# scipy.stats.siegelslopes

scipy.stats.siegelslopes(y, x=None, method='hierarchical')

  Computes the Siegel estimator for a set of points (x, y).

  `siegelslopes` implements a method for robust linear regression using repeated medians (see [1]) to fit a line to the points (x, y). The method is robust to outliers with an asymptotic breakdown point of 50%.

  Parameters:
    y array_like

      Dependent variable.

    x array_like or None, optional

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

    method{‘hierarchical’, ‘separate’}

      If ‘hierarchical’, estimate the intercept using the estimated slope `slope` (default option). If ‘separate’, estimate the intercept independent of the estimated slope. See Notes for details.

  Returns:
    result `SiegelslopesResult` instance

      The return value is an object with the following attributes:

      slopefloat

        Estimate of the slope of the regression line.

      interceptfloat

        Estimate of the intercept of the regression line.

  See also

  `theilslopes`

    a similar technique without repeated medians

  Notes

  With `n = len(y)`, compute `m_j` as the median of the slopes from the point `(x[j], y[j])` to all other n-1 points. `slope` is then the median of all slopes `m_j`. Two ways are given to estimate the intercept in [1] which can be chosen via the parameter `method`. The hierarchical approach uses the estimated slope `slope` and computes `intercept` as the median of `y - slope*x`. The other approach estimates the intercept separately as follows: for each point `(x[j], y[j])`, compute the intercepts of all the n-1 lines through the remaining points and take the median `i_j`. `intercept` is the median of the `i_j`.

  The implementation computes n times the median of a vector of size n which can be slow for large vectors. There are more efficient algorithms (see [2]) which are not implemented here.

  For compatibility with older versions of SciPy, the return value acts like a `namedtuple` of length 2, with fields `slope` and `intercept`, so one can continue to write:

      slope, intercept = siegelslopes(y, x)

  References

  [1] (1,2)

  A. Siegel, “Robust Regression Using Repeated Medians”, Biometrika, Vol. 69, pp. 242-244, 1982.

  [2]

  A. Stein and M. Werman, “Finding the repeated median regression line”, Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 409-413, 1992.

  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 and intercept. For comparison, also compute the least-squares fit with `linregress`:

      >>> res = stats.siegelslopes(y, x)
      >>> lsq_res = stats.linregress(x, y)

  Plot the results. The Siegel regression line is shown in red. 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, lsq_res[1] + lsq_res[0] * x, 'g-')
      >>> plt.show()