LOWESS : Locally weighted regression

lowess.py

"""
This module implements the Lowess function for nonparametric regression.

Functions:
lowess Fit a smooth nonparametric regression curve to a scatterplot.

For more information, see

William S. Cleveland: "Robust locally weighted regression and smoothing
scatterplots", Journal of the American Statistical Association, December 1979,
volume 74, number 368, pp. 829-836.

William S. Cleveland and Susan J. Devlin: "Locally weighted regression: An
approach to regression analysis by local fitting", Journal of the American
Statistical Association, September 1988, volume 83, number 403, pp. 596-610.
"""

# Authors: Alexandre Gramfort <[email protected]>
#
# License: BSD (3-clause)

from math import ceil
import numpy as np
from scipy import linalg


def lowess(x, y, f=2. / 3., iter=3):
    """lowess(x, y, f=2./3., iter=3) -> yest

    Lowess smoother: Robust locally weighted regression.
    The lowess function fits a nonparametric regression curve to a scatterplot.
    The arrays x and y contain an equal number of elements; each pair
    (x[i], y[i]) defines a data point in the scatterplot. The function returns
    the estimated (smooth) values of y.

    The smoothing span is given by f. A larger value for f will result in a
    smoother curve. The number of robustifying iterations is given by iter. The
    function will run faster with a smaller number of iterations.
    """
    n = len(x)
    r = int(ceil(f * n))
    h = [np.sort(np.abs(x - x[i]))[r] for i in range(n)]
    w = np.clip(np.abs((x[:, None] - x[None, :]) / h), 0.0, 1.0)
    w = (1 - w ** 3) ** 3
    yest = np.zeros(n)
    delta = np.ones(n)
    for iteration in range(iter):
        for i in range(n):
            weights = delta * w[:, i]
            b = np.array([np.sum(weights * y), np.sum(weights * y * x)])
            A = np.array([[np.sum(weights), np.sum(weights * x)],
                          [np.sum(weights * x), np.sum(weights * x * x)]])
            beta = linalg.solve(A, b)
            yest[i] = beta[0] + beta[1] * x[i]

        residuals = y - yest
        s = np.median(np.abs(residuals))
        delta = np.clip(residuals / (6.0 * s), -1, 1)
        delta = (1 - delta ** 2) ** 2

    return yest

if __name__ == '__main__':
    import math
    n = 100
    x = np.linspace(0, 2 * math.pi, n)
    y = np.sin(x) + 0.3 * np.random.randn(n)

    f = 0.25
    yest = lowess(x, y, f=f, iter=3)

    import pylab as pl
    pl.clf()
    pl.plot(x, y, label='y noisy')
    pl.plot(x, yest, label='y pred')
    pl.legend()
    pl.show()

Good code. But a minor suggestion that might take into consideration is that if input x is integers, the w matrix might have problem and result in singular matrix in solving linear systems. I would add a conversion sentence such as x = x.astype('float') before implementing w = np.clip(np.abs((x[:,None] - x[None,:]) / h), 0.0, 1.0). Or simply force h to be float.

Hi Alex,
Do you mind adding a licence?
Cheers,
N

done

Thanks for the code.
Is it possible to extend this to 2D? I'm looking for a solution that works on data in the format of maps/images. Is it possible to allow the local subset to be non-square, e.g. 10 grid wide x 6 grid height?

This is really cool, but it doesn't quite do what I need. I guess what I'm looking for is a gist that estimates beta[1] and the standard error for beta[1]. If I were a little smarter I could probably figure out how to modify this.

Thanks, this is very nice. However, I have an issue. When I change f I often get this error:
numpy.linalg.linalg.LinAlgError: Singular matrix

What can I do about it?

Thanks for the script. Just a comment: according to Cleveland (1979), "h_i is the rth smallest number among | x_i - x_j |, for j = 1, ..., n". So I think it would be necessary to add np.unique() in line 42, in the computation of h, as follows:

h = [np.unique(np.sort(np.abs(x - x[i])))[r] for i in range(n)]

to consider the case of equally separated values in x. Am I wrong?

Thanks again.

I don't know and I don't have much time to look. Feel free to fork and update the code.

Nice implemented.
It would be great to have the functionality that deals with unequally weighted data points, so called weighted regression

For what it is worth with @tjof2, we have tweaked this version to avoid the numpy.linalg.linalg.LinAlgError: Singular matrix and speed it up with numba; see https://gist.github.com/ericpre/7a4dfba660bc8bb7499e7d96b8bdd4bb

THis is very nice

Thanks @agramfort, this was really handy in my research.

@ericpre, to remove the Singular matrix error is the only change required: beta = np.linalg.lstsq(A, b)[0]?

@AyrtonB, I don't remember, you will need to figure it out yourself, sorry!

No worries @ericpre, think I've managed to work it out.

I've implemented an sklearn compatible version of this that also allows quantile predictions and confidence intervals to be calculated. I've also added the option to specify the locations at which the local regressions are calculated which significantly reduces the run-time.

Details can be found here

Thanks you very much!