Weighted and Robust Regression

Sometimes weighted regression is a way of preserving the conceptual simplicity of linear models. 

In principle, weighing factors w, can be based upon the information of the measurement error of y at position x,-. If the error has been estimated from n[ measurements at each calibration position x  

In analytical chemistry the highest variance often is associated with the highest values of the concentration. Then it is possible to obtain reasonable results by weighing the least squares of the residuals in the following way  

In a previous section we mentioned that outliers and highly deviating values in a series of measurements are known to have a severe elfect on most tests. In regression models also, the parameters are most sensitive to the response values near the borders of the calibration range. In order to moderate the influence of possible outliers one should try robust techniques. These so-called nonparametric regression statistics start from the common model  

If all x, are different they can form m = n (n - l)/2 pairs of values from which the following constants can be computed  

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K. Baumann, Regression and calibration for analytical separation techniques. Part II Validation, weighted and robust regression. Process Control and Quality 10 (1997), 75-112. 

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