Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse

Fast solving of Weighted Pairing Least-Squares systems

This paper presents a generalization of the "weighted least-squares" (WLS), named "weighted pairing least-squares" (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual "Cholesky factorization-backward substitution" method for solving linear systems

A kernel-based approach for fault diagnosis in batch processes

This article explores the potential of kernel-based techniques for discriminating on-specification and off-specification batch runs, combining kernel-partial least squares discriminant analysis and three common approaches to analyze batch data by means of bilinear models: landmark features extraction, batchwise unfolding, and variablewise unfolding. Gower s idea of pseudo-sample projection is exploited to recover the contribution of the initial variables to the final model and visualize those having the highest discriminant power. The results show that the proposed approach provides an efficient fault discrimination and enables a correct identification of the discriminant variables in the considered case studies.Vitale, R.; De Noord, OE.; Ferrer, A. (2014). A kernel-based approach for fault diagnosis in batch processes. Journal of Chemometrics. 28(8):697-707. doi:10.1002/cem.2629S697707288Cao, D.-S., Liang, Y.-Z., Xu, Q.-S., Hu, Q.-N., Zhang, L.-X., & Fu, G.-H. (2011). 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IEEE Transactions on Geoscience and Remote Sensing, 46(10), 2872-2881. doi:10.1109/tgrs.2008.918765Sun, R., & Tsung, F. (2003). A kernel-distance-based multivariate control chart using support vector methods. International Journal of Production Research, 41(13), 2975-2989. doi:10.1080/1352816031000075224Lee, J.-M., Yoo, C., Choi, S. W., Vanrolleghem, P. A., & Lee, I.-B. (2004). Nonlinear process monitoring using kernel principal component analysis. Chemical Engineering Science, 59(1), 223-234. doi:10.1016/j.ces.2003.09.012Kewley, R. H., Embrechts, M. J., & Breneman, C. (2000). Data strip mining for the virtual design of pharmaceuticals with neural networks. IEEE Transactions on Neural Networks, 11(3), 668-679. doi:10.1109/72.846738Üstün, B., Melssen, W. J., & Buydens, L. M. C. (2007). Visualisation and interpretation of Support Vector Regression models. Analytica Chimica Acta, 595(1-2), 299-309. doi:10.1016/j.aca.2007.03.023Krooshof, P. W. T., Üstün, B., Postma, G. J., & Buydens, L. M. C. (2010). Visualization and Recovery of the (Bio)chemical Interesting Variables in Data Analysis with Support Vector Machine Classification. Analytical Chemistry, 82(16), 7000-7007. doi:10.1021/ac101338yGOWER, J. C., & HARDING, S. A. (1988). Nonlinear biplots. Biometrika, 75(3), 445-455. doi:10.1093/biomet/75.3.445Postma, G. J., Krooshof, P. W. T., & Buydens, L. M. C. (2011). Opening the kernel of kernel partial least squares and support vector machines. Analytica Chimica Acta, 705(1-2), 123-134. doi:10.1016/j.aca.2011.04.025Smolinska, A., Blanchet, L., Coulier, L., Ampt, K. A. M., Luider, T., Hintzen, R. Q., … Buydens, L. M. C. (2012). Interpretation and Visualization of Non-Linear Data Fusion in Kernel Space: Study on Metabolomic Characterization of Progression of Multiple Sclerosis. PLoS ONE, 7(6), e38163. doi:10.1371/journal.pone.0038163Camacho, J., Picó, J., & Ferrer, A. (2008). Bilinear modelling of batch processes. Part I: theoretical discussion. 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Straight monotonic embedding of data sets in Euclidean spaces

International audienceThis paper presents a fast incremental algorithm for embedding data sets belonging to various topological spaces in Euclidean spaces. This is useful for networks whose input consists of non-Euclidean (possibly non-numerical) data, for the on-line computation of spatial maps in autonomous agent navigation problems, and for building internal representations from empirical similarity data. (C) 2002 Elsevier Science Ltd. All rights reserved

Symbolic computation of Hankel determinants and matrix generalized inverses

In this thesis, existing methods for symbolic computation of Hankel deteriminants and matrix generalized inverses are modified and new are introducted. There are derived closed-form expressions for Hankel determinants of different classes of sequences. It is constructed the method for rapid computation of generalized inverses whose complexity reaches theoretical lower bound. There are also constructed new methods for computation of generalized inverses of rational and polynomial matrices