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Sparse principal component analysis (sparse PCA) is a specialised 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 adding sparsity constraint on the input variables. Ordinary principal component analysis (PCA) uses a vector space transform to reduce multidimensional data sets to lower dimensions. It finds linear combinations of input variables, and transforms them into new variables (called principal components) that correspond to directions of maximal variance in the data. The number of new variables created by these linear combinations is usually much lower than the number of input variables in the original dataset, while still explaining most of the variance present in the data. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables. == Mathematical Formulation == Consider a data matrix, ''X'', where each of the ''p'' columns represent an input variable, and each of the ''n'' rows represents an independent sample from data population. We assume each column of ''X'' has mean zero, otherwise we can subtract column-wise mean from each element of ''X''. Let ''Σ=XTX'' be the empirical covariance matrix of ''X'', which has dimension ''p×p''. Given an integer ''k'' with ''1≤k≤p'', the sparse PCA problem can be formulated as maximizing the variance along a direction represented by vector while constraining its cardinality: : The first constraint specifies that ''v'' is a unit vector. In the second constraint, represents the L0 norm of ''v'', which is defined as the number of its non-zero components. So the second constraint specifies that the number of non-zero components in ''v'' is less than or equal to ''k'', which is typically an integer that is much smaller than dimension ''p''. The optimal value of is known as the ''k''-sparse largest eigenvalue. If we take ''k=p'', the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance matrix ''Σ''. After finding the optimal solution ''v'', we deflate ''Σ'' to obtain a new matrix : and iterate this process to obtain further principal components. However, unlike PCA, sparse PCA cannot guarantee that different principal components are orthogonal. In order to achieve orthogonality, additional constraints must be enforced. Because of the cardinality constraint, the maximization problem is hard to solve exactly, especially when dimension ''p'' is high. In fact, the sparse PCA problem in is NP-hard in the strong sense.〔 〕 Several alternative approaches have been proposed, including * a regression framework,〔 〕 * a convex relaxation/semidefinite programming framework,〔 〕 * a generalized power method framework〔 〕 * an alternating maximization framework〔 〕 * forward/backward greedy search and exact methods using branch-and-bound techniques,〔 〕 * Bayesian formulation framework.〔 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sparse PCA」の詳細全文を読む スポンサード リンク
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