Nonlinear principal components

Kramer, M. A., Nonlinear principal component analysis using autoassociative neural networks. AIChE J. 37, 233-243 (1991). 

Nonlinear principal component Nonlinear projection, nonlocal Adaptive shape [a, ], minimum input prediction error... 

Input-output analysis methods that project the inputs on a nonlocal hypersurface have also been developed, such as BPNs with multiple hidden layers and regression based on nonlinear principal components. 

Dong, D., and McAvoy, T. J., Nonlinear principal component analysis—based on principal curves and neural networks, Comput. Chem. Eng. 20(1), 65 (1996). 

The 0-NLPCA network has 8-6-10-12 neurons in each layer, yielding a prototype model with 6 principal components (PCs). For comparison, the linear PCA was also applied to the same data. As a performance criterion, the root mean square of error (RMSE) was evaluated to compare the prediction ability of the developed PCA and O-NLPCA models on the training and validation data. While the linear PCA gave 0.3021 and 0.3227 RMSE on training and validation data sets, respectively, the O-NLPCA provided 0.2526 and 0.2244 RMSE. This suggests that to capture the same amount of information, the linear PCA entails utilization of more principal components than its nonlinear counterpart. As a result, the information embedded in the nonlinear principal components addresses the underlying events more efficiently than the linear ones. 

R Shao, F Jia, EB Martin, and AJ Morris. Wavelets and nonlinear principal components analysis for process monitoring. Control Engg. Practice, 7 865-879, 1999. 

O-NLPCA Orthogonal nonlinear principal components analysis... 

Dong D, McAvoy TJ, Nonlinear principal component analysis-based on principal curves and neural networks, Computers in Chemical Engineering, 1996, 20, 65-78. 

A. R. Webb, Neural Networks, 12, 339 (1999). A Loss Function Approach to Model Selection in Nonlinear Principal Components. 

Harkat M.F., Djelel S., Doghmane N., Benouaret M., 2007, Sensor Fault Detection, Isolation and Reconstruction Using Nonlinear Principal Component Analysis , International Journal of Automation and Computing, vol. 4, No. 2, pp. 149-155. 

Mirgolbabaei, H., Echekki, T., Smaoui, N. A nonlinear principal component analysis approach for turbulent combustion composition space. Int. J. Hydrogen Energy 39, 4622—4633 (2014) Mitsos, A., Oxberry, G.M., Barton, P.I., Gretai, W.H. Optimal automatic reactirm and species elimination in kinetic mechanisms. Combust. Flame 155, 118—132 (2008)... 

Features Extraction from Satellite Data, Fig. 3 Original Landsat image left) and two computed nonlinear principal components center and right)... 

Many methods have been developed to tackle the issue of high dimensionality of hyperspectral data (Serpico and Bruzzone 1994). In summary, we may say that feature-reduction methods can be divided into two classes feature-selection algorithms (which suitably select a suboptimal subset of the original set of features while discarding the remaining ones) and feature extraction by data transformation which projects the original data space onto a lower-dimensional feature subspace that preserves most of the information, such as nonlinear principal component analysis (NLPCA Licciardi and Del Prate 2011). 

---