Regression methods, assumptions

When the assumption [G]o = [G] can not be applied, other approximation or regression methods have to be employed. Here, the regression method is shown. Typical examples of the regression methods are the Rose-Drago [17], Nakano [18], and Greswell-Allred [19] methods. Because of its wide applicability, a practical guide based on the Rose-Drago method is presented here for an example from UV/vis spectroscopy. 

The prediction of Y-data of unknown samples is based on a regression method where the X-data are correlated to the Y-data. The multivariate methods, usually used for such a calibration, are principal component regression (PCR) and partial least squares regression (PLS). Both methods are based on the assumption of linearity and can deal with co-linear data. The problem of co-linearity is solved in the same way as the formation of a PCA plot. The X-variables are added together into latent variables, score vectors. These vectors are independent since they are orthogonal to each other and they can therefore be used to create a calibration model. 

The problem of over-optimistic estimates of model quality is a general one for all multivariate regression methods, and a number of model quality diagnostics have been developed that do not rely on parametric assumptions, to both limit the model fitting process and to assess the ability of the model to generalise beyond the training set. 

The quantity Y (at the points x ) shows a normal distribution (see Fig. 2, left) and is outlier-free (the latter situation is tested with the Dixon test [13], [21] if the assumption is not confirmed, a robust regression method can be used, for example)... 

Compared with linear and nonlinear regression methods, the advantage of ANN is its ability to correlate a nonlinear function without assumption about the form of this function beforehand. And the trained ANN can be used for unknown prediction. Therefore, ANN has been widely used in data processing of SAR. But if we use ANN solely, sometimes the results of prediction may be not very reliable. Experimental results indicate that some of the test samples predicted by ANN as optimal samples are really not true optimal samples. This is a typical example of so-called overfitting that makes the prediction results of trained ANN not reliable enough. Since the data files in many practical problems usually have strong noise and non-uniform sample point distribution, the overfitting problem may lead to more serious mistake in these practical problems. 

These assumptions are not overly restrictive. Since the value of u is due to many factors acting in opposite directions, it should be expected that small values of u occur more frequently than large values, and that is a variable with a probability distribution centered at zero and having a finite variance o. This is true when the form of Eq. (7.112) is close to the correct relationship. Because of the many factors involved, the central limit theorem would further suggest that u has a normal distribution, which gives the parameter estimates the desirable property of being maximum-likelihood estimates. Later on in the discussion, it will be shown that the regression method can handle cases where o is not constant, and where u is not independent of X. 

The MSM is appropriate for predicting plant responses to herbicide mixtures whose components act by different modes of action. Use of the MSM requires that experimental observations be expressed as percentages or proportions of a hypothetical maximum measure of activityUnder these assumptions, the MSM equates the proportion of the observed response of a test plant to the action of a herbicide mixture to the product of the corresponding proportions observed when the two components of the mixture are administered separately. Methods developed for the prediction of plant response to herbicide mixtures using the MSM include Colby s equation, the MSM equation developed by Morse, and the regression method of Nash. " Recently, Flint et devised a method for the statistical treatment of Colby s equation. 

An alternative way to estimate tmeness is to analyze a series of test samples by the method based on the developed sensor and by another method (preferably a reference/validated method) based on different physical/chemical principles. This because it is quite unlikely that the second method can suffer from the same systematic errors. Method comparison must be performed by specific regression techniques since, in its case, assumption (1) of the OLS method is not valid anymore both variables are affected by experimental errors (see Sect. 18.2.1) and Model II regression methods are mandatory. The relevant literature information is unexpectedly plentiful, since disciplines interested in such a kind of comparison span from statistics to astronomy, geology, physics, chemistry, biology, allometry, industrial pharmacology, and medicine (references (2i-23) (4S-50) ... 

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