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scipy.stats.

# circvar

scipy.stats.circvar(samples, high=6.283185307179586, low=0, axis=None, nan_policy='propagate', *, keepdims=False)

  Compute the circular variance of a sample of angle observations.

  Given \(n\) angle observations \(x_1, \cdots, x_n\) measured in radians, their circular variance is defined by ([2], Eq. 2.3.3)

  \[1 - \left| \frac{1}{n} \sum_{k=1}^n e^{i x_k} \right|\]

  where \(i\) is the imaginary unit and \(|z|\) gives the length of the complex number \(z\). \(|z|\) in the above expression is known as the mean resultant length.

  Parameters:
    samples array_like

      Input array of angle observations. The value of a full angle is equal to `(high - low)`.

    high float, optional

      Upper boundary of the principal value of an angle. Default is `2*pi`.

    low float, optional

      Lower boundary of the principal value of an angle. Default is `0`.

    axis int or None, default: None

      If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If `None`, the input will be raveled before computing the statistic.

    nan_policy{‘propagate’, ‘omit’, ‘raise’}

      Defines how to handle input NaNs.

      - `propagate`: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `omit`: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `raise`: if a NaN is present, a `ValueError` will be raised.

    keepdims bool, default: False

      If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

  Returns:
    circvar float

      Circular variance. The returned value is in the range `[0, 1]`, where `0` indicates no variance and `1` indicates large variance.

      If the input array is empty, `np.nan` is returned.

  See also

  `circmean`

    Circular mean.

  `circstd`

    Circular standard deviation.

  Notes

  In the limit of small angles, the circular variance is close to half the ‘linear’ variance if measured in radians.

  Beginning in SciPy 1.9, `np.matrix` inputs (not recommended for new code) are converted to `np.ndarray` before the calculation is performed. In this case, the output will be a scalar or `np.ndarray` of appropriate shape rather than a 2D `np.matrix`. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or `np.ndarray` rather than a masked array with `mask=False`.

  Array API Standard Support

  `circvar` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  `circvar` also accepts MArrays backed by the backends indicated above; masked values will be treated as though they were not present.

  See Support for the array API standard for more information.

  References

  [1]

  Fisher, N.I. Statistical analysis of circular data. Cambridge University Press, 1993.

  [2]

  Mardia, K. V. and Jupp, P. E. Directional Statistics. John Wiley & Sons, 1999.

  Examples

      >>> import numpy as np
      >>> from scipy.stats import circvar
      >>> import matplotlib.pyplot as plt
      >>> samples_1 = np.array([0.072, -0.158, 0.077, 0.108, 0.286,
      ...                       0.133, -0.473, -0.001, -0.348, 0.131])
      >>> samples_2 = np.array([0.111, -0.879, 0.078, 0.733, 0.421,
      ...                       0.104, -0.136, -0.867,  0.012,  0.105])
      >>> circvar_1 = circvar(samples_1)
      >>> circvar_2 = circvar(samples_2)

  Plot the samples.

      >>> fig, (left, right) = plt.subplots(ncols=2)
      >>> for image in (left, right):
      ...     image.plot(np.cos(np.linspace(0, 2*np.pi, 500)),
      ...                np.sin(np.linspace(0, 2*np.pi, 500)),
      ...                c='k')
      ...     image.axis('equal')
      ...     image.axis('off')
      >>> left.scatter(np.cos(samples_1), np.sin(samples_1), c='k', s=15)
      >>> left.set_title(f"circular variance: {np.round(circvar_1, 2)!r}")
      >>> right.scatter(np.cos(samples_2), np.sin(samples_2), c='k', s=15)
      >>> right.set_title(f"circular variance: {np.round(circvar_2, 2)!r}")
      >>> plt.show()