Polynomial PLS

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

At the simplest level the orientational correlation of molecular pairs can be characterised by the averages of the even Legendre polynomials Pl(cos J ij) where is the angle between the symmetry axes of molecules i and j separated by a distance r. This correlation coefficient is denoted by... 

Since the work of Baxter et al. [75,76] around 1990, we have not found many more recent applications and it was not until 2003 that Felipe-Sotelo et al. [77] presented another application. They considered a problem where a major element (Fe) caused spectral and chemical interferences on a minor element (Cr), which had to be quantified in natural waters. They demonstrated that linear PLS handled (eventual) nonlinearities since polynomial PLS and locally weighted regression (nonlinear models) did not outperform its results. Further, it was found that linear PLS was able to model three typical effects which currently occur in ETAAS peak shift, peak enhancement (depletion) and random noise. 

No quantitative applications of neural networks to quantitative analysis of linear systems have been reported where the results have been significantly better than those obtained by PLS, as would be expected since PLS (and indeed, the other multivariate methods described in this chapter) have been designed explicitly to handle linear systems. Analogous techniques, such as polynomial PLS or spline PLS, have been designed for nonlinear systems. It is interesting that with nonlinear data, neural networks have been shown to outperform any of the linear or nonlinear PLS techniques [27]. 

M= 0 because there is no < -dependence owing to the assumed axial symmetry of the problem with respect to the Z axis. Thus, the spherical basis vectors transform under rotation as the Legendre polynomials Pl(cosO), see Eq. (10.23) below, where classically 9 is the angle between J and the Z axis, see Figure 10.7. The spherical basis vectors also satisfy the orthonormaUty condition, T Ty = Sy. Then N may be expanded in terms of the spherical basis set as... 

Although the actual cycle decomposition (as well as the tree structure) of a particular graph is determined exactly by the set of elementary divisors i(a ), much of the general form of the possible dynamics may be extracted from Pl x) itself. All graphs whf)se characteristic polynomials Pii=P Yi=i Pi AY (mod q), for. some fixed P ( / ), for example, mu.st share the following properties ... 

A typical performance behaviour is shown in Fig. 44.16b. The increase of the NSE for the monitoring set is a phenomenon that is called overtraining. This phenomenon can be compared to fitting a curve with a polynomial of a too high order or with a PCR or PLS model with too many latent variables. It is caused by the fact that after a certain number of iterations, the noise present in the training set is modelled by the network. The network acts then as a memory, able to recall... 

The point being that, as our conclusions indicate, this is one case where the use of latent variables is not the best approach. The fact remains that with data such as this, one wavelength can model the constituent concentration exactly, with zero error - precisely because it can avoid the regions of nonlinearity, which the PCA/PLS methods cannot do. It is not possible to model the constituent better than that, and even if PLS could model it just as well (a point we are not yet convinced of since it has not yet been tried -it should work for a polynomial nonlinearity but this nonlinearity is logarithmic) with one or even two factors, you still wind up with a more complicated model, something that there is no benefit to. 

There are also forms of nonlinear PCR and PLS where the linear PCR or PLS factors are subjected to a nonlinear transformation during singular value decomposition the nonlinear transformation function can be varied with the nonlinearity expected within the data. These forms of PCR/PLS utilize a polynomial inner relation as spline fit functions or neural networks. References for these methods are found in [7], A mathematical description of the nonlinear decomposition steps in PLS is found in [8],... 

Exercise 7.4 Illustrate Proposition 7.3 hy finding a basis ofP consisting of five harmonic polynomials and one polynomial with a factor of r. Find a basis of Pl consisting of seven harmonic polynomials and three polynomials with a factor ofr. ... 

The calculated gradient curve, C, = 179exp[(3.6 r) /4.2], is shown as curve (a) in Figure 9.7. Curve (b) is the calculated average concentration of etched particles, based on the gradient in curve (a), which agrees with the measurement. Curve (c) is the polynomial simulation of the measured PL relative intensities. A 140% increase of the PL intensity was observed from the particles with 1 pm of the top surface etched away. 

Among the nonlinear methods, there are, besides nonlinear least squares regression, i.e. polynomial regression, the nonlinear PLS method. Alternating Conditional Expectations ACE), SMART, and MARS. Moreover, some Artificial Neural Networks techniques have also to be considered among nonlinear regression methods, such as the back-propagation method. 

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