code: def theilslopes(y, x=None, alpha=0.95):
      r"""
      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".
  
      Returns
      -------
      medslope : float
          Theil slope.
      medintercept : float
          Intercept of the Theil line, as ``median(y) - medslope*median(x)``.
      lo_slope : float
          Lower bound of the confidence interval on `medslope`.
      up_slope : float
          Upper bound of the confidence interval on `medslope`.
  
      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) -
      medslope*median(x)``, which is given in [3]_. Other definitions of
      the intercept exist in the literature. A confidence interval for
      the intercept is not given as this question is not addressed in
      [1]_.
  
      References
      ----------
      .. [1] 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.
  
      Examples
      --------
      >>> 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)
      >>> 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()
  
      """
      # We copy both x and y so we can use _find_repeats.
      y = np.array(y).flatten()
      if x is None:
          x = np.arange(len(y), dtype=float)
      else:
          x = np.array(x, dtype=float).flatten()
          if len(x) != len(y):
              raise ValueError(
                  "Incompatible lengths ! (%s<>%s)" % (len(y), len(x))
              )
  
      # Compute sorted slopes only when deltax > 0
      deltax = x[:, np.newaxis] - x
      deltay = y[:, np.newaxis] - y
      slopes = deltay[deltax > 0] / deltax[deltax > 0]
      slopes.sort()
      medslope = np.median(slopes)
      medinter = np.median(y) - medslope * np.median(x)
      # Now compute confidence intervals
      if alpha > 0.5:
          alpha = 1.0 - alpha
  
      z = distributions.norm.ppf(alpha / 2.0)
      # This implements (2.6) from Sen (1968)
      _, nxreps = _find_repeats(x)
      _, nyreps = _find_repeats(y)
      nt = len(slopes)  # N in Sen (1968)
      ny = len(y)  # n in Sen (1968)
      # Equation 2.6 in Sen (1968):
      sigsq = (
          1
          / 18.0
          * (
              ny * (ny - 1) * (2 * ny + 5)
              - sum(k * (k - 1) * (2 * k + 5) for k in nxreps)
              - sum(k * (k - 1) * (2 * k + 5) for k in nyreps)
          )
      )
      # Find the confidence interval indices in `slopes`
      sigma = np.sqrt(sigsq)
      Ru = min(int(np.round((nt - z * sigma) / 2.0)), len(slopes) - 1)
      Rl = max(int(np.round((nt + z * sigma) / 2.0)) - 1, 0)
      delta = slopes[[Rl, Ru]]
      return medslope, medinter, delta[0], delta[1]
  
file_path: catboost/contrib/python/scipy/py3/scipy/stats/_stats_mstats_common.py
imports: import numpy
  import scipy.stats.stats
  from .None import distributions
  from .._lib._bunch import _make_tuple_bunch
label_desc: Use image node_id 2 for calling a global function with example usage: theilslopes(y, x, alpha) and returns: medslope,  medinter,  delta,  delta