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Register a new userDiagonally-Dominant Principal Component Analysis
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Added by
Fan Yang
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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We consider the problem of decomposing a large covariance matrix into the sum of a low-rank matrix and a diagonally dominant matrix, and we call this problem the Diagonally-Dominant Principal Component Analysis (DD-PCA). DD-PCA is an effective tool for designing statistical methods for strongly correlated data. We showcase the use of DD-PCA in two statistical problems: covariance matrix estimation, and global detection in multiple testing. Using the output of DD-PCA, we propose a new estimator for estimating a large covariance matrix with factor structure. Thanks to a nice property of diagonally dominant matrices, this estimator enjoys the advantage of simultaneous good estimation of the covariance matrix and the precision matrix (by a plain inversion). A plug-in of this estimator to linear discriminant analysis and portfolio optimization yields appealing performance in real data. We also propose two new tests for testing the global null hypothesis in multiple testing when the $z$-scores have a factor covariance structure. Both tests first use DD-PCA to adjust the individual $p$-values and then plug in the adjusted $p$-values to the Higher Criticism (HC) test. These new tests significantly improve over the HC test and compare favorably with other existing tests. For computation of DD-PCA, we propose an iterative projection algorithm and an ADMM algorithm.
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Principal component analysis (PCA) is an important tool in exploring data. The conventional approach to PCA leads to a solution which favours the structures with large variances. This is sensitive to outliers and could obfuscate interesting underlying structures. One of the equivalent definitions of PCA is that it seeks the subspaces that maximize the sum of squared pairwise distances between data projections. This definition opens up more flexibility in the analysis of principal components which is useful in enhancing PCA. In this paper we introduce scales into PCA by maximizing only the sum of pairwise distances between projections for pairs of datapoints with distances within a chosen interval of values [l,u]. The resulting principal component decompositions in Multiscale PCA depend on point (l,u) on the plane and for each point we define projectors onto principal components. Cluster analysis of these projectors reveals the structures in the data at various scales. Each structure is described by the eigenvectors at the medoid point of the cluster which represent the structure. We also use the distortion of projections as a criterion for choosing an appropriate scale especially for data with outliers. This method was tested on both artificial distribution of data and real data. For data with multiscale structures, the method was able to reveal the different structures of the data and also to reduce the effect of outliers in the principal component analysis.
Motivation: Although principal component analysis is frequently applied to reduce the dimensionality of matrix data, the method is sensitive to noise and bias and has difficulty with comparability and interpretation. These issues are addressed by improving the fidelity to the study design. Principal axes and the components for variables are found through the arrangement of the training data set, and the centers of data are found according to the design. By using both the axes and the center, components for an observation that belong to various studies can be separately estimated. Both of the components for variables and observations are scaled to a unit length, which enables relationships to be seen between them. Results: Analyses in transcriptome studies showed an improvement in the separation of experimental groups and in robustness to bias and noise. Unknown samples were appropriately classified on predetermined axes. These axes well reflected the study design, and this facilitated the interpretation. Together, the introduced concepts resulted in improved generality and objectivity in the analytical results, with the ability to locate hidden structures in the data.
A principal component analysis based on the generalized Gini correlation index is proposed (Gini PCA). The Gini PCA generalizes the standard PCA based on the variance. It is shown, in the Gaussian case, that the standard PCA is equivalent to the Gini PCA. It is also proven that the dimensionality reduction based on the generalized Gini correlation matrix, that relies on city-block distances, is robust to outliers. Monte Carlo simulations and an application on cars data (with outliers) show the robustness of the Gini PCA and provide different interpretations of the results compared with the variance PCA.
Sparse Principal Component Analysis (SPCA) is widely used in data processing and dimension reduction; it uses the lasso to produce modified principal components with sparse loadings for better interpretability. However, sparse PCA never considers an additional grouping structure where the loadings share similar coefficients (i.e., feature grouping), besides a special group with all coefficients being zero (i.e., feature selection). In this paper, we propose a novel method called Feature Grouping and Sparse Principal Component Analysis (FGSPCA) which allows the loadings to belong to disjoint homogeneous groups, with sparsity as a special case. The proposed FGSPCA is a subspace learning method designed to simultaneously perform grouping pursuit and feature selection, by imposing a non-convex regularization with naturally adjustable sparsity and grouping effect. To solve the resulting non-convex optimization problem, we propose an alternating algorithm that incorporates the difference-of-convex programming, augmented Lagrange and coordinate descent methods. Additionally, the experimental results on real data sets show that the proposed FGSPCA benefits from the grouping effect compared with methods without grouping effect.
We present a novel technique for sparse principal component analysis. This method, named Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA), is based on the recently detailed formula for computing normed, squared eigenvector loadings of a Hermitian matrix from the eigenvalues of the full matrix and associated sub-matrices. Relative to the state-of-the-art LASSO-based sparse PCA method of Witten, Tibshirani and Hastie, the EESPCA technique offers a two-orders-of-magnitude improvement in computational speed, does not require estimation of tuning parameters, and can more accurately identify true zero principal component loadings across a range of data matrix sizes and covariance structures. Importantly, EESPCA achieves these performance benefits while maintaining a reconstruction error close to that generated by the Witten et al. approach. EESPCA is a practical and effective technique for sparse PCA with particular relevance to computationally demanding problems such as the analysis of large data matrices or statistical techniques like resampling that involve the repeated application of sparse PCA.
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