Regression weighted

In the previous sections of Chapter 2 it was assumed that the standard deviation obtained for a series of repeat measurements at a concentration X would be the same no matter which x was chosen this concept is termed homoscedacity (homogeneous scatter across the observed range). 

Under certain combinations of instrument type and operating conditions the proceeding assumption is untenable signal noise depends on the analyte concentration. A very common form of heteroscedacity is presented in Fig. 2.17. 

Curve-fitting need not be abandoned in this case, but some modifications are necessary so that precisely measured points influence to a greater degree the form of the curve, more so than a similar number of less precisely measured ones. Thus, a weighting scheme is introduced. There are different ways of doing this the most accepted model makes use of the experimental standard deviation,namely  

How does one obtain the necessary s -values There are two ways  

One performs so many repeat measurements at each concentration point that standard deviations can be reasonably calculated, e.g., as in validation work the statistical weights w, are then taken to be inversely proportional to the local variance. The proportionality constant k is estimated from the data. 

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For the weighted regression the standard deviation was modeled as i(x) = 100 + 5 x this information stems from experience with the analytical technique. Intermediate results and regression parameters are given in Tables 2.13 and 2.14. Table 2.15 details the contributions the individual residuals make. 

The third item in the menu bar is (Options), which lists the program-specific selections, here (Font), (Scale), (Specification Limits), (Select p), (LOD), (Residuals), (Interpolate Y = /(x)), (Interpolate X = j y)), (Clear Interpolation), respectively (Weighted Regression). 

Weighted Regression) requires the user to dehne a signal-dependent model of the measurement error, e.g., sy = a + b x, which is then used to calculate the weighting factors 1/Vy at every abscissa x,-. For an example on how to enter the model, see Algebraic Function, ... 

This GLS estimator is akin to inverse variance-weighted regression discussed in Section 8.2.3. Again there is a limitation V can be inverted only when the number of calibration samples is larger than the number of predictor variables, i.e. spectral wavelengths. Thus, one either has to work with a limited set of selected wavelengths or one must apply other solutions which have been proposed for tackling this problem [5]. 

T. Naes, T. Isaksson and B. Kowalski, Locally weighted regression in NIR analysis. Anal. Chem., 2 (1990) 664-673. 

Weighted regression of U- " U- °Th- Th isotope data on three or more coeval samples provides robust estimates of the isotopic information required for age calculation. Ludwig (2003) details the use of maximum likelihood estimation of the regression parameters in either coupled XY-XZ isochrons or a single three dimensional XYZ isochron, where A, Y and Z correspond to either (1) U/ Th, °Th/ Th and... 

If a weighted regression is used, the expression for the Mahalanobis Distance becomes equation 74-5b ... 

Local flux-density profile, 23 816 Localized molecular orbital (LMO) calculations, 10 633 Locally weighted regression, 6 53 Local oscillator (LO), 23 142, 143 Local toxicity, 25 202 Locard Exchange Principle, 12 99 Lochett, W., 11 8... 

Naes, T., Isaksson, T., Kowalski, B. R. Anal. Chem. 62, 1990, 664—673. Locally weighted regression and scatter correction for near-infrared reflectance data. 

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. 

It is often assumed in regression calculations that the experimental error only affects the y value and is independent for the concentration, which is typically placed on the x axis. Should this not be the case, the data points used to estimate the best parameters for a straight line do not have the same quality. In such cases, a coefficient Wj is applied to each data point and a weighted regression is used. A variety of formulae have been proposed for this method. 

Solow, A.R. (1988) Detecting changes through time in the variance of a long-term hemispheric temperature record an application of robust locally weighted regression. J.Climate 1,290-296. 

The variance of y at each point xt should be equal, i.e. constant over the whole working range of x, or, in other words, the errors in measuring y are independent of the values of x. This property is called homoscedasticity and can be tested by the COCHRAN test or by other tests (see [ISO 5725, clause 12]). If this condition is not met, weighted regression models may be considered. 

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

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