Robust Locally-Weighted Regression

Using the LOWESS of Residuals as a Diagnostic Tool

• Residual plots can be used as a useful diagnostic tool for controlling the LOWESS fitting process (Cleveland, W. S. (1993): Visualizing Data. Hobart Press, Summit, NJ). This is particularly important for using the appropriate value for the local smoothing width (Alpha).

• Example for Illustrating the Effect of Alpha on the LOWESS Fitting Process

The effect of Alpha for controlling the LOWESS fitting process is shown below for a given bivariate data set, using Alpha values of 0.1, 0.6, and 0.99.

• In choosing Alpha, we want a curve that is as smooth as possible without introducing lack of fit, i.e., the main structure within the data should be by and large represented, with no remaining residual dependence.
Alpha=0.1
• Comparing the original robust LOWESS fit with Alpha=0.1 (the left-hand side image) with the LOWESS fit to the corresponding residuals (the right-hand side image) shows:
• There is no remaining residual dependence but the original fitted LOWESS curve has excessive, unnecessary undulation from the noise components of the data.
• For the given data set, the local smoothing width Alpha=0.1 generates a curve that is highly affected by the noise components of the data.

Alpha=0.6
• Comparing the original robust LOWESS fit with Alpha=0.6 (the left-hand side image) with the LOWESS fit to the corresponding residuals (the right-hand side image) shows:
• There is no remaining residual dependence and the original fitted LOWESS curve tracks adequately the data pattern.
• For the given data set, the local smoothing width Alpha=0.6 is a good fit; i.e., the main structure within the data is adequately represented, and there is no remaining residual dependence.

Alpha=0.99
• Comparing the original robust LOWESS fit with Alpha=0.99 (the left-hand side image) with the LOWESS fit to the corresponding residuals (the right-hand side image) shows:
• There is remaining residual dependence, and the original fitted LOWESS curve is over-smoothed.
• For the given data set, the local smoothing width Alpha=0.99 results in an over-smoothed curve with remaining residual dependence (i.e., lack of fit).