Negative Exponentially-Weighted Fitting

The negative exponentially-weighted fitting (smoothing) procedure is based on a polynomial regression algorithm similar to the one used in the Distance-Weighted Least Squares procedure. However, the weights that determine the influence of individual data points on consecutive segments of the curve (depending on their distance from the given segment) are calculated according to a negative (decreasing) exponential function.

The choice of that function is not entirely arbitrary because research experience confirms that, in many circumstances, exponential weighting offers an adequate balance between preventing points at remote subregions from biasing the curve while not ignoring them entirely. McLain (1974) argues that negative exponential weighting receives "some slight theoretical support" from the fact that in cubic splines (which are proven to provide adequate approximations of real-life sequential data sets) the effect of remote data points decreases approximately exponentially with distance.

Applications

There are two general classes of applications for this and similar (e.g., spline, distance-weighted least squares) 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).

See also the Stiffness option in the plot Fitting dialog box topic and Exploratory data analysis and data mining techniques.