code: def theilslopes(y, x=None, alpha=0.95, method="separate"):
      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".
      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'.
  
          .. versionadded:: 1.8.0
  
      Returns
      -------
      result : ``TheilslopesResult`` instance
          The return value is an object with the following attributes:
  
          slope : float
              Theil slope.
          intercept : float
              Intercept of the Theil line.
          low_slope : float
              Lower bound of the confidence interval on `slope`.
          high_slope : float
              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] 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()
  
      """
      if method not in ["joint", "separate"]:
          raise ValueError(
              "method must be either 'joint' or 'separate'."
              f"'{method}' is invalid."
          )
      # We copy both x and y so we can use _find_repeats.
      y = np.array(y, dtype=float, copy=True).ravel()
      if x is None:
          x = np.arange(len(y), dtype=float)
      else:
          x = np.array(x, dtype=float, copy=True).ravel()
          if len(x) != len(y):
              raise ValueError(
                  f"Incompatible lengths ! ({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]
      if not slopes.size:
          msg = "All `x` coordinates are identical."
          warnings.warn(msg, RuntimeWarning, stacklevel=2)
      slopes.sort()
      medslope = np.median(slopes)
      if method == "joint":
          medinter = np.median(y - medslope * x)
      else:
          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`
      try:
          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]]
      except (ValueError, IndexError):
          delta = (np.nan, np.nan)
  
      return TheilslopesResult(
          slope=medslope,
          intercept=medinter,
          low_slope=delta[0],
          high_slope=delta[1],
      )
  
file_path: scipy/scipy/stats/_stats_mstats_common.py
imports: import warnings
  import numpy
  import scipy.stats._stats_py
  from .None import distributions
  from .._lib._bunch import _make_tuple_bunch
  from ._stats_pythran import siegelslopes
label_desc: Use image node_id 2 for calling a global function with example usage: theilslopes(y, x, alpha, method) and returns: TheilslopesResult