angle that has the minimal mean deviation.
  • count: find the angle that has the equally devide the number of points on the right and left of it.

For grouped data, we use the method described in Mardia (1972).

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None
method str
  • For ungrouped data, there are two ways
  • To compute the medians:
    • deviation
    • count
  • Set to none to return np.nan.
 'deviation'
return_average bool

Return the average of the median

 True
average_method str
  • all: circular mean of all medians
  • unique: circular mean of unique medians
 'all'

Returns:

Name Type Description
median float or NaN
References
  • For ungrouped data: Section 2.3.2 of Fisher (1993)
  • For grouped data: Mardia (1972)
Source code in pycircstat2/descriptive.py 
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def circ_median(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
    method: str = "deviation",
    return_average: bool = True,
    average_method: str = "all",
) -> Union[float, np.ndarray]:
    r"""
    Circular median.

    Two ways to compute the circular median for ungrouped data (Fisher, 1993):

    - `deviation`: find the angle that has the minimal mean deviation.
    - `count`: find the angle that has the equally devide the number of points on the right and left of it.

    For grouped data, we use the method described in Mardia (1972).

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    method: str
        - For ungrouped data, there are two ways
        - To compute the medians:
            - deviation
            - count
        - Set to `none` to return np.nan.
    return_average: bool
        Return the average of the median
    average_method: str
        - all: circular mean of all medians
        - unique: circular mean of unique medians

    Returns
    -------
    median: float or NaN

    References
    ----------
    - For ungrouped data: Section 2.3.2 of Fisher (1993)
    - For grouped data: Mardia (1972)
    """

    if w is None:
        w = np.ones_like(alpha)

    # grouped data
    if not np.all(w == 1):
        median = _circ_median_grouped(alpha, w)
    # ungrouped data
    else:
        # find which data point that can divide the dataset into two half
        if method == "count":
            median = _circ_median_count(alpha)
        # find the angle that has the minimal mean deviation
        elif method == "deviation":
            median = _circ_median_mean_deviation(alpha)
        elif method == "none" or method is None:
            median = np.nan
        else:
            raise ValueError(
                f"Method `{method}` for `circ_median` is not supported.\nTry `deviation` or `count`"
            )

    if return_average:
        if average_method == "all":
            # Circular mean of all medians
            median = circ_mean(alpha=median)
        elif average_method == "unique":
            # Circular mean of unique medians
            median = circ_mean(alpha=np.unique(median))
        else:
            raise ValueError(
                f"Average method `{average_method}` is not supported.\nTry `all` or `unique`."
            )

    return angmod(median)

_circ_median_mean_deviation(alpha)

Note

Implementation of Section 2.3.2 of Fisher (1993)

Source code in pycircstat2/descriptive.py 
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def _circ_median_mean_deviation(alpha: np.array) -> float:
    """
    Note
    ----
    Implementation of Section 2.3.2 of Fisher (1993)
    """

    # get pairwise circular mean deviation
    if len(alpha) > 10000:
        angdist = circ_mean_deviation_chuncked(alpha, alpha)
    else:
        # get pairwise circular mean deviation
        angdist = circ_mean_deviation(alpha, alpha)
    # data point(s) with minimal circular mean deviation is/are potential median(s);
    idx_candidates = np.where(angdist == angdist.min())[0]
    # if number of potential median is the same as the number of data point
    # meaning that the data is more or less uniformly distributed. Retrun Nan.
    if len(idx_candidates) == len(alpha):
        median = np.nan
    # if number of potential median is 1, return it as median
    elif len(idx_candidates) == 1:
        median = alpha[idx_candidates][0]
    # if there are more than one potential median, return them all
    else:
        median = alpha[idx_candidates]

    return median

circ_mean_deviation_chuncked(alpha, beta, chunk_size=1000)

Optimized circular mean deviation with chunking.

\[ \delta = \pi - \frac{1}{n} \sum^{n}_{1}\left| \pi - \left| \alpha - \beta \right| \right| \]

Parameters:

Name Type Description Default
alpha ndarray

Data in radians.

 required
beta ndarray

Reference angles in radians.

 required
chunk_size int

Number of rows to process in chunks.

 1000

Returns:

Type Description
ndarray

Circular mean deviation.
Source code in pycircstat2/descriptive.py 
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def circ_mean_deviation_chuncked(
    alpha: Union[np.ndarray, float, int, list],
    beta: Union[np.ndarray, float, int, list],
    chunk_size=1000,
):
    r"""
    Optimized circular mean deviation with chunking.

    $$
    \delta = \pi - \frac{1}{n} \sum^{n}_{1}\left| \pi - \left| \alpha - \beta \right| \right|
    $$

    Parameters
    ----------
    alpha : np.ndarray
        Data in radians.
    beta : np.ndarray
        Reference angles in radians.
    chunk_size : int
        Number of rows to process in chunks.

    Returns
    -------
    np.ndarray
        Circular mean deviation.
    """
    if not isinstance(alpha, np.ndarray):
        alpha = np.array([alpha])

    if not isinstance(beta, np.ndarray):
        beta = np.array([beta])

    n = len(beta)
    result = np.zeros(n)

    for i in range(0, n, chunk_size):
        beta_chunk = beta[i : i + chunk_size]
        angdist = np.pi - np.abs(np.pi - np.abs(alpha - beta_chunk[:, None]))
        result[i : i + chunk_size] = np.mean(angdist, axis=1).round(5)

    return result

circ_mean_deviation(alpha, beta)

Circular mean deviation.

\[ \delta = \pi - \left| \pi - \left| \alpha - \beta \right| \right| / n \]

It is the mean angular distance from one data point to all others. The circular median of a set of data should be the point with minimal circular mean deviation.

Parameters:

Name Type Description Default
alpha Union[ndarray, float, int, list]

Data in radian.

 required
beta Union[ndarray, float, int, list]

reference angle in radian.

 required

Returns:

Type Description
circular mean deviation: np.array
Note

eq 2.32, Section 2.3.2, Fisher (1993)

Source code in pycircstat2/descriptive.py 
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def circ_mean_deviation(
    alpha: Union[np.ndarray, float, int, list],
    beta: Union[np.ndarray, float, int, list],
) -> np.ndarray:
    r"""
    Circular mean deviation.

    $$
    \delta = \pi - \left| \pi - \left| \alpha - \beta \right| \right| / n
    $$

    It is the mean angular distance from one data point to all others.
    The circular median of a set of data should be the point with minimal
    circular mean deviation.

    Parameters
    ---------
    alpha: np.array, int or float
        Data in radian.
    beta: np.array, int or float
        reference angle in radian.

    Returns
    -------
    circular mean deviation: np.array

    Note
    ----
    eq 2.32, Section 2.3.2, Fisher (1993)
    """
    if not isinstance(alpha, np.ndarray):
        alpha = np.array([alpha])

    if not isinstance(beta, np.ndarray):
        beta = np.array([beta])

    return (np.pi - np.mean(np.abs(np.pi - np.abs(alpha - beta[:, None])), 1)).round(5)

circ_mean_ci(alpha=None, w=None, mean=None, r=None, n=None, ci=0.95, method='approximate', B=2000)

Confidence interval of circular mean.

There are three methods to compute the confidence interval of circular mean:

  • approximate: for n > 8
  • bootstrap: for 8 < n < 25
  • dispersion: for n >= 25
Approximate Method

For n as small as 8, and r \(\le\) 0.9, r \(>\) \(\sqrt{\chi^{2}_{\alpha, 1}/2n}\), the confidence interval can be approximated by:

\[ \delta = \arccos\left(\sqrt{\frac{2n(2R^{2} - n\chi^{2}_{\alpha, 1})}{4n - \chi^{2}_{\alpha, 1}}} /R \right) \]

For r \(ge\) 0.9,

\[ \delta = \arccos \left(\sqrt{n^2 - (n^2 - R^2)e^{\chi^2_{\alpha, 1}/n} } /R \right) \]
Bootstrap Method

For 8 \(<\) n \(<\) 25, the confidence interval can be computed by bootstrapping the data.

Dispersion Method

For n \(\ge\) 25, the confidence interval can be computed by the circular dispersion:

\[ \hat\sigma = \hat\delta / n\]

where \(\hat\delta\) is the sample circular dispersion (see circ_dispersion). The confidence interval is then:

\[(\hat\mu - \sin^-1(z_{\frac{1}{2}\alpha}\hat\sigma),\space \hat\mu + \sin^-1(z_{\frac{1}{2}\alpha} \hat\sigma))\]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
mean Optional[float]

Precomputed circular mean.

 None
r Optional[float]

Precomputed resultant vector length.

 None
n Union[int, None]

Sample size.

 None
ci float

Confidence interval (default is 0.95).

 0.95
method str
  • approximate: for n > 8
  • bootstrap: for n < 25
  • dispersion: for n >= 25
 'approximate'
B int

Number of samples for bootstrap.

 2000

Returns:

Name Type Description
lower_bound float

Lower bound of the confidence interval.
upper_bound float

Upper bound of the confidence
References
  • Section 26.7, Zar (2010)
  • Section 4.4.4a/b, Fisher (1993)
Source code in pycircstat2/descriptive.py 
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def circ_mean_ci(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    mean: Optional[float] = None,
    r: Optional[float] = None,
    n: Union[int, None] = None,
    ci: float = 0.95,
    method: str = "approximate",
    B: int = 2000,  # number of samples for bootstrap
) -> tuple[float, float]:
    r"""
    Confidence interval of circular mean.

    There are three methods to compute the confidence interval of circular mean:

    - `approximate`: for n > 8
    - `bootstrap`: for 8 < n < 25
    - `dispersion`: for n >= 25

    ### Approximate Method

    For n as small as 8, and r $\le$ 0.9, r $>$ $\sqrt{\chi^{2}_{\alpha, 1}/2n}$, the confidence interval can be approximated by:

    $$
    \delta = \arccos\left(\sqrt{\frac{2n(2R^{2} - n\chi^{2}_{\alpha, 1})}{4n - \chi^{2}_{\alpha, 1}}} /R \right)
    $$

    For r $ge$ 0.9,

    $$
    \delta = \arccos \left(\sqrt{n^2 - (n^2 - R^2)e^{\chi^2_{\alpha, 1}/n} } /R \right)
    $$

    ### Bootstrap Method

    For 8 $<$ n $<$ 25, the confidence interval can be computed by bootstrapping the data.

    ### Dispersion Method

    For n $\ge$ 25, the confidence interval can be computed by the circular dispersion:

    $$ \hat\sigma = \hat\delta / n$$

    where $\hat\delta$ is the sample circular dispersion (see `circ_dispersion`). The confidence interval is then:

    $$(\hat\mu - \sin^-1(z_{\frac{1}{2}\alpha}\hat\sigma),\space \hat\mu + \sin^-1(z_{\frac{1}{2}\alpha} \hat\sigma))$$

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    mean: float or None
        Precomputed circular mean.
    r: float or None
        Precomputed resultant vector length.
    n: int or None
        Sample size.
    ci: float
        Confidence interval (default is 0.95).
    method: str
        - approximate: for n > 8
        - bootstrap: for n < 25
        - dispersion: for n >= 25
    B: int
        Number of samples for bootstrap.

    Returns
    -------
    lower_bound: float
        Lower bound of the confidence interval.
    upper_bound: float
        Upper bound of the confidence

    References
    ----------
    - Section 26.7, Zar (2010)
    - Section 4.4.4a/b, Fisher (1993)
    """

    #  n > 8, according to Ch 26.7 (Zar, 2010)
    if method == "approximate":
        (lb, ub) = _circ_mean_ci_approximate(
            alpha=alpha, w=w, mean=mean, r=r, n=n, ci=ci
        )

    # n < 25, according to 4.4.4a (Fisher, 1993, P75)
    elif method == "bootstrap":
        (lb, ub) = _circ_mean_ci_bootstrap(alpha=alpha, B=B, ci=ci)

    # n >= 25, according to 4.4.4b (Fisher, 1993, P75)
    elif method == "dispersion":
        (lb, ub) = _circ_mean_ci_dispersion(alpha=alpha, w=w, mean=mean, ci=ci)

    else:
        raise ValueError(
            f"Method `{method}` for `circ_mean_ci` is not supported.\nTry `dispersion`, `approximate` or `bootstrap`"
        )

    return angmod(lb), angmod(ub)

_circ_mean_ci_dispersion(alpha, w=None, mean=None, ci=0.95)

Confidence intervals based on circular dispersion.

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w Optional[ndarray]

Frequencies or weights

 None
mean Optional[float]

Precomputed circular mean.

 None
ci float

Confidence interval (default is 0.95).

 0.95

Returns:

Name Type Description
lower_bound float

Lower bound of the confidence interval.
upper_bound float

Upper bound of the confidence interval.
Note

Implementation of Section 4.4.4b (Fisher, 1993)

Source code in pycircstat2/descriptive.py 
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def _circ_mean_ci_dispersion(
    alpha: np.ndarray,
    w: Optional[np.ndarray] = None,
    mean: Optional[float] = None,
    ci: float = 0.95,
) -> tuple[float, float]:
    r"""Confidence intervals based on circular dispersion.

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    mean: float or None
        Precomputed circular mean.
    ci: float
        Confidence interval (default is 0.95).


    Returns
    -------
    lower_bound: float
        Lower bound of the confidence interval.
    upper_bound: float
        Upper bound of the confidence interval.


    Note
    ----
    Implementation of Section 4.4.4b (Fisher, 1993)
    """

    if w is None:
        w = np.ones_like(alpha)
    if mean is None:
        mean, r = circ_mean_and_r(alpha, w)

    n = np.sum(w)
    if n < 25:
        raise ValueError(
            f"n={n} is too small (< 25) for computing CI with circular dispersion."
        )

    # TODO: sometime return nan because x in arcsin(x) is larger than 1.
    # Should we centered the data here? <- No.
    d = np.arcsin(
        np.sqrt(circ_dispersion(alpha=alpha, w=w) / n) * norm.ppf(1 - 0.5 * (1 - ci))
    )
    lb = mean - d
    ub = mean + d

    if not is_within_circular_range(mean, lb, ub):
        lb, ub = ub, lb

    return (lb, ub)

_circ_mean_ci_approximate(alpha=None, w=None, mean=None, r=None, n=None, ci=0.95)

Confidence Interval of circular mean.

\[ \displaylines{ \delta = \arccos\left(\sqrt{\frac{2n(2R^{2} - n\chi^{2}_{alpha, 1})}{4n - \chi^{2}_{\alpha, 1}}}\right)/R \cr } \]

Parameters:

Name Type Description Default
alpha Optional[ndarray]

Angles in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None
mean Optional[float]

Precomputed circular mean.

 None
r Optional[float]

Precomputed resultant vector length.

 None
n Union[int, None]

Sample size.

 None
ci float

Confidence interval (default is 0.95).

 0.95

Returns:

Name Type Description
lower_bound float

Lower bound of the confidence interval.
upper_bound float

Upper bound of the confidence interval.
Note

Implementation of Example 26.6 (Zar, 2010)

Source code in pycircstat2/descriptive.py 
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def _circ_mean_ci_approximate(
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    mean: Optional[float] = None,
    r: Optional[float] = None,
    n: Union[int, None] = None,
    ci: float = 0.95,
) -> tuple:
    r"""
    Confidence Interval of circular mean.

    $$
    \displaylines{
    \delta = \arccos\left(\sqrt{\frac{2n(2R^{2} - n\chi^{2}_{alpha, 1})}{4n - \chi^{2}_{\alpha, 1}}}\right)/R \cr
    }
    $$

    Parameters
    ----------
    alpha: np.array (n, )
        Angles in radian.
    w: np.array (n,) or None
        Frequencies or weights
    mean: float or None
        Precomputed circular mean.
    r: float or None
        Precomputed resultant vector length.
    n: int or None
        Sample size.
    ci: float
        Confidence interval (default is 0.95).

    Returns
    -------
    lower_bound: float
        Lower bound of the confidence interval.
    upper_bound: float
        Upper bound of the confidence interval.

    Note
    ----
    Implementation of Example 26.6 (Zar, 2010)
    """

    if r is None:
        assert isinstance(
            alpha, np.ndarray
        ), "If `r` is None, then `alpha` (and `w`) is needed."
        if w is None:
            w = np.ones_like(alpha)
        n = np.sum(w)
        mean, r = circ_mean_and_r(alpha, w)

    if n is None:
        raise ValueError("Sample size `n` is missing.")

    R = n * r

    # Zar cited (Upton, 1986) that n can be as small as 8
    # but I encountered a few cases where n<11 will result in negative
    # value within `inner`.
    if n >= 8:  #
        if r <= 0.9:  # eq(26.24)
            inner = (
                np.sqrt(
                    (2 * n * (2 * R**2 - n * chi2.isf(0.05, 1)))
                    / (4 * n - chi2.isf(1 - ci, 1))
                )
                / R
            )
        else:  # eq(26.25)
            inner = np.sqrt(n**2 - (n**2 - R**2) * np.exp(chi2.isf(1 - ci, 1) / n)) / R

        d = np.arccos(inner)
        lb = mean - d
        ub = mean + d

        if not is_within_circular_range(mean, lb, ub):
            lb, ub = ub, lb

        return (lb, ub)

    else:
        raise ValueError(
            f"n={n} is too small (<= 8) for computing CI with approximation method."
        )

_circ_mean_ci_bootstrap(alpha, B=2000, ci=0.95)

Implementation of Section 8.3 (Fisher, 1993, P207)

Source code in pycircstat2/descriptive.py 
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def _circ_mean_ci_bootstrap(alpha, B=2000, ci=0.95):
    """
    Implementation of Section 8.3 (Fisher, 1993, P207)
    """

    # Precompute z0 and v0 from original data
    # algo 1
    X = np.cos(alpha)
    Y = np.sin(alpha)
    z1 = np.mean(X)  # eq(8.24)
    z2 = np.mean(Y)
    z0 = np.array([z1, z2])

    # algo 2
    u11 = np.mean((X - z1) ** 2)  # eq(8.25)
    u22 = np.mean((Y - z2) ** 2)
    u12 = u21 = np.mean((X - z1) * (Y - z2))  # eq(8.26)

    β = (u11 - u22) / (2 * u12) - np.sqrt(
        (u11 - u22) ** 2 / (4 * u12**2 + 1)
    )  # eq(8.27)
    t1 = np.sqrt(β**2 * u11 + 2 * β * u12 + u22) / np.sqrt(1 + β**2)  # eq(8.28)
    t2 = np.sqrt(u11 - 2 * β * u12 + β**2 * u22) / np.sqrt(1 + β**2)  # eq(8.29)
    v11 = (β**2 * t1 + t2) / (1 + β**2)  # eq(8.30)
    v22 = (t1 + β**2 * t2) / (1 + β**2)
    v12 = v21 = β * (t1 - t2) / (1 + β**2)  # eq(8.31)
    v0 = np.array([[v11, v12], [v21, v22]])

    beta = np.array([_circ_mean_resample(alpha, z0, v0) for i in range(B)]).flatten()

    # here we use HDI instead of the percentile method
    lb, ub = compute_hdi(beta, ci=ci)

    mean = circ_mean(beta)
    if not is_within_circular_range(mean, lb, ub):
        lb, ub = ub, lb

    return lb, ub

_circ_mean_resample(alpha, z0, v0)

Implementation of Section 8.3.5 (Fisher, 1993, P210)

Source code in pycircstat2/descriptive.py 
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def _circ_mean_resample(alpha, z0, v0):
    """
    Implementation of Section 8.3.5 (Fisher, 1993, P210)
    """

    θ = np.random.choice(alpha, len(alpha), replace=True)
    X = np.cos(θ)
    Y = np.sin(θ)

    # algo 1
    z1 = np.mean(X)  # eq(8.24)
    z2 = np.mean(Y)
    zB = np.array([z1, z2])

    u11 = np.mean((X - z1) ** 2)  # eq(8.25)
    u22 = np.mean((X - z2) ** 2)
    u12 = u21 = np.mean((X - z1) * (Y - z2))  # eq(8.26)

    # algo 3
    β = (u11 - u22) / (2 * u12) - np.sqrt(
        (u11 - u22) ** 2 / (4 * u12**2 + 1)
    )  # eq(8.27)
    t1 = np.sqrt(1 + β**2) / np.sqrt(β**2 * u11 + 2 * β * u12 + u22)  # eq(8.33)
    t2 = np.sqrt(1 + β**2) / np.sqrt(u11 - 2 * β * u12 + β**2 * u22)  # eq(8.34)
    w11 = (β**2 * t1 + t2) / (1 + β**2)  # eq(8.35)
    w22 = (t1 + β**2 * t2) / (1 + β**2)
    w12 = w21 = β * (t1 - t2) / (1 + β**2)  # eq(8.36)

    wB = np.array([[w11, w12], [w21, w22]])

    Cbar, Sbar = z0 + v0 @ wB @ (zB - z0)
    Cbar = np.power(Cbar**2 + Sbar**2, -0.5) * Cbar
    Sbar = np.power(Cbar**2 + Sbar**2, -0.5) * Sbar

    m = np.arctan2(Sbar, Cbar)

    return angmod(m)

circ_median_ci(median=None, alpha=None, w=None, method='deviation', ci=0.95)

Confidence interval for circular median

For n > 15, the confidence interval can be computed by:

\[ m = 1 + \text{integer part of} \frac{1}{2} n^{1/2} z_{\frac{1}{2}\alpha} \]

For n \(\le\) 15, the confidence interval can be selected from the table in Fisher (1993).

Parameters:

Name Type Description Default
median float

Circular median.

 None
alpha Optional[ndarray]

Data in radian.

 None
w Optional[ndarray]

Frequencies or weights

 None

Returns:

Type Description
lower, upper, ci: tuple

confidence intervals and alpha-level
Note

Implementation of section 4.4.2 (Fisher,1993)

Source code in pycircstat2/descriptive.py 
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def circ_median_ci(
    median: float = None,
    alpha: Optional[np.ndarray] = None,
    w: Optional[np.ndarray] = None,
    method: str = "deviation",
    ci: float = 0.95,
) -> tuple:
    r"""Confidence interval for circular median

    For n > 15, the confidence interval can be computed by:

    $$
    m = 1 + \text{integer part of} \frac{1}{2} n^{1/2} z_{\frac{1}{2}\alpha}
    $$

    For n $\le$ 15, the confidence interval can be selected from the table in Fisher (1993).

    Parameters
    ----------
    median: float or None
        Circular median.
    alpha: np.array or None
        Data in radian.
    w: np.array or None
        Frequencies or weights

    Returns
    -------
    lower, upper, ci: tuple
        confidence intervals and alpha-level

    Note
    ----
    Implementation of section 4.4.2 (Fisher,1993)
    """

    if median is None:
        assert isinstance(
            alpha, np.ndarray
        ), "If `median` is None, then `alpha` (and `w`) is needed."
        if w is None:
            w = np.ones_like(alpha)
        median = circ_median(alpha=alpha, w=w, method=method)

    if alpha is None:
        raise ValueError(
            "`alpha` is needed for computing the confidence interval for circular median."
        )

    n = len(alpha)
    alpha = np.sort(alpha)

    if n > 15:
        z = norm.ppf(1 - 0.5 * (1 - ci))

        offset = int(1 + np.floor(0.5 * np.sqrt(n) * z))  # fisher:eq(4.19)

        # idx_median = np.where(alpha.round(5) < np.round(median, 5))[0][-1]
        arr = np.where(alpha.round(5) < np.round(median, 5))[0]
        if len(arr) == 0:
            # That means median is smaller than alpha[0] (to 5 decimals).
            # In a circular sense, the “closest index below” is alpha[-1].
            idx_median = len(alpha) - 1
        else:
            idx_median = arr[-1]

        idx_lb = idx_median - offset + 1
        idx_ub = idx_median + offset
        if np.round(median, 5) in alpha.round(5):  # don't count the median per se
            idx_ub += 1

        if idx_ub > n:
            idx_ub = idx_ub - n

        if idx_lb < 0:
            idx_lb = n + idx_lb

        lower, upper = alpha[int(idx_lb)], alpha[int(idx_ub)]

        if not is_within_circular_range(median, lower, upper):
            lower, upper = upper, lower

    # selected confidence intervals for the median direction for n < 15
    # from A6, Fisher, 1993.
    # We only return the widest CI if there are more than one in the table.

    elif n == 3:
        lower, upper = alpha[0], alpha[2]
        ci = 0.75
    elif n == 4:
        lower, upper = alpha[0], alpha[3]
        ci = 0.875
    elif n == 5:
        lower, upper = alpha[0], alpha[4]
        ci = 0.937
    elif n == 6:
        lower, upper = alpha[0], alpha[5]
        ci = 0.97
    elif n == 7:
        lower, upper = alpha[0], alpha[6]
        ci = 0.984
    elif n == 8:
        lower, upper = alpha[0], alpha[7]
        ci = 0.992
    elif n == 9:
        lower, upper = alpha[0], alpha[8]
        ci = 0.996
    elif n == 10:
        lower, upper = alpha[1], alpha[8]
        ci = 0.978
    elif n == 11:
        lower, upper = alpha[1], alpha[9]
        ci = 0.99
    elif n == 12:
        lower, upper = alpha[2], alpha[9]
        ci = 0.962
    elif n == 13:
        lower, upper = alpha[2], alpha[10]
        ci = 0.978
    elif n == 14:
        lower, upper = alpha[3], alpha[10]
        ci = 0.937
    elif n == 15:
        lower, upper = alpha[2], alpha[12]
        ci = 0.965
    else:
        lower, upper = np.nan, np.nan

    return (angmod(lower), angmod(upper), ci)

circ_kappa(r, n=None)

Estimate kappa by approximation.

\[ \hat\kappa_{ML} = \begin{cases} 2r + r^3 + 5r^5/6, , & \text{if } r < 0.53 \\ -0.4 + 1.39 r + 0.43 / (1 - r) , & \text{if } 0.53 \le r < 0.85\\ 1 / (r^3 - 4r^2 + 3r), & \text{if } r \ge 0.85 \end{cases} \]

For \(n \le 15\):

\[ \hat\kappa = \begin{cases} \max\left(\hat\kappa - \frac{2}{n\hat\kappa}, 0\right), & \text{if } \hat\kappa < 2 \\ \frac{(n - 1)^3 \hat\kappa}{n^3 + n}, & \text{if } \hat\kappa \ge 2 \end{cases} \]

Parameters:

Name Type Description Default
r float

Resultant vector length

 required
n Union[int, None]

Sample size. If n is not None, the adjustment for small sample size will be applied.

 None

Returns:

Name Type Description
kappa float

Concentration parameter
Reference

Section 4.5.5 (P88, Fisher, 1993)

Source code in pycircstat2/descriptive.py 
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def circ_kappa(r: float, n: Union[int, None] = None) -> float:
    r"""Estimate kappa by approximation.

    $$
    \hat\kappa_{ML} =
    \begin{cases}
     2r + r^3 + 5r^5/6, , & \text{if } r < 0.53  \\
     -0.4 + 1.39 r + 0.43 / (1 - r) , & \text{if } 0.53 \le r < 0.85\\
        1 / (r^3 - 4r^2 + 3r), & \text{if } r \ge 0.85
    \end{cases}
    $$

    For $n \le 15$:

    $$
    \hat\kappa =
    \begin{cases}
        \max\left(\hat\kappa - \frac{2}{n\hat\kappa}, 0\right), & \text{if } \hat\kappa < 2 \\
        \frac{(n - 1)^3 \hat\kappa}{n^3 + n}, & \text{if } \hat\kappa \ge 2
    \end{cases}
    $$


    Parameters
    ----------
    r: float
        Resultant vector length
    n: int or None
        Sample size. If n is not None, the adjustment for small sample size will be applied.

    Returns
    -------
    kappa: float
        Concentration parameter

    Reference
    ---------
    Section 4.5.5 (P88, Fisher, 1993)
    """

    # eq 4.40
    if r < 0.53:
        kappa = 2 * r + r**3 + 5 * r**5 / 6
    elif r < 0.85:
        kappa = -0.4 + 1.39 * r + 0.43 / (1 - r)
    else:
        nom = r**3 - 4 * r**2 + 3 * r
        if nom != 0:
            kappa = 1 / nom
        else:
            # not sure how to handle this...
            kappa = 1e-16

    # eq 4.41
    if n is not None:
        if n <= 15 and r < 0.7:
            if kappa < 2:
                kappa = np.max([kappa - 2 * 1 / (n * kappa), 0])
            else:
                kappa = (n - 1) ** 3 * kappa / (n**3 + n)

    return kappa

circ_dist(x, y)

Compute the pairwise circular difference \(x_i - y_i\) using complex representation.

Parameters:

Name Type Description Default
x array - like

Sample of circular data (radians).

 required
y array - like

Sample of circular data (radians) or a single angle.

 required

Returns:

Type Description
array

Circular differences wrapped to [-pi, pi].
References
  • Section 27.7 (Zar, 2010, P642)
Source code in pycircstat2/descriptive.py 
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def circ_dist(x: np.ndarray, y: np.ndarray) -> np.ndarray:
    r"""
    Compute the pairwise circular difference $x_i - y_i$ using complex representation.

    Parameters
    ----------
    x : array-like
        Sample of circular data (radians).
    y : array-like
        Sample of circular data (radians) or a single angle.

    Returns
    -------
    array
        Circular differences wrapped to [-pi, pi].

    References
    ----------
    - Section 27.7 (Zar, 2010, P642)

    """
    return np.angle(np.exp(1j * x) / np.exp(1j * y))

circ_pairdist(x, y=None)

Compute all pairwise circular differences \(x_i - y_j\) around the circle.

Parameters:

Name Type Description Default
x array - like

Sample of circular data (radians).

 required
y array - like

Sample of circular data (radians). If None, computes all pairwise differences within x.

 None

Returns:

Type Description
ndarray

Matrix of pairwise circular differences wrapped to [-pi, pi].
References
  • Section 27.7 (Zar, 2010, P642)
Source code in pycircstat2/descriptive.py 
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def circ_pairdist(x: np.ndarray, y: Optional[np.ndarray]=None) -> np.ndarray:
    r"""
    Compute all pairwise circular differences $x_i - y_j$ around the circle.

    Parameters
    ----------
    x : array-like
        Sample of circular data (radians).
    y : array-like, optional
        Sample of circular data (radians). If None, computes all pairwise 
        differences within x.

    Returns
    -------
    ndarray
        Matrix of pairwise circular differences wrapped to [-pi, pi].

    References
    ----------
    - Section 27.7 (Zar, 2010, P642)
    """
    x = np.asarray(x)

    if y is None:
        y = x  # Compute pairwise distances within x itself

    y = np.asarray(y)

    # Broadcasting-friendly complex exponentiation method
    return np.angle(np.exp(1j * x[:, None]) / np.exp(1j * y[None, :]))

convert_moment(mp)

Convert complex moment to polar coordinates.

Parameters:

Name Type Description Default
mp complex

Complex moment

 required

Returns:

Name Type Description
u float

Angle in radian
r float

Magnitude
Source code in pycircstat2/descriptive.py 
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def convert_moment(
    mp: complex,
) -> Tuple[float, float]:
    """
    Convert complex moment to polar coordinates.

    Parameters
    ----------
    mp: complex
        Complex moment

    Returns
    -------
    u: float
        Angle in radian
    r: float
        Magnitude

    """

    u = angmod(np.angle(mp))
    r = np.abs(mp)

    return u, r

compute_C_and_S(alpha, w, p=1, mean=0.0)

Compute the intermediate values Cbar and Sbar.

\[ \displaylines{ \bar{C}_{p} = \frac{\sum_{i=1}^{n} w_{i} \cos(p(\alpha_{i} - \mu))}{n} \\ \bar{S}_{p} = \frac{\sum_{i=1}^{n} w_{i} \sin(p(\alpha_{i} - \mu))}{n} } \]

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian.

 required
w ndarray

Frequencies or weights.

 required
p int

Order of the moment (default is 1, for the first moment).

 1
mean Union[float, ndarray]

Mean angle (μ) to center the computation (default is 0.0).

 0.0

Returns:

Name Type Description
Cbar float

Weighted mean cosine for the given moment.
Sbar float

Weighted mean sine for the given moment.
Source code in pycircstat2/descriptive.py 
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def compute_C_and_S(
    alpha: np.ndarray,
    w: np.ndarray,
    p: int = 1,
    mean: Union[float, np.ndarray] = 0.0,
) -> Tuple[float, float]:
    r"""
    Compute the intermediate values Cbar and Sbar.

    $$
    \displaylines{
    \bar{C}_{p} = \frac{\sum_{i=1}^{n} w_{i} \cos(p(\alpha_{i} - \mu))}{n} \\
    \bar{S}_{p} = \frac{\sum_{i=1}^{n} w_{i} \sin(p(\alpha_{i} - \mu))}{n}
    }
    $$

    Parameters
    ----------
    alpha: np.ndarray
        Angles in radian.
    w: np.ndarray
        Frequencies or weights.
    p: int, optional
        Order of the moment (default is 1, for the first moment).
    mean: float, optional
        Mean angle (μ) to center the computation (default is 0.0).

    Returns
    -------
    Cbar: float
        Weighted mean cosine for the given moment.
    Sbar: float
        Weighted mean sine for the given moment.
    """
    n = np.sum(w)
    Cbar = np.sum(w * np.cos(p * (alpha - mean))) / n
    Sbar = np.sum(w * np.sin(p * (alpha - mean))) / n

    return Cbar, Sbar

compute_hdi(samples, ci=0.95)

Compute the Highest Density Interval (HDI) for circular data.

Parameters:

Name Type Description Default
samples ndarray

Bootstrap samples of the circular mean in radians.

 required
ci float

Credible interval (default is 0.95 for 95% HDI).

 0.95

Returns:

Name Type Description
hdi tuple

Lower and upper bounds of the HDI in radians.
Source code in pycircstat2/descriptive.py 
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def compute_hdi(samples, ci=0.95):
    """
    Compute the Highest Density Interval (HDI) for circular data.

    Parameters
    ----------
    samples : np.ndarray
        Bootstrap samples of the circular mean in radians.
    ci : float, optional
        Credible interval (default is 0.95 for 95% HDI).

    Returns
    -------
    hdi : tuple
        Lower and upper bounds of the HDI in radians.
    """
    # Wrap samples to [0, 2π) for circular consistency
    wrapped_samples = angmod(samples)

    # Sort the samples
    sorted_samples = np.sort(wrapped_samples)

    # Number of samples in the HDI
    n_samples = len(sorted_samples)
    interval_idx = int(np.floor(ci * n_samples))
    if interval_idx == 0:
        raise ValueError("Insufficient data to compute HDI.")

    # Find the shortest interval
    hdi_width = np.inf
    hdi_bounds = (None, None)
    for i in range(n_samples - interval_idx):
        lower = sorted_samples[i]
        upper = sorted_samples[i + interval_idx]
        width = angmod(upper - lower)  # Handle wrapping for circularity
        if width < hdi_width:
            hdi_width = width
            hdi_bounds = (lower, upper)

    return hdi_bounds

compute_smooth_params(r, n)

Parameters:

Name Type Description Default
r float

resultant vector length

 required
n int

sample size

 required

Returns:

Name Type Description
h float

smoothing parameter
Reference

Section 2.2 (P26, Fisher, 1993)

Source code in pycircstat2/descriptive.py 
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def compute_smooth_params(r: float, n: int) -> float:
    """
    Parameters
    ----------
    r: float
        resultant vector length
    n: int
        sample size

    Returns
    -------
    h: float
        smoothing parameter

    Reference
    ---------
    Section 2.2 (P26, Fisher, 1993)
    """

    kappa = circ_kappa(r, n)
    l = 1 / np.sqrt(kappa)  # eq 2.3
    h = np.sqrt(7) * l / np.power(n, 0.2)  # eq 2.4

    return h

nonparametric_density_estimation(alpha, h, radius=1)

Nonparametric density estimates with a quartic kernel function.

Parameters:

Name Type Description Default
alpha ndarray

Angles in radian

 required
h float

Smoothing parameters

 required
radius float

radius of the plotted circle

 1

Returns:

Name Type Description
x ndarray(100)

grid
f ndarray(100)

density
Reference

Section 2.2 (P26, Fisher, 1993)

Source code in pycircstat2/descriptive.py 
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def nonparametric_density_estimation(
    alpha: np.ndarray,  # angles in radian
    h: float,  # smoothing parameters
    radius: float = 1,  # radius of the plotted circle
) -> tuple:
    """Nonparametric density estimates with
    a quartic kernel function.

    Parameters
    ----------
    alpha: np.ndarray (n, )
        Angles in radian
    h: float
        Smoothing parameters
    radius: float
        radius of the plotted circle

    Returns
    -------
    x: np.ndarray (100, )
        grid
    f: np.ndarray (100, )
        density

    Reference
    ---------
    Section 2.2 (P26, Fisher, 1993)
    """

    # vectorized version of step 3
    a = alpha
    n = len(a)
    x = np.linspace(0, 2 * np.pi, 100)
    d = np.abs(x[:, None] - a)
    e = np.minimum(d, 2 * np.pi - d)
    e = np.minimum(e, h)
    sum = np.sum((1 - e**2 / h**2) ** 2, 1)
    f = 0.9375 * sum / n / h

    f = radius * np.sqrt(1 + np.pi * f) - radius

    return x, f

circ_range(alpha)

Compute the circular range of angular data.

The circular range is the difference between the maximum and minimum angles in the dataset, adjusted for circular continuity.

Parameters:

Name Type Description Default
alpha ndarray

Angles in radians.

 required

Returns:

Type Description
float

Circular range, a measure of clustering (higher = more clustered).
Reference

P162, Section 7.2.3 of Jammalamadaka, S. Rao and SenGupta, A. (2001)

Source code in pycircstat2/descriptive.py 
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def circ_range(alpha: np.ndarray) -> float:
    """
    Compute the circular range of angular data.

    The circular range is the difference between the maximum and minimum angles 
    in the dataset, adjusted for circular continuity.

    Parameters
    ----------
    alpha : np.ndarray
        Angles in radians.

    Returns
    -------
    float
        Circular range, a measure of clustering (higher = more clustered).

    Reference
    ---------
    P162, Section 7.2.3 of Jammalamadaka, S. Rao and SenGupta, A. (2001)
    """
    alpha = np.sort(alpha % (2 * np.pi))  # Convert to [0, 2π) and sort
    spacings = np.diff(alpha, prepend=alpha[-1] - 2 * np.pi)  # Compute spacings
    return 2 * np.pi - np.max(spacings)  # Circular range

circ_quantile(alpha, probs=np.array([0, 0.25, 0.5, 0.75, 1.0]), type=7)

Compute quantiles for circular data.

This function computes quantiles for circular data by shifting the data to be centered around the circular median, applying a linear quantile function, and then shifting back.

Parameters:

Name Type Description Default
alpha ndarray

Sample of circular data (radians).

 required
probs float or ndarray

Probabilities at which to compute quantiles. Default is [0, 0.25, 0.5, 0.75, 1.0].

 array([0, 0.25, 0.5, 0.75, 1.0])
type int

Quantile algorithm type (default 7, matches R’s default quantile type).

 7

Returns:

Type Description
ndarray

Circular quantiles.
References
  • R's quantile.circular from the circular package.
  • Fisher (1993), Section 2.3.2.
Source code in pycircstat2/descriptive.py 
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def circ_quantile(
    alpha: np.ndarray,
    probs: Union[float, np.ndarray] = np.array([0, 0.25, 0.5, 0.75, 1.0]),
    type: int = 7,
) -> np.ndarray:
    """
    Compute quantiles for circular data.

    This function computes quantiles for circular data by shifting the 
    data to be centered around the circular median, applying a linear quantile function,
    and then shifting back.

    Parameters
    ----------
    alpha : np.ndarray
        Sample of circular data (radians).
    probs : float or np.ndarray, optional
        Probabilities at which to compute quantiles. Default is `[0, 0.25, 0.5, 0.75, 1.0]`.
    type : int, optional
        Quantile algorithm type (default `7`, matches R’s default quantile type).

    Returns
    -------
    np.ndarray
        Circular quantiles.

    References
    ----------
    - R's `quantile.circular` from the `circular` package.
    - Fisher (1993), Section 2.3.2.
    """

    # Convert to numpy array
    alpha = np.asarray(alpha)
    probs = np.atleast_1d(probs)

    # Compute circular median
    circular_median = circ_median(alpha)

    # If the median is NaN (e.g., uniform data), return NaNs
    if np.isnan(circular_median):
        return np.full_like(probs, np.nan)

    # Transform data relative to circular median
    shifted_alpha = (alpha - circular_median) % (2 * np.pi)
    shifted_alpha = np.where(shifted_alpha > np.pi, shifted_alpha - 2 * np.pi, shifted_alpha)

    # Compute linear quantiles on transformed data
    linear_quantiles = np.quantile(shifted_alpha, probs, method="linear" if type == 7 else "midpoint")

    # Transform back to original circular space
    circular_quantiles = (linear_quantiles + circular_median) % (2 * np.pi)

    return circular_quantiles