Nonlinear techniques

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

Although ANNs can, in theory, model any relation between predictors and predictands, it was found that common regression methods such as PLS can outperform ANN solutions when linear or slightly nonlinear problems are considered [1-5]. In fact, although ANNs can model linear relationships, they require a long training time since a nonlinear technique is applied to linear data. Despite, ideally, for a perfectly linear and noise-free data set, the ANN performance tends asymptotically towards the linear model performance, in practical situations ANNs can reach a performance qualitatively similar to that of linear methods. Therefore, it seems not too reasonable to apply them before simpler alternatives have been considered. 

E-state indices, counts of atoms determined for E-state atom types, and fragment (SMF) descriptors. Individual structure-complexation property models obtained with nonlinear methods demonstrated a significantly better performance than the models built using MLR. However, the consensus models calculated by averaging several MLR models display a prediction performance as good as the most efficient nonlinear techniques. The use of SMF descriptors and E-state counts provided similar results, whereas E-state indices led to less significant models. For the best models, the RMSE of the log A- predictions is 1.3-1.6 for Ag+and 1.5-1.8 for Eu3+. 

In this section mainly results of linear Raman spectroscopy of gases and vapours have been considered and selected examples of the results of nonlinear techniques were included, e.g. CARS or stimulated and inverse Raman spectroscopy, by which much higher resolution can be achieved. Further such investigations have been reviewed elsewhere (Esherick and Owyoung, 1982 Schrotter et al., 1988a, 1988b, 1990 Lavorel et al., 1992). 

Aside from CARS, a second-order nonlinear technique, such as second harmonic generation or sum frequency generation, can also be a suitable spectroscopic technique because of its higher surface selective detection [103], which is arising from... 

Degenerate four-wave mixing (DFWM) Degenerate four-wave mixing (DFWM) is another well-known nonlinear technique based on third-order nonlinear susceptibility similar to CARS... 

Nonlinear Raman spectroscopy The nonlinear techniques include stimulated Raman scattering (SRS), hyper Raman, stimulated Raman gain (SRG), inverse... 

Waals dimers [266, 267, 302, 328], The theoretically predicted dimer Raman spectra [262, 263] have hitherto not been observed in the laboratory, presumably because of the substantial width (%10GHz) of the 5145- and 4880-A argon laser lines commonly employed in such work and because the high gas densities used tend to broaden the rotational features and render them indiscernible [262], Nevertheless, the envelopes of the rotational van der Waals dimer bands have been seen in several gases and mixtures [229, 230, 266, 267, 328]. Under more favorable jet expansion conditions, high-resolution argon dimer spectra have been obtained recently [280], There seems to be little room for doubt that eventually van der Waals dimer spectra will be obtained with ordinary Raman or third-order nonlinear techniques [676]. 

The order-reduction of the process model could offer a solution. Several linear [1] and nonlinear techniques [2] have been developed and their application to different case studies reported. Although significant reduction of the number of equations is achieved, the benefit is often partial, because the structure of the problem is destroyed, the physical meaning of the model variables is lost and there is little or no decrease of the solution time [3]. 

Multivariate methods are more suitable for the data analysis of multispecies toxicity tests. No one multivariate technique is always best. Given that many responses of multispecies toxicity tests are nonlinear, techniques that do not assume linear relationships may allow a more accurate interpretation of the test system. 

These later two models of bioavailability as a continuous variable are linear since they used stepwise multiple linear regression (M LR) as the modeling tool. An obvious alternative, which may offer improved performance, is a nonlinear technique and such a model using an artificial neural network (ANN) was reported by Turner and colleagues [30], This study employed 167 compounds characterized by several descriptor types, ID, 2D, and 3D, and resulted in a 10-term model. Although the predictive performance was judged adequate, it was felt that the model was better able to differentiate qualitatively between poorly and highly bioavailable compounds. 

Another nonlinear technique that is potentially applicable to thermometric measurements is DPWM [7,9]. Por instance, a Boltzmann plot constructed out of the relative line intensities of a DPWM spectrum can lead to temperature predictions that can be as accurate as CARS in some cases. An alternative method is to fit theoretical simulations to the experimental spectrum. Nonetheless, the versatility of CARS is not equaled by DPWM. In effect, single pulse measurements seems to be limited to some radical species and mode fluctuations of conventional lasers perturb the data severely. To avoid troubles with such laser intensity fluctuations, saturated DPWM is often employed, but the difficulties of spectral S5mthe-sis remain a serious hindrance to a major role of DPWM thermometry. 

Taking advantage of the S5mimetry changes induced by the presence of a surface. Many nonlinear techniques rely on the fact that the surface breaks the centrosymmetrical nature of the bulk. The use of polarized light can also discriminate among dipole moments in different orientations. 

Nonlinear techniques have been used to overeome some of the drawbacks of conventional Raman spectroscopy, particularly its low dficienev, its limitation to the visible and near-ultraviolet regions, and its susceptibility to interference from fluorescence. A major disadvantage of nonlinear methods is that they lend to be analyte specific and often require several different tunable lasers to be applicable lo diverse species. I o dale, none of the nonlinear methods has found widespread application among nonspccialisls. However, many of these methods have shown considerable promise. As less expensive and more routinely useful lasers become available, nonlinear Raman methods, particu-larlv CARS, should become more widely used. 

More research is needed with both spectroscopic tools, especially with the recent advances in the field of Fourier transform Raman spectroscopy and the tremendously improved sensitivity and stability of the IR interferometers. In addition, the availability of new algorithms such as the ratio method (32), factor analysis (20, 33), and recently developed nonlinear techniques (34), coupled with an easy access to fast computers, will advance spectral analysis tremendously and make it more precise and reliable. 

The next step in the protocol answers the question about what is the best method to estimate the reactivity ratios. Historically, because of its simplicity, linearization techniques such as the Fineman-Ross, Kelen-Tudos, and extended Kelen-Tudos methods have been used. Easily performed on a simple calculator, these techniques suffer from inaccuracies due to the linearization of the inherently nonlinear Mayo-Lewis model. Such techniques violate basic assumptions of linear regression and have been repeatedly shown to be invalid [117, 119, 126]. Nonlinear least squares (NLLS) techniques and other more advanced nonlinear techniques such as the error-in-variables-model (EVM) method have been readily available for several decades [119, 120, 126, 127]. 

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