# scipy.stats.theilslopes

scipy.stats.theilslopes(y, x=None, alpha=0.95, method='separate')

  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’.

      New in version 1.8.0.

  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)

  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()