Nonlinear Iterative Partial Least Squares

The nonlinear iterative partial least-squares (NIPALS) algorithm, also called power method, has been popular especially in the early time of PCA applications in chemistry an extended version is used in PLS regression. The algorithm is efficient if only a few PCA components are required because the components are calculated step-by-step. 

NIPALS Nonlinear iterative partial least-squares... 

Wold, H. Soft Modeling by Latent Variables the Nonlinear Iterative Partial Least Squares Approach," Ed. J. Gani, in Perspective in Probability and Statistics - Papers in Honor of M. S. Bartlett, Academic Press, London 1975, pp. 117-142... 

Wold, H., Soft modeling by latent variables the nonlinear iterative partial least squares approach, in Perspectives in Probability and Statistics, Papers in Honor of M.S. Bartlett, Gani, J., Ed., Academic Press, London, 1975. 

A straightforward method for PCA is the NIPALS (nonlinear iterative partial least squares) algorithm it is quickly implemented and can be applied to large datasets [58, 79],... 

Wold H. Soft modelling with latent variables The nonlinear iterative partial least squares approach. In Gani J, editor, Perspectives in probability and statistics Papers in honour of M.S. Barlett. London Academic Press, 1975. p. 114-42. 

Wold H., Lyttkens E.. Nonlinear iterative partial least squares (NIPALS) estimation procedures in Bull. Intern. Statist. Inst. Proc., 37th session, London -15 1969. 

The simplest method for PCA used in analytics is the iterative nonlinear iterative partial least squares (NIPALS) algorithm explained in Example 5.1. More powerful methods are based on matrix diagonalization, such as SVD, or bidiagonalization, such as the partial least squares (PLS) method. 

During the calibration step, the PLS technique assumes that the spectral data set X can be decomposed in the form of Equation 6.16. PLS factors are then computed with the help of iterative numerical procedures, such as the popular nonlinear iterative partial least squares (NIPALS) algorithm, as described in standard texts [46,74,75]. The PLS factors can be regarded as rotations of the PCA factors computed in... 

CV = cross-validation MLR = multiple linear regression mp = melting point NIPALS = nonlinear iterative partial least squares NN = neural networks PCA = principal... 

The PLS approach was developed around 1975 by Herman Wold and co-workers for the modeling of complicated data sets in terms of chains of matrices (blocks), so-called path models . Herman Wold developed a simple but efficient way to estimate the parameters in these models called NIPALS (nonlinear iterative partial least squares). This led, in turn, to the acronym PLS for these models, where PLS stood for partial least squares . This term describes the central part of the estimation, namely that each model parameter is iteratively estimated as the slope of a simple bivariate regression (least squares) between a matrix column or row as the y variable, and another parameter vector as the x variable. So, for instance, in each iteration the PLS weights w are re-estimated as u X/(u u). Here denotes u transpose, i.e., the transpose of the current u vector. The partial in PLS indicates that this is a partial regression, since the second parameter vector (u in the... 

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