Linearize Iterate

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. 

In many instances non-linear functions can be linearised and in this way a non-linear, iterative fitting procedure can be reduced to an explicit linear fit. A typical example is the exponential decay of the intensity of the emission of a radioactive sample. We use the data already used for Figure 4-4, produced by the function Data Decay. m. 

MCRs of up to seven different starting materials have been described in the past [3]. MCRs have numerous advantages over classical approaches (linear, iterative or divergent synthesis) in assembling useful chemical products. The advantage of convergence over a divergent synthetic approach is well appreciated in the syn-... 

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. 

Fine wall mesh schemes have been used to avoid this patching process. It is critical to use a good implicit-difference scheme in this case. Mellor (Ml) developed a good linearized iteration technique which has since been adopted by others. 

Most published computations have dealt with boundary layers. The numerical techniques employed have varied considerably, and hence the computational costs initially varied widely among programs. But now most workers have adopted implicit-difference schemes, with special wall-region treatment as outlined above, and/or a linearized iteration technique (Ml), so that run times are now reasonably uniform. A typical two-dimensional compressible boundary layer can now be treated in under one minute on a typical large computer. 

Allylic alcohol (166) is the product of right-to-left linear iteration by this process. Not only is the tri-substituted alkene accessible with high stereochemical control, but also the ( )-disubstituted alkene is readily prepared. These linear polyenes play an important role in the biomimetic synthesis of steroids and higher terpenes. ... 

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... 

Equation (2) also can be solved by other methods without direct implementation of matrix inverse transformation K for example, by means of linear iterations ... 

There are a number of the methods that use linear iterations, for example, the known Jacobi and Gauss-Seidel techniques, steepest descent method, etc. differing by the definition of matrix Hp. This matrix should provide convergence of the iterations to a solution aP a attaining equality in Eq. 

In contrast to Eqs. (19-20), certain types of non-linear iterations invert linear systems and naturally provide non-negative solutions. For example, in atmospheric optics, the relaxation techniques [14, 17] often are considered as alternatives to linear methods (e.g., see discussions [1,19]). 

Using this equation, Eq. (30) can be re-written for non-linear iterations ... 

After this modification, the camera model is established and the system of equations defined by equation 7 is solved, having previously performed the change of variable proposed. After the first calculation stage and based on the solution p and using a linear iterative resolution process we will solve the entire camera system, including the determination of distortion coefficients. We will minimize the normalized distance between the center of grid distortion target dots... 

NIPALS non-linear iterative partial least squares... 

We now present and discuss the basic steps in standard stationary linear iterative methods needed to compute the electrical forces. The current discussion concentrates on the general representation of the two-dimensional Poisson s equation for simplicity... 

Mossbauer Absorption Spectroscopy. Spectra were acquired at room temperature in a constant acceleration spectrometer using a Co in Rh source. Isomer shifts are relative to the NBS standard sodium nitroprusside. Magnetic hyperfine fields were calibrated with the 515 kOe field of a-FcjO, at RT. Mossbauer parameters were determined by fitting the collected spectra with reference sub-spectra consisting of Lorentzian-shaped lines using a non-linear iterative minimization routine. 

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. 

Butte, W. and Blum, J.K. 1984. Calculation of bioconcentration factors from kinetic data by non-linear iterative least-squares regression analysis using a programmable minicalculator. Chemosphere 13 151-160. 

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