Nonlinear statistics

Neural nets can also be used for modeling physical systems whose behavior is poorly understood, as an alternative to nonlinear statistical techniques, eg, to develop empirical relationships between independent and dependent variables using large amounts of raw data. 

Neural networks (NN) represent, as opposed to PLS and MLR, a nonlinear statistical analysis technique [43]. As is the case for both PLS and MLR, several aspects of NN should be considered when using this type of analysis technique ... 

Disadvantages are that these response surface models are not available in standard software packages. Like all nonlinear statistical methods, the methodology is still subject to research, which has 2 important consequences. First, correlation structure of the parameters in these nonlinear models is usually not addressed. Second, the assessment of the test statistic is based on approximate statistical procedures. The statistical analyses can probably be improved through bootstrap analysis or permutation tests. 

Barton CN. 1993. Nonlinear statistical models for the joint action of toxins. Biometrics 49 95-105. 

We observe at this point that Eq. (S.S9) supplemented by Eq. (S.S8) expresses the most recent analytical result obtained to account for the effects of nonlinear excitation. Note, however, that the perturbation approach behind this equation means that it is unable to account for large deviations from linear response theory. In other words, both the intrinsically nonlinear statistics of the system under study and the intensity of the external excitation have to be assumed to be quite small. The rotational coimterpart of Eq. (S.S8) (< is replaced by the angular velocity comparison with the results obtained by applying the continued fraction procedure (CPF) (see Chapters III and IV). It has been shown that the deviation of the linear response theory from the CFP is intermediate between that predicted by Eq. (5.59) and that based on Suzuki s mean held approximation (Chapter V). (In agreement with the CFP, however, both predict that the decay of becomes slower with increases in the excitation parameter r = ( )exc/ " )eq -1.)... 

Chapter II addresses another fundamental problem under what physical conditions can the Fokker-Planck equation provide a reliable picture of fluctuation-dissipation processes Aware as they are of the technical and conceptual difidculties involved in nonlinear statistics, the authors share Zwanzig s optimistic view that use of the Fokker-Planck equation is practicable and advantageous. Chapter II contains a brief description of rules to construct, via a suitable procedure, equations of Fokker-Planck type for the slow variables of the system under study. This theoretical method eliminates explicit analytic dependence on fast variables and thereby produces a significant simplification of the problem under discussion. 

We acknowledge the benefit we have received from past discussions with numerous people who have been active in nonlinear statistical analysis and Burnett data reduction. Also we would like to express our... 

Sakiyama, Y. (2009) The use of machine learning and nonlinear statistical tools for ADME prediction. Expert Opinion on Drug Metabolism S, Toxicology, 5 (2), 49-169. 

At present, investigations for applying neural networks in analytics are done without reference to statistical methods. Future studies will show which applications of the networks could be superior to existing multivariate methods including the nonlinear statistical methods. 

Multidimensional scaling is a nonlinear statistical mapping technique that can be used in data reduction and compression. Its use by the chemical community appears to be virtually nonexistent, and we merely refer the reader to the text by Schiffman et al. ... 

Wang, Y. (2008). Nonlinear statistics to improve signal detection in generalized Gaussian Noise, Digital Signal Processing, vol. 18, Issue 3, pp. 444-449. 

Most physicochemical properties and biological activities are multidimensional in essence. Their study in terms of structure-property or structure-activity relationships requires the examination of multidimensional spaces which are hardly perceivable by humans. Under these conditions, numerous linear and nonlinear methods are routinely used in environmental QSAR for data reduction and graphical display (Figure 2). In the same way, even if in most cases linear statistical methods allow the derivation of powerful QSAR models, it has been shown that numerous environmental phenomena are better simulated by means of nonlinear statistical tools such as artificial neural networks. ... 

Variational inequalitties (3),(7) are the compact formulations of the nonlinear statistical and dynamical problems of a body with its crack. The criterion of no propagation (or propagation) of the crack is expressed by (6) (or (12)). In general case the derived criterion is different from the well-known criterions [4-7]... 

---