Projection pursuit method

These automatic projection pursuit methods using a series of low-dimensional projections have made impressive gains in the problem of multidimensional data analysis, but they have limitations. One of the most important problems is the difficulty in interpreting the solutions from the automatic projection pursuit. Since the axes are the linear/nonlinear combination of the variables (or dimensions) of the original data, it is hard to determine what the projection actually means to users. Conversely, this is one of the reasons that axis-parallel projections (projection methods in category 2) are used in many multidimensional analysis tools (Guo, 2003 Ward, 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). 

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. 

Theory. PP is also a variable reduction method, very similar to PCA. In fact, PP can be considered a generalization of classical PCA (6,14-18). While in PCA the PCs are determined by maximizing variance, in PP, the latent variables, called the projection pursuit features (PPFs), are obtained by optimizing a given projection index that describes the inhomogeneity of the data, instead of its variance (6,18). In the literature, many PP indices have been described. 

These methods are aimed at projecting the original data set from a high-dimensional space onto a line, a plane, or a three-dimensional coordinate system. Perhaps the best way would be to have a mathematical procedure that allows you to sit before the computer screen pursuing the rotation of the data into all possible directions and stopping this process when the best projection, that is, optimal clustering of data groups, has been found. In fact, such methods of projection pursuits already exist in statistics and are tested within the field of chemometrics. 

Friedman and Tukey (1974) devised a method to automate the task of projection pursuit. They dehned interesting projections as ones deviating from the normal distribution and provided a numerical index to evaluate the interestingness of the projection. When an interesting projection is found, the features on the projection are extracted and projection pursuit is continued until there is no remaining feature found. 

In 2009, Liu et al. modelled, once again, the groups of melts previously studied by different groups, in which the influence of the anion is ignored, it being the same (bromide) in all cases. Descriptors were calculated with CODESSA, and the sole novelty of this piece of work is the use of a Projection Pursuit Regression (PPR) to derive the model, along with CODESSA built-in Heuristic Method (HM), preceded by Principal Component Analysis (PCA). The authors concluded that PPR performed better than HM... 

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