Projection pursuit

Projection pursuit regression Linear projection Adaptive shape, supersmoother [a, 0, d], minimum output prediction error... 

Friedman, J. H., and Stuetzle, W., Projection pursuit regression, J. Amer. Stat. Assoc. 76, 817-823 (1981). 

Hwang, J. N., Lay, M., Martin, R. D., and Schimert, J., Regression modeling in back-propagation and projection pursuit learning, IEEE Trans. Near. Networks 5 (1994). 

Roosen, C. B., and Hastie, T. J., Automatic smoothing spline projection pursuit. J. Comput. Graph. Stat., 3, 235 (1994). 

Projection FPDs, 22 259 Projection Pursuit (PP), nonlinear method, 6 53... 

There are essentially two different procedures for robust PCA, a method based on robust estimation of the covariance, and a method based on projection pursuit. For the covariance-based procedure the population covariance matrix X has to be... 

Methods of robust PCA are less sensitive to outliers and visualize the main data structure one approach for robust PCA uses a robust estimation of the covariance matrix, another approach searches for a direction which has the maximum of a robust variance measure (projection pursuit). 

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

Du, Y., Liang, Y., Yun, D. J. Chem. Inf. Comput. Sci. 42, 2002, 1283-1292. Data mining for seeking an accurate quantitative relationship between molecular structure and GC retention indices of alkenes by projection pursuit. 

K.R. Beebe and B.R. Kowalski, Nonlinear calibration using projection pursuit regression Application to an array of ion-selective electrodes. Anal. Chem., 60, 2273-2278 (1988). 

Many other affine equivariant and robust estimators of location and scatter have been presented in the literature. The first such estimator was proposed independently by Stahel [11] and Donoho [12] and investigated by Tyler [13] and Maronna and Yohai [14], Multivariate M-estimators [15] have a relatively low breakdown value due to possible implosion of the estimated scatter matrix. Together with the MCD estimator, Rousseeuw [16] introduced the minimum-volume ellipsoid. Davies [17] also studied one-step M-estimators. Other classes of robust estimators of multivariate location and scatter include S-estimators [6, 18], CM-estimators [19], T-estimators [20], MM-estimators [21], estimators based on multivariate ranks or signs [22], depth-based estimators [23-26], methods based on projection pursuit [27], and many others. 

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. 

Another approach to robust PCA has been proposed by Hubert et al. [52] and is called ROBPCA. This method combines ideas of both projection pursuit and robust covariance estimation. The projection pursuit part is used for the initial dimension reduction. Some ideas based on the MCD estimator are then applied to this lowerdimensional data space. Simulations have shown that this combined approach yields more accurate estimates than the raw projection pursuit algorithm RAPCA. The complete description of the ROBPCA method is quite involved, so here we will only outline the main stages of the algorithm. 

Li, G. and Chen, Z., Projection-pursuit approach to robust dispersion matrices and principal components primary theory and Monte Carlo, J. Am. Stat. Assoc., 80, 759-766, 1985. 

Croux, C. and Ruiz-Gazen, A., A fast algorithm for robust principal components based on projection pursuit, in COMPSTAT1996 (Barcelona), Physica, Heidelberg, 1996, pp. 211-217. 

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

Projection pursuit algorithm for robust principal component analysis... 

CLASSIFICATION OF MATERIALS BY PROJECTION PURSUIT BASED ON GENERALIZED SIMULATED ANNEALING... 

---