# Robust outlier detection in curve fitting with correlated errors

Assume I have data originating from a model $$y_i=f(t_i)+e(t_i)$$ with $f\in C_2(\mathbb{R})$.

The only thing I know about the errors is that they roughly happen to be of to different sources:

• Some more or less random noise of varying magnitude over time. The noise might just oscillate between different values or may look rather random
• Massive outliers. Those usually are correlated over a few time steps, mostly even lying on a straight line (e.g. $e(t_i)=at_i+b$ for $i=k,k+1,k+2$). In the presence of outliers, the noise usually can be ignored or is even non-existant.

As long as there are no outliers involved, I am quite successful fitting B-Splines through the data. However, this approach fails if the outliers have not been removed.

All fitting has to be done highly automated, thus manual analysis is no option.

One Idea I have for detecting the outliers is to estimate a local measure of dispersion and declare all datapoints as outliers that are much further away form a local estimate on position than that measure.

To identify my estimate on position, I have been very successful using a local median as robust estimate. A straightforward extension would be to use a local MAD as measure for dispersion.

However, now my theoretical knowledge is stretched to its limits. I do not really have the background to understand what I am doing from an analytical and mathematical point of view.

Thus, my question is twofold:

• Do you have a better idea on how to identify outliers? (Just drop me a thought, I will gladly do the research)
• Can you recommend literature giving me theoretical background on rubust dispresion estimation? Even better for models like the one I have sketched.

P.S.: While I had quite a few lectures on statistics at university, my background is based on numerical analysis, so please be gentle.

• There is a whole book [outliers in statistical data amazon.com/Outliers-Statistical-Series-Probability-Statistics/… which is old, but has a lot of useful information. It should provide a starting point for research. An idea off the top of my head - since you are using splines anyway - you could see how adjusting the span of the spline affects things. A very smooth spline won't be affected by outliers unless they are very common (in which case, they aren't outliers). – Peter Flom - Reinstate Monica Nov 11 '12 at 11:38
• The problem is the strong correlation of the outliers. Of course you can make the spline as rigid as necessary to ignore outliers. However, at the same time you reduce it's ability to match features you want to preserve. I also tried iterative methods (IRLS), but they also suffer of the strong correlation. – Thilo Nov 11 '12 at 11:49
• Ah, and thanks for the book recommendation. Looks promising at a first glance. – Thilo Nov 11 '12 at 11:50
• Have you already looked at robust GAM fitting procedures? – user603 Nov 11 '12 at 17:29