Robust principal component analysis

Croux, C., Filzmoser, P., Oliveira, M. R. Chemom. Intell. Lab. Syst. 87, 2007, 218-225. Algorithms for projection-pursuit robust principal component analysis. 

Robust principal component analysis and constrained background bilinearization for quantitative analysis... 

ROBUST PRINCIPAL COMPONENT ANALYSIS BY PROJECTION PURSUIT AND SIMULATED ANNEALING... 

Projection pursuit algorithm for robust principal component analysis... 

Robust principal component analysis by projection pursuit and... 

The third group of robust approaches to PCA combines the PP idea with the construction of robust covariance. An example of such a hybrid method is robust principal component analysis (ROBPCA) proposed by Hubert et al. [27]. 

Xie YL, Wang JH, Liang Y-Z, Sun LX, Song XH, Yu RQ. Robust principal component analysis by projection pursuit. 1 Chemometr 1993 7 527-41. 

Hubert M, Rousseeuw PJ, Vanden Branden K. ROBPCA a new approach to robust principal component analysis. Technometrics 2005 47 64—79. 

Croux C, Filzmoser P, Oliveira MR. Algorithms for Projection-Pursuit robust principal component analysis. Chemometr Intell Lab Syst 2007 87 218-25. 

Finally, approaches are emerging within the data reconciliation problem, such as Bayesian approaches and robust estimation techniques, as well as strategies that use Principal Component Analysis. They offer viable alternatives to traditional methods and provide new grounds for further improvement. 

In Chapter 11 some recent approaches for dealing with different aspects of the data reconciliation problem are discussed. A more general formulation in terms of a probabilistic framework is first introduced and its application in dealing with gross error is discussed in particular. In addition, robust estimation approaches are considered, in which the estimators are designed so they that are insensitive to outliers. Finally, an alternative strategy that uses Principal Component Analysis is reviewed. 

Cundari, T.R., Sarbu, C. and Pop, H.F. (2002) Robust fuzzy principal component analysis (FPCA). A comparative study concerning interaction of carbon-hydrogen bonds with molybdenum—oxo bonds. /. Chem. Inf. Comp. Sci., 42, 1363. 

Thus, multilinear models were introduced, and then a wide series of tools, such as nonlinear models, including artificial neural networks, fuzzy logic, Bayesian models, and expert systems. A number of reviews deal with the different techniques [4-6]. Mathematical techniques have also been used to keep into account the high number (up to several thousands) of chemical descriptors and fragments that can be used for modeling purposes, with the problem of increase in noise and lack of statistical robustness. Also in this case, linear and nonlinear methods have been used, such as principal component analysis (PCA) and genetic algorithms (GA) [6]. 

A second and orthogonally equivariant approach to robust PCA uses projection pursuit (PP) techniques. These methods maximize a robust measure of spread to obtain consecutive directions on which the data points are projected. In Hubert et al. [46], a projection pursuit (PP) algorithm is presented, based on the ideas of Li and Chen [47] and Croux and Ruiz-Gazen [48], The algorithm is called RAPCA, which stands for reflection algorithm for principal components analysis. 

Croux, C. and Haesbroeck, G., Principal components analysis based on robust estimators of the covariance or correlation matrix influence functions and efficiencies, Biometrika, 87, 603-618, 2000. 

After the optimization, the robustness of the Pareto front has to be assessed. For each final solution of the optimization, a simulation of 609 days is performed with the influent proposed for BSM1 LT. This influent is typical of the events which perturb the WWTPs like rains, storms and holidays. After this long-term simulation, daily means are computed for both objectives. Then, for the evaluation of the robustness, a Principal Components Analysis (PCA) of these 609 daily mean values is performed. This provides the two directions of variations. 10 and 90 percentiles of the projections of... 

In the first mentioned type of application, electrophoretic data are subjected to exploratory analysis techniques, such as principal component analysis (PCA) (5-8), robust PCA (rPCA) (9-13), projection pursuit (PP) (6,14-18), or cluster analysis (8, 19, 20). They all result in a simple low-dimensional visualization of the multivariate data. As a consequence, it will be easier for the analyst to get insight in the data in order to see whether there is a given... 

To assess the robustness of any conclusions drawn from the Principal Components Analysis (PCA) results, four sets of calculations were undertaken using different types of data and different calculation methods ... 

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