Outliers and robust fits#

Least squares gives every point full weight, so a single outlier can pull the fit away from the bulk of the data. A robust loss downweights points that lie far from the curve:

result = fw.fit(model, x, y, yerr=dy, loss="huber")
print(result)
result.plot()          # downweighted points are circled
result.downweighted    # boolean mask of the outliers

How it works#

First an ordinary fit runs, with the usual automatic choices (direct solution, starting values, multi-start).
From that solution, Trust Region Reflective minimizes the robust loss. Residuals larger than loss_scale are downweighted.

`loss`	Behaviour
`"soft_l1"`, `"huber"`	mild: outliers count linearly instead of quadratically
`"cauchy"`, `"arctan"`	strong: far outliers have almost no influence

Scale. With yerr, loss_scale is measured in units of yerr (default 1.5). Without yerr, the default is 1.5 times a robust estimate of the scatter (1.4826 · MAD) of the ordinary fit. Set loss_scale explicitly if you know at what deviation a point becomes suspicious.

Uncertainties#

The covariance is that of the equivalent weighted least-squares problem, with the robust weights at the solution. Without yerr, it is scaled by the robust scatter estimate, so the outliers do not inflate it. The report shows this robust estimate as χ²/ndf (robust) with yerr, or as σ_res (robust) without. With correct yerr it is close to 1. The χ² test and its warning are skipped, because outliers are expected by design.

Monte Carlo check (straight line, 60 points, 5 % outliers of 10–30 σ, 300 data sets): the pulls (fit − truth)/σ have a mean within ±0.1. Their spread is 0.95–1.09 with yerr for all four losses, and 0.96–1.17 without yerr. The somewhat larger spread without yerr comes from the statistical noise of the scatter estimate itself.

Note

A robust fit is no substitute for looking at the data. If many points are downweighted, the model may be wrong, not the data.