Quadratic PLS

Baffi, G., Martin, E.B. Morris, A.J. (1999a). Non-linear projection to latent structures revisited The quadratic PLS algorithm. Computers and Chemical Engineering, Vol. 23, pp. 395-411. 

In general time-consuming optimization procedures have to be applied to find the best parameters of the non-linear regression function. A further disadvantage of the method is that the type of non-linearity must be known a priori. Quadratic PLS (QPLS) for instance uses a quadratic inner relationship between the scores of the x-space and the scores of the y-space. 

The selection of the number of PLS-components to be included in the model was done according to the PRESS criterion (Section 36.3). Note that the result is comparable to the one which we obtained earlier by means of the simple Hansch analysis (Section 37.1.1). Hence, in this case, there is no obvious benefit to include a quadratic term of log P in the model. 

Similarly, extending linear PLS to nonlinear PLS involves using nonlinear basis functions. A variety of nonlinear basis functions have been used to model the inner relationship indicated in Eq. (22), including quadratic... 

At low excitation intensities, the quadratic term in (68) can be neglected, and the PL efficiency... 

It is desirable that the PLS model covers the X space as well as possible. This imphes that the experimental design is very important (as always ). In this example, a composite design was used which made it possible also to consider quadratic variations. 

Modulo a prime power, pL where p 2, a number y is a quadratic residue if and only if it is one modulo p. Furthermore, each quadratic residue has two square roots again. This can be seen by considering the isomorphism with the additive group modulo 0 = (p - l)p If g is a generator, exactly the elements with an even exponent e are the quadratic residues, and g and are the roots. 

Also non-linear regression, that is, using quadratic terms such as (logP) and cross terms, may be used. However, as described in detail, there are a number of pitfalls to this method. A statistically more robust method which could be used instead of MLR is the PLS regression method. 

A study of the Zeeman effect of the PL lines of the excitons bound to group-III acceptors in silicon by Karasyuk et al. [75] has allowed us to obtain the ground-state (/-factors and the quadratic coefficients (72 and (73 of these acceptors. 

Fig. 3.3. Stern-Volmer plot obtained at 23°C for a film prepared by drop casting 50 pL of a solution containing PtOEP PS at a ratio of 1 50. The fine represents a best quadratic fit...

Recently, a nonlinear version of PLS analysis was described [616]. The CARSO (computer-aided response surface optimization) procedure [617, 618] aims at obtaining response surfaces for non-designed data sets. Quadratic terms and interaction terms are generated for each independent variable and PLS analysis is used to model the data, due to the fact that regression analysis will fail for data sets with many highly interrelated variables in the X block. 

Figure 14.3. TPL generated from GNRs, using a femtosecond-pulsed Ti sapphire laser, (a) Excitation intensities superimposed onto GNR absorption spectrum (b) Quadratic dependence of PL intensity with excitation power (c) TPL emission spectra from GNRs in aqueous solution, excited at 730, 780 and 830 nm respectively. ...

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