Non-linear iterative partial least squares NIPALS

H. Wold, Soft modelling by latent variables the non-linear iterative partial least squares (NIPALS) algorithm. In Perspectives in Probability and Statistics, J. Gani (Ed.). Academic Press, London, 1975, pp. 117-142. 

There are a variety of methods used to obtain the loading and scores matrix in Eq. (15). Perhaps, the most common methods employed are non-linear iterative partial least squares (NIPALS), and the singular value decomposition (SVD). Being an iterative method, NIPALS allows the user to calculate a minimum number of factors, whereas the SVD is more accurate and robust, but in most implementations provides all the factors, thus can be slow with large data sets. During SVD the data matrix can be expressed as... 

The deeomposition in eqn (4.30) is general for PCR, PLS and other regression methods. These methods differ in the criterion (and the algorithm) used for ealeulating P and, hence, they characterise the ealibrators by different scores T. In PCR, T and P are found from the PCA of the data matrix R. Both the non-linear iterative partial least-squares (NIPALS) algorithm and the singular-value deeomposition (SVD) (much used, see Appendix) of R can be used to obtain the T and P used in PCA/PCR. In PLS, other algorithms are used to obtain T and P (see Chapter 5). 

How can one relate T, U, P and Q in such a way First, our previous knowledge of the problem and the analytical technique suggests that these blocks of data, which represent two different aspects of the same true materials (solutions, slurries, etc.), must be related (we do not know how, but they must ). The algorithm developed by H. Wold (called non-linear iterative partial least squares , NIPALS sometimes it is also termed non-iterative partial least squares ) started from this idea and was formulated as presented below. The following ideas have roots in works by Geladi (and co-workers) and Otto. We consider seven major steps. 

The main algorithms used for eigenvectors/eigenvalues computation differ in two aspects the matrix to work on, either X X (eigenvalue decomposition (EVD) and the POWER method) or X (singular value decomposition (SVD) and non-linear iterative partial least squares (NIPALS)). However SVD may work as well on X X (giving the same results as eigenvalue decomposition). Another difference is whether PCs are obtained simultaneously (EVD and SVD) or sequentially (POWER and NIPALS) for details and comparison of efficiency see Wu et al. [38]. In all the cases for which rows dimension I is much smaller than columns dimension /, one can operate on XX instead (EVD, POWER, SVD), and on X (NIPALS). 

A more complex method is described by WOLD [1978], who used cross-validation to estimate the number of factors in FA and PCA. WOLD applied the NIPALS (non linear iterative partial least squares) algorithm and also mentioned its usefulness in cases of incomplete data. 

NIPALS non-linear iterative partial least squares... 

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