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The PCR method may be broadly divided into three major steps: 1. {\displaystyle \;\;} Perform PCA on the observed data matrix for the explanatory variables to obtain the principal components, and then (usually) select a subset, based on some appropriate criteria, of the principal components so obtained for further use.
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing.. The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified.
The 2014 guaranteed algorithm for the robust PCA problem (with the input matrix being = +) is an alternating minimization type algorithm. [12] The computational complexity is () where the input is the superposition of a low-rank (of rank ) and a sparse matrix of dimension and is the desired accuracy of the recovered solution, i.e., ‖ ^ ‖ where is the true low-rank component and ^ is the ...
In ()-(), L1-norm ‖ ‖ returns the sum of the absolute entries of its argument and L2-norm ‖ ‖ returns the sum of the squared entries of its argument.If one substitutes ‖ ‖ in by the Frobenius/L2-norm ‖ ‖, then the problem becomes standard PCA and it is solved by the matrix that contains the dominant singular vectors of (i.e., the singular vectors that correspond to the highest ...
Output after kernel PCA, with a Gaussian kernel. Note in particular that the first principal component is enough to distinguish the three different groups, which is impossible using only linear PCA, because linear PCA operates only in the given (in this case two-dimensional) space, in which these concentric point clouds are not linearly separable.
Functional principal component analysis (FPCA) is a statistical method for investigating the dominant modes of variation of functional data.Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space L 2 that consists of the eigenfunctions of the autocovariance operator.
Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables.
An implementation of several whitening procedures in R, including ZCA-whitening and PCA whitening but also CCA whitening, is available in the "whitening" R package [7] published on CRAN. The R package "pfica" [8] allows the computation of high-dimensional whitening representations using basis function systems (B-splines, Fourier basis, etc.).