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scipy.stats.

# gstd

scipy.stats.gstd(a, axis=0, ddof=1, *, keepdims=False, nan_policy='propagate')

  Calculate the geometric standard deviation of an array.

  The geometric standard deviation describes the spread of a set of numbers where the geometric mean is preferred. It is a multiplicative factor, and so a dimensionless quantity.

  It is defined as the exponential of the standard deviation of the natural logarithms of the observations.

  Parameters:
    a array_like

      An array containing finite, strictly positive, real numbers.

    axis int, tuple or None, optional

      Axis along which to operate. Default is 0. If None, compute over the whole array a.

    ddof int, optional

      Degree of freedom correction in the calculation of the geometric standard deviation. Default is 1.

    keepdims boolean, optional

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

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

      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.

  Returns:
    gstd ndarray or float

      An array of the geometric standard deviation. If axis is None or a is a 1d array a float is returned.

  See also

  `gmean`

    Geometric mean

  `numpy.std`

    Standard deviation

  `gzscore`

    Geometric standard score

  Notes

  Mathematically, the sample geometric standard deviation \(s_G\) can be defined in terms of the natural logarithms of the observations \(y_i = \log(x_i)\):

  \[s_G = \exp(s), \quad s = \sqrt{\frac{1}{n - d} \sum_{i=1}^n (y_i - \bar y)^2}\]

  where \(n\) is the number of observations, \(d\) is the adjustment ddof to the degrees of freedom, and \(\bar y\) denotes the mean of the natural logarithms of the observations. Note that the default `ddof=1` is different from the default value used by similar functions, such as `numpy.std` and `numpy.var`.

  When an observation is infinite, the geometric standard deviation is NaN (undefined). Non-positive observations will also produce NaNs in the output because the natural logarithm (as opposed to the complex logarithm) is defined and finite only for positive reals. The geometric standard deviation is sometimes confused with the exponential of the standard deviation, `exp(std(a))`. Instead, the geometric standard deviation is `exp(std(log(a)))`.

  `gstd` 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

  See Support for the array API standard for more information.

  References

  [1]

  “Geometric standard deviation”, Wikipedia, https://en.wikipedia.org/wiki/Geometric_standard_deviation.

  [2]

  Kirkwood, T. B., “Geometric means and measures of dispersion”, Biometrics, vol. 35, pp. 908-909, 1979

  Examples

  Find the geometric standard deviation of a log-normally distributed sample. Note that the standard deviation of the distribution is one; on a log scale this evaluates to approximately `exp(1)`.

      >>> import numpy as np
      >>> from scipy.stats import gstd
      >>> rng = np.random.default_rng()
      >>> sample = rng.lognormal(mean=0, sigma=1, size=1000)
      >>> gstd(sample)
      2.810010162475324

  Compute the geometric standard deviation of a multidimensional array and of a given axis.

      >>> a = np.arange(1, 25).reshape(2, 3, 4)
      >>> gstd(a, axis=None)
      2.2944076136018947
      >>> gstd(a, axis=2)
      array([[1.82424757, 1.22436866, 1.13183117],
             [1.09348306, 1.07244798, 1.05914985]])
      >>> gstd(a, axis=(1,2))
      array([2.12939215, 1.22120169])