Alternating regression

The rows with the highest purities are estimates of the row factors, i.e. the purest spectra from the data set, which are refined afterward by alternating regression. 

In summary, the selection procedure consists of three steps (1) compare each spectrum in X with all spectra already selected by applying eq. (34.14). Initially, when no spectrum has been selected, the spectra are compared with the average spectrum of matrix X (2) plot of the dissimilarity values as a function of the retention time (dissimilarity plot) and (3) select the spectrum with the highest dissimilarity value by including it as a reference in matrix Y,-. The selection of the spectra is finished when the dissimilarity plot shows a random pattern. It is considered that there are as many compounds as there are spectra. Once the purest spectra are available, the data matrix X can be resolved into its spectra and elution profiles by Alternating Regression explained in Section 34.3.1. 

E.J. Karjalainen, Spectrum reconstruction in GC/MS. The robustness of the solution found with alternating regression, in E.J. Karjalainen (Ed.), Scientific Computing and Automation (Europe). Elsevier, Amsterdam, 1990, pp. 477-488. 

In their paper, Tangen and Koch (2001) concentrate primarily on the analysis of contrasts of particular interest. The approach is thus in the spirit of that described as the disconnected treatment approach in Section 20.2.1. For an alternative regression approach, an extremely interesting paper is that of Pool et al. (1997). This reports and analyses a 4 x 4 factorial trial of hydrochlorothiazide and fosinopril in hypertension in 550 patients. They fit a quadratic surface model in both doses to the... 

To solve the matrix equation the method of the alternating regression is used. Initially the mass spectra are filled with random numbers. The peak shape curve (concentration curve) is calculated from the overlapped specttum. In the next step, the peak shape curve is treated as given and from this the mass spectra are calculated. In a not resolved region, the mass spectra should not change at all. In the iteration step from the newly obtained mass spectra, the peak shapes are calculated. 

Karjalainen, E.J., Karjalainen, U.P. Simultaneous analysis of multiple chromatographic runs and samples with alternating regression. Chemometr. InteU. Lab. Syst. 14(1-3), 423 27 (1992)... 

Dhanoa, M.S., R. Sanderson, S. Lopez, J. Dijkstra, E. Kebreab and J. France, 2007. Alternative regression approaches when modelling energy components. In Ortigues-Marty, I., N. Miraux, and W. Brand-Williams (eds.), Wageningen Academic Publishers, the Netherlands, EAAP Publication No. 124 593-594. 

Alternatives to Multiple Linear Regression Discriminant Analysis, Neural Networks and Classification Methods... 

If a standard method is available, the performance of a new method can be evaluated by comparing results with those obtained with an approved standard method. The comparison should be done at a minimum of three concentrations to evaluate the applicability of the new method for different amounts of analyte. Alternatively, we can plot the results obtained by the new method against those obtained by the approved standard method. A linear regression analysis should give a slope of 1 and ay-intercept of 0 if the results of the two methods are equivalent. 

Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modehng apphcations, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 

One shortcoming of Schild analysis is an overemphasized use of the control dose-response curve (i.e., the accuracy of every DR value depends on the accuracy of the control EC o value). An alternative method utilizes nonlinear regression of the Gaddum equation (with visualization of the data with a Clark plot [10], named for A. J. Clark). This method, unlike Schild analysis, does not emphasize control pECS0, thereby giving a more balanced estimate of antagonist affinity. This method, first described by Lew and Angus [11], is robust and theoretically more sound than Schild analysis. On the other hand, it is not as visual. Schild analysis is rapid and intuitive, and can be used to detect nonequilibrium steady states in the system that can corrupt... 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

To benchmark our learning methodology with alternative conventional approaches, we used the same 500 (x, y) data records and followed the usual regression analysis steps (including stepwise variable selection, examination of residuals, and variable transformations) to find an approximate empirical model, / (x), with a coefficient of determination = 0.79. This model is given by... 

An alternative and illuminating explanation of reduced rank regression is through a principal component analysis of Y, the set of fitted F-variables resulting from an unrestricted multivariate multiple regression. This interpretation reveals the two least-squares approximations involved projection (regression) of Y onto X, followed by a further projection (PCA) onto a lower dimensional subspace. 

Principal covariates regression (PCovR) is a technique that recently has been put forward as a more flexible alternative to PLS regression [17]. Like CCA, RRR, PCR and PLS it extracts factors t from X that are used to estimate Y. These factors are chosen by a weighted least-squares criterion, viz. to fit both Y and X. By requiring the factors to be predictive not only for Y but also to represent X adequately, one introduces a preference towards the directions of the stable principal components of X. 

Finally, another alternative to continuum regression has been put forward by Wise and de Jong [18]. Their continuum power-PLS (CP-PLS) method modifies the matrix X = USV into X " = i.e. the singular values are raised to a... 

S. de Jong, SIMPLS an alternative approach to partial least squares regression. Chemom. Intell. Lab. Syst., 18 (1993) 251-263. 

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