scipy.stats.siegelslopes(y, x=None, method='hierarchical')[source]#

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%.


Dependent variable.

xarray_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.

resultSiegelslopesResult instance

The return value is an object with the following attributes:


Estimate of the slope of the regression line.


Estimate of the intercept of the regression line.

See also


a similar technique without repeated medians


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)


[1] (1,2)

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


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.


>>> 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-')