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Outliers

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  1. Outliers • Because we square errors, a few points with large errors can have large effect on fitted response surface • Both in simulations and in experiments there is potential for preposterous results due to failures of algorithms or tests • Points with large deviations from fit are called outliers. • Key question is how to distinguish between outliers that should be removed and ones that should be kept.

  2. Weighted least squares • Weighted least squares was developed to allow us to assign weights to data based on confidence or relevance. • Most popular use of weighted least squares is for moving least squares, where we refit data for each prediction point with high weights for nearby data. • Linear interpolation from a table is an extreme form. • Error measure • Normal equations

  3. Determination of weights • One of the simplest is Huber’s • Once you add the weight, the fit and the errors will change and you will need to iterate, hence the name of the method, iteratively reweighted least squares. • If the point is an outlier, the fit will progressively move away from it until its weight is very small • Matlab’srobustfit implements Huber and a few other alternatives. For estimating it uses = MAD/0.6745 • MAD is the median absolute deviation of the absolute errors from their median • It also adjusts by multiplying it by

  4. Example x = (1:10)'; y = 10 - 2*x + randn(10,1); y(10) = 0; bls = regress(y,[ones(10,1) x]) brob = robustfit(x,y) bls = 7.8518 -1.3644 brob = 8.4504 -1.5278 >> scatter(x,y,'filled'); grid on; hold on plot(x,bls(1)+bls(2)*x,'r','LineWidth',2); plot(x,brob(1)+brob(2)*x,'g','LineWidth',2) legend('Data','Ordinary Least Squares','Robust Regression')

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