Continuum regression methods

We have seen that PLS regression (covariance criterion) forms a compromise between ordinary least squares regression (OLS, correlation criterion) and principal components regression (variance criterion). This has inspired Stone and Brooks [15] to devise a method in such a way that a continuum of models can be generated embracing OLS, PLS and PCR. To this end the PLS covariance criterion, cov(t,y) = s, s. r, is modified into a criterion T = r. (For 

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 

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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). 

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... 

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. 

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

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. 

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