# Distance-Weighted Least Squares Fitting

Unlike some other fitting procedures, the distance-weighted least squares procedure does not fit to the data one function that can be easily described by a single formula and plotted independently from the data.

The distance-weighted least squares method fits a curve to the data by using the following procedure. A polynomial (second-order) regression is calculated for each value on the X variable scale to determine the corresponding Y value such that the influence of the individual data points on the regression (i.e., the weight, see the Stiffness option on the plot Fitting dialog) decreases with their distance from the particular X value (an algorithm similar to the one used by this procedure is described by McLain, 1974).

Applications

There are two general classes of applications for this and similar (e.g., spline, negative exponentially-weighted) smoothing procedures. First, it provides a sensitive method for revealing non-salient overall patterns of data. Due to measurement error, such patterns can be hard to identify by simply looking at the scatterplot, although if revealed, they may turn out to be interpretable and reliable.

Another type of application is to use the identified pattern to develop quantitative models of the investigated phenomenon. Specifically, the curve revealed by the smoothing procedure often consists of segments that cannot easily be described by one function (e.g., such as a particular polynomial or logarithmic function). However, the segmentation, and the nature of the component curves (in the consecutive segments) may contain interpretable information about the investigated phenomenon, and a linear or nonlinear piecewise regression function can be developed to account for (and predict) the process in question. The model suggested by the results of such analyses can then be quantitatively verified in a nonlinear estimation analysis (note that the Nonlinear Estimation module offers flexible facilities to estimate user-defined piecewise regression and other models).