Continuum regression

In MLR the equations are constructed so as to maximize the explanation of the correlation between the dependent variable and the independent variables. Variance in the independent set is ignored, regression coefficients are simply calculated on the basis of the fit of y to the x variables. PCR, on the other hand, concentrates on the explanation of variance in the descriptor set. The first step in PCR is the generation of the PCs, regression coefficients are calculated on the basis of explanation of the correlation between y and these components. 

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Section 35.4), reduced rank regression (Section 35.5), principal components regression (Section 35.6), partial least squares regression (Section 35.7) and continuum regression methods (Section 35.8). 

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... 

M. Stone and R.J. Brooks, Continuum regression cross-validated sequentially constructed prediction embracing ordinary least sqaures, partial least squares, and principal component regression. J. Roy. Stat. Soc. B52 (1990) 237-269. 

R.J. Brooks and M. Stone, Joint Continuum Regression for multiple predictands. J. Am. Stat. Assoc., 89 (1994) 1374-1377. 

A general requirement for P-matrix analysis is n = rank(R). Unfortcmately, for most practical cases, the rank of R is greater than the number of components, i.e., rank(R) > n, and rank(R) = min(m, p). Thus, P-matrix analysis is associated with the problem of substituting R with an R that produces rank(R ) = n. This is mostly done by orthogonal decomposition methods, such as principal components analysis, partial least squares (PLS), or continuum regression [4]. Dimension requirements of involved matrices for these methods are m > n, and p > n. If the method of least squares is used, additional constraints on matrix dimensions are needed [4]. The approach of P-matrix analysis does not require quantitative concentration information of all constituents. Specifically, calibration samples with known concentrations of analytes under investigation satisfy the calibration needs. The method of PLS will be used in this chapter for P-matrix analysis. 

The development of new data analysis methods is also an important area of QSAR research. Several methods have been developed in recent years, and these include kernel partial least squares (K-PLS) [92], robust continuum regression [93], local lazy regression [94], fuzzy interval number -nearest neighbor (FINkNN) [95], and fast projection plane classifier (FPPC) [96], These methods have been shown to be useful for the prediction of a wide variety of target properties, which include moisture, oil, protein and starch... 

Semeels S, Filzmoser P, Croux C, Van Espen PJ. Robust continuum regression. Chemom Intell Lab Sys 2005 76 197-204. 

Malpass J A 1994 Continuum Regression Optimised Prediction of Biological Activity PhD thesis, Umversity of Portsmouth, UK... 

In chemometrics, PCR and PLS seem to be the most widely used method for building a calibration model. Recently, we developed a method, called elastic component regression (ECR), which utilizes a tuning parameter a [0,l] to supervise the decomposition of X-matrix [36], which falls into the category of continuum regression [37-40]. It is demonstrated theoretically that the elastic component resulting from ECR coincides with principal components of PC A when a = 0 and also coincides with PLS components when a = 1. In this context, PCR and PLS occupy the two ends of ECR and a (0,l) will lead to an infinite number of transitional models which collectively uncover the model path from PCR to PLS. The source codes implementing ECR in MATLAB are freely available at [41]. In this section, we would like to compare the predictive performance of PCR, PLS and an ECR model with a = 0.5. 

M. Stone, R.J. Brooks, Continuum Regression Cross-Validated Sequentially... 

A. Bjorkstrom, R. Sundberg, A generalized view on continuum regression, Scand. J. 

PCR), partial least squares (PLS), and continuum regression (CR) methods. 

Principal Component Regression, Partial Least Squares, and Continuum Regression... 

Chemometric Methods in Molecular Design, Vol. 3, H. van de Waterbeemd, Ed., VCH Publishers, Weinheim, Germany, 1995, pp. 163-189. Continuum Regression A New Algorithm for the Prediction of Biological Activity. 

Malpass, J. A., Salt, D. W., Wynn, E. W. et al (1995) Prediction of biological activity using continuum regression, in Trends in QSAR and Molecular Modelling 92 (ed. C. G. Wermuth), ESCOM,pp. 314-16. 

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