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Angular Correlation

aacorr(a, b, method='fl', test=False, strict=True)

Angular-Angular Correlation.

Parameters:

Name Type Description Default
a Union[Type[Circular], ndarray]

Angles in radian

 required
b Union[Type[Circular], ndarray]

Angles in radian

 required
method str
  • 'fl' (Fisher & Lee)
  • 'js' (Jammalamadaka & SenGupta)
  • 'nonparametric'
 'fl'
test bool

Return significant test results.

 False
strict bool

Strict mode. If True, raise an error when mean direction is not significant. Only for Jammalamadaka & SenGupta (2001)

 True

Returns:

Name Type Description
r float

Correlation coefficient.
reject bool

Return significant test if test is set to True.
Source code in pycircstat2/correlation.py 
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def aacorr(
    a: Union[Type[Circular], np.ndarray],
    b: Union[Type[Circular], np.ndarray],
    method: str = "fl",
    test: bool = False,
    strict: bool = True,
) -> tuple:
    """
    Angular-Angular Correlation.

    Parameters
    ----------
    a: Circular or np.ndarray
        Angles in radian
    b: Circular or np.ndarray
        Angles in radian
    method: str
        - 'fl' (Fisher & Lee)
        - 'js' (Jammalamadaka & SenGupta)
        - 'nonparametric'
    test: bool
        Return significant test results.
    strict: bool
        Strict mode. If True, raise an error when mean direction is
        not significant. Only for Jammalamadaka & SenGupta (2001)

    Returns
    -------
    r: float
        Correlation coefficient.
    reject: bool
        Return significant test if `test` is set to True.
    """

    if method == "fl":  # Fisher & Lee (1983)
        _corr = _aacorr_fl
    elif method == "js":  # Jammalamadaka & SenGupta (2001)
        _corr = _aacorr_js
    elif method == "nonparametric":
        _corr = _aacorr_np

    r = _corr(a, b, strict)

    if test:
        if isinstance(a, Circular):
            a = a.alpha
        if isinstance(b, Circular):
            b = b.alpha
        assert len(a) == len(b), "`a` and `b` must be the same length."

        if method == "nonparametric":
            # assuming α=0.05, critical values from P661, Zar, 2010
            n = len(a)
            reject = (n - 1) * r > 2.99 + 2.16 / n
        else:
            # jackknife test (Fingleton, 1989)
            n = len(a)
            raas = [
                _corr(np.delete(a, i), np.delete(b, i), strict)
                for i in range(n)
            ]
            m_raas = np.mean(raas)
            s2_raas = np.var(raas, ddof=1)
            z = norm.ppf(0.975)
            lb = n * r - (n - 1) * m_raas - z * np.sqrt(s2_raas / n)
            ub = n * r - (n - 1) * m_raas + z * np.sqrt(s2_raas / n)

            reject = ~(lb <= 0 <= ub)

        return r, reject
    else:
        return r

alcorr(a, x)

Angular-Linear Correlation based on Mardia (1972)

Parameters:

Name Type Description Default
a Union[Type[Circular], ndarray]

Angles in radian

 required
x ndarray

Linear variable

 required

Returns:

Name Type Description
ral float

correlation coefficient.
pval float
Reference

P658-659, Section 27.15(b) of Example 27.21 (Zar, 2010).

Source code in pycircstat2/correlation.py 
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def alcorr(
    a: Union[Type[Circular], np.ndarray],
    x: np.ndarray,
) -> float:
    """Angular-Linear Correlation based on Mardia (1972)

    Parameters
    ----------
    a: Circular or np.ndarray
        Angles in radian
    x: np.ndarray
        Linear variable

    Returns
    -------
    ral: float
        correlation coefficient.
    pval: float

    Reference
    ----
    P658-659, Section 27.15(b) of Example 27.21 (Zar, 2010).
    """

    if isinstance(a, Circular):
        a = a.alpha
    assert len(a) == len(x), "`a` and `x` must be the same length."

    n = len(a)

    rxc = np.corrcoef(np.cos(a), x)[0, 1]
    rxs = np.corrcoef(x, np.sin(a))[0, 1]
    rcs = np.corrcoef(np.sin(a), np.cos(a))[0, 1]

    num = rxc**2 + rxs**2 - 2 * rxc * rxs * rcs
    den = 1 - rcs**2
    ral = np.sqrt(num / den)

    pval = 1 - chi2(df=2).cdf(n * ral**2)

    return ral, pval