Nonlinear multivariate data

The EDBD variant of BP was used in several chemical kinetic studies. 2-154 This method gave much better estimates of kinetic analytical parameters than either nonlinear regression or principal components regression. EDBD has also been applied to multicomponent kinetic determinations and to the estimation of kinetic compartmental model parameters.i The EDBD method was found to offer increased modeling power for nonlinear multivariate data compared to partial least squares and principal components regression, provided the training set is extensive enough to adequately sample the nonlinear features of the data.i55 Finally, EDBD has been successfully applied to the prediction of retention indices, i ... 

PPR is a linear projection-based method with nonlinear basis functions and can be described with the same three-layer network representation as a BPN (see Fig. 16). Originally proposed by Friedman and Stuetzle (1981), it is a nonlinear multivariate statistical technique suitable for analyzing high-dimensional data, Again, the general input-output relationship is again given by Eq. (22). In PPR, the basis functions 9m can adapt their shape to provide the best fit to the available data. 

In general, the evaluation of interlaboratory studies can be carried out in various ways (Danzer et al. [1991]). Apart from z-scores, multivariate data analysis (nonlinear mapping, principal component analysis) and information theory (see Sect. 9.2) have been applied. 

A great variety of different methods for multivariate classification (pattern recognition) is available (Table 5.6). The conceptually most simply one is fc-NN classification (Section 5.3.3), which is solely based on the fundamental hypothesis of multivariate data analysis, that the distance between objects is related to the similarity of the objects. fc-NN does not assume any model of the object groups, is nonlinear, applicable to multicategory classification, and mathematically very simple furthermore, the method is very similar to spectral similarity search. On the other hand, an example for a rather sophisticated classification method is the SVM (Section 5.6). 

One such study details the effects of temperature variation on substrate and metabolite concentration predictions, and used an artificial neural network creating a nonlinear multivariate model to improve concentration predictions. Another study notes the effects of temperature on the mid-infrared spectral data as well, but also noted that the sensor was not affected by reactor operating conditions such as agitation, airflow and backpressure. ... 

Nonlinear PCA To address the nonlinearity in the identity mapping of multivariate data, a nonlinear counterpart of the PCA can be used (see Section 3.6.1). As the versions of NLPCA make use of the neural network (NN) concept to address the nonlinearity, they suffer from the known overparameterization problem in the case of noise corrupted data. Data with small SNR will also give rise to extensive computations during the training of the network. Shao et al. [266] used wavelet filtering to pre-process the data followed by IT-net to detect the non-conforming trends in an industrial spray drier. 

Lucic, B., Amic, D. andTrinajstic, N. (2000) Nonlinear multivariate regression outperforms several concisely designed neural networks on three QSPR data sets./. Chem. Inf. Comput. Sci., 40, 403—413. 

T. B. Blank and S. D. Brown, Anal. Chem., 65, 3081 (1993). Nonlinear Multivariate Mapping of Chemical Data Using Feed-Forward Neural Networks. 

To take account of interactions between individual components (association, nonlinearities), calibration using multivariate data analysis is often also carried out with mixtures rather than pure substances. Despite this fact, limitations to this method of assessment are encountered quickly. Therefore, the so-called inverse method using the g-matrix is employed, and either principal component regression (PCR) or the partial least squares (PLS) method is used [6, [114], [116]. In both methods, calibration is carried out not with pure substances, but with various mixtures, which must cover the expected concentration range of all components. Within limits, this can allow for non-linearities ... 

Livingstone et al. have employed a particular neural net architecture called a reversible nonlinear dimensionality reduction (ReNDeR) net for a low-dimensional display of multivariate data sets (160). The method makes use of the activity values of the hidden neurons in a trained three-layer feedforward network to produce the low-dimensional display. It was claimed that, in contrast to con-... 

J. Zhang, J.-H. Jiang, P. Liu, Y.-Z. Liang and R.-Q. Yu, Multivariate nonlinear modelling of fluorescence data by neural network with hidden node pruning algorithm. Anal. Chim. Acta, 344(1997) 29 0. 

Techniques for multivariate input analysis reduce the data dimensionality by projecting the variables on a linear or nonlinear hypersurface and then describe the input data with a smaller number of attributes of the hypersurface. Among the most popular methods based on linear projection is principal component analysis (PCA). Those based on nonlinear projection are nonlinear PCA (NLPCA) and clustering methods. 

Among nonlocal methods, those based on linear projection are the most widely used for data interpretation. Owing to their limited modeling ability, linear univariate and multivariate methods are used mainly to extract the most relevant features and reduce data dimensionality. Nonlinear methods often are used to directly map the numerical inputs to the symbolic outputs, but require careful attention to avoid arbitrary extrapolation because of their global nature. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Unsupervised multivariate statistical methods [CA, principal components analysis, Kohonen s self-organizing maps (SOMs), nonlinear mapping, etc.], which perform spontaneous data analysis without the need for special training (learning), levels of knowledge, or preliminary conditions. 

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