Locally weighted regression models

Barton, F.E., II, Shenk, J.S., Westerhaus, M.O. and Funk, D.B. (2000) The development of near infrared wheat quality models by locally weighted regressions. Journal of Near Infrared Spectroscopy 8, 201-208. 

Since the work of Baxter et al. [75,76] around 1990, we have not found many more recent applications and it was not until 2003 that Felipe-Sotelo et al. [77] presented another application. They considered a problem where a major element (Fe) caused spectral and chemical interferences on a minor element (Cr), which had to be quantified in natural waters. They demonstrated that linear PLS handled (eventual) nonlinearities since polynomial PLS and locally weighted regression (nonlinear models) did not outperform its results. Further, it was found that linear PLS was able to model three typical effects which currently occur in ETAAS peak shift, peak enhancement (depletion) and random noise. 

The locally weighted regression (LWR) philosophy assumes that the data can be efficiently modeled over a short span with linear methods. The first step in LWR is to determine the N samples that are most similar with the unknown sample to be analyzed. Similarity can be defined by distance between samples in the spectral space [25] by projections into the principal component space [26] and by employing estimates of the property of interest [27]. Once the N nearest standards are determined, either PLS or PCR is employed to calculate the calibration model. 

Nonlinear Calibration Approaches Spectral data can respond nonlinearly to process perturbations due to deviations of the Lambert-Beer law, to the nonlinear characteristics of light detectors or to interactions among analytes. Sources of nonlinear behavior and techniques for the detection of important nonlinear effects in spectral responses have been discussed in the literature [25, 76]. In order to cope with the nonlinear features of spectral data sets, different approaches have been applied to build calibration models. These calibration approaches have almost always been based on NN models and locally weighted regression (LWR) models. 

Finally, we note that the use of a discrete spectrum involves the use of an empirical equation to fit data. The resulting constants have no physical significance, and the resulting function will have local features that are artifacts of the model and do not reflect the structure of the polymer. This can cause trouble, for example if this function is used to infer the molecular weight distribution. For such a purpose, it may thus be preferable to work with a continuous spectrum function such as H(t) or L(t). Honerkamp and Weese [42] reported a nonlinear regression with regularization technique (NLRG) that takes into account noise in the data and yields a smooth relaxation spectrum. 

---