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scipy.stats.

# theilslopes

scipy.stats.theilslopes(y, x=None, alpha=0.95, method='separate', *, axis=None, nan_policy='propagate', keepdims=False)

  Computes the Theil-Sen estimator for a set of points (x, y).

  `theilslopes` implements a method for robust linear regression. It computes the slope as the median of all slopes between paired values.

  Parameters:
    y array_like

      Dependent variable.

    x array_like or None, optional

      Independent variable. If None, use `arange(len(y))` instead.

    alpha float, optional

      Confidence degree between 0 and 1. Default is 95% confidence. Note that `alpha` is symmetric around 0.5, i.e. both 0.1 and 0.9 are interpreted as “find the 90% confidence interval”.

    method{‘joint’, ‘separate’}, optional

      Method to be used for computing estimate for intercept. Following methods are supported,

      - ‘joint’: Uses `np.median(y - slope * x)` as intercept.

      - ‘separate’: Uses `np.median(y) - slope * np.median(x)`

          as intercept.

      The default is ‘separate’.

      Added in version 1.8.0.

    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:
    result `TheilslopesResult` instance

      The return value is an object with the following attributes:

      slopefloat

        Theil slope.

      interceptfloat

        Intercept of the Theil line.

      low_slopefloat

        Lower bound of the confidence interval on slope.

      high_slopefloat

        Upper bound of the confidence interval on slope.

  See also

  `siegelslopes`

    a similar technique using repeated medians

  Notes

  The implementation of `theilslopes` follows [1]. The intercept is not defined in [1], and here it is defined as `median(y) - slope*median(x)`, which is given in [3]. Other definitions of the intercept exist in the literature such as `median(y - slope*x)` in [4]. The approach to compute the intercept can be determined by the parameter `method`. A confidence interval for the intercept is not given as this question is not addressed in [1].

  For compatibility with older versions of SciPy, the return value acts like a `namedtuple` of length 4, with fields `slope`, `intercept`, `low_slope`, and `high_slope`, so one can continue to write:

      slope, intercept, low_slope, high_slope = theilslopes(y, x)

  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

  `theilslopes` 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

  `theilslopes` 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] (1,2,3)

  P.K. Sen, “Estimates of the regression coefficient based on Kendall’s tau”, J. Am. Stat. Assoc., Vol. 63, pp. 1379-1389, 1968.

  [2]

  H. Theil, “A rank-invariant method of linear and polynomial regression analysis I, II and III”, Nederl. Akad. Wetensch., Proc. 53:, pp. 386-392, pp. 521-525, pp. 1397-1412, 1950.

  [3]

  W.L. Conover, “Practical nonparametric statistics”, 2nd ed., John Wiley and Sons, New York, pp. 493.

  [4]

  https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator

  Examples

      >>> import numpy as np
      >>> from scipy import stats
      >>> import matplotlib.pyplot as plt

      >>> x = np.linspace(-5, 5, num=150)
      >>> y = x + np.random.normal(size=x.size)
      >>> y[11:15] += 10  # add outliers
      >>> y[-5:] -= 7

  Compute the slope, intercept and 90% confidence interval. For comparison, also compute the least-squares fit with `linregress`:

      >>> res = stats.theilslopes(y, x, 0.90, method='separate')
      >>> lsq_res = stats.linregress(x, y)

  Plot the results. The Theil-Sen regression line is shown in red, with the dashed red lines illustrating the confidence interval of the slope (note that the dashed red lines are not the confidence interval of the regression as the confidence interval of the intercept is not included). The green line shows the least-squares fit for comparison.

      >>> fig = plt.figure()
      >>> ax = fig.add_subplot(111)
      >>> ax.plot(x, y, 'b.')
      >>> ax.plot(x, res[1] + res[0] * x, 'r-')
      >>> ax.plot(x, res[1] + res[2] * x, 'r--')
      >>> ax.plot(x, res[1] + res[3] * x, 'r--')
      >>> ax.plot(x, lsq_res[1] + lsq_res[0] * x, 'g-')
      >>> plt.show()