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Register a new userBayesian Functional Principal Components Analysis via Variational Message Passing
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Added by
Tui Nolan
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Functional principal components analysis is a popular tool for inference on functional data. Standard approaches rely on an eigendecomposition of a smoothed covariance surface in order to extract the orthonormal functions representing the major modes of variation. This approach can be a computationally intensive procedure, especially in the presence of large datasets with irregular observations. In this article, we develop a Bayesian approach, which aims to determine the Karhunen-Lo`eve decomposition directly without the need to smooth and estimate a covariance surface. More specifically, we develop a variational Bayesian algorithm via message passing over a factor graph, which is more commonly referred to as variational message passing. Message passing algorithms are a powerful tool for compartmentalizing the algebra and coding required for inference in hierarchical statistical models. Recently, there has been much focus on formulating variational inference algorithms in the message passing framework because it removes the need for rederiving approximate posterior density functions if there is a change to the model. Instead, model changes are handled by changing specific computational units, known as fragments, within the factor graph. We extend the notion of variational message passing to functional principal components analysis. Indeed, this is the first article to address a functional data model via variational message passing. Our approach introduces two new fragments that are necessary for Bayesian functional principal components analysis. We present the computational details, a set of simulations for assessing accuracy and speed and an application to United States temperature data.
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Functional principal component analysis (FPCA) has been widely used to capture major modes of variation and reduce dimensions in functional data analysis. However, standard FPCA based on the sample covariance estimator does not work well in the presence of outliers. To address this challenge, a new robust functional principal component analysis approach based on the functional pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced where we propose estimation procedures for both eigenfunctions and eigenvalues with and without measurement error. Compared to existing robust FPCA methods, the proposed one requires weaker distributional assumptions to conserve the eigenspace of the covariance function. In particular, a class of distributions called the weakly functional coordinate symmetric (weakly FCS) is introduced that allows for severe asymmetry and is strictly larger than the functional elliptical distribution class, the latter of which has been well used in the robust statistics literature. The robustness of the PASS FPCA is demonstrated via simulation studies and analyses of accelerometry data from a large-scale epidemiological study of physical activity on older women that partly motivates this work.
Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alternative approach to FPCA using penalized rank one approximation to the data matrix. Our contributions are four-fold: (1) by considering invariance under scale transformation of the measurements, the new formulation sheds light on how regularization should be performed for FPCA and suggests an efficient power algorithm for computation; (2) it naturally incorporates spline smoothing of discretized functional data; (3) the connection with smoothing splines also facilitates construction of cross-validation or generalized cross-validation criteria for smoothing parameter selection that allows efficient computation; (4) different smoothing parameters are permitted for different FPCs. The methodology is illustrated with a real data example and a simulation.
We present a new functional Bayes classifier that uses principal component (PC) or partial least squares (PLS) scores from the common covariance function, that is, the covariance function marginalized over groups. When the groups have different covariance functions, the PC or PLS scores need not be independent or even uncorrelated. We use copulas to model the dependence. Our method is semiparametric; the marginal densities are estimated nonparametrically by kernel smoothing and the copula is modeled parametrically. We focus on Gaussian and t-copulas, but other copulas could be used. The strong performance of our methodology is demonstrated through simulation, real data examples, and asymptotic properties.
The principal submatrix localization problem deals with recovering a $Ktimes K$ principal submatrix of elevated mean $mu$ in a large $ntimes n$ symmetric matrix subject to additive standard Gaussian noise. This problem serves as a prototypical example for community detection, in which the community corresponds to the support of the submatrix. The main result of this paper is that in the regime $Omega(sqrt{n}) leq K leq o(n)$, the support of the submatrix can be weakly recovered (with $o(K)$ misclassification errors on average) by an optimized message passing algorithm if $lambda = mu^2K^2/n$, the signal-to-noise ratio, exceeds $1/e$. This extends a result by Deshpande and Montanari previously obtained for $K=Theta(sqrt{n}).$ In addition, the algorithm can be extended to provide exact recovery whenever information-theoretically possible and achieve the information limit of exact recovery as long as $K geq frac{n}{log n} (frac{1}{8e} + o(1))$. The total running time of the algorithm is $O(n^2log n)$. Another version of the submatrix localization problem, known as noisy biclustering, aims to recover a $K_1times K_2$ submatrix of elevated mean $mu$ in a large $n_1times n_2$ Gaussian matrix. The optimized message passing algorithm and its analysis are adapted to the bicluster problem assuming $Omega(sqrt{n_i}) leq K_i leq o(n_i)$ and $K_1asymp K_2.$ A sharp information-theoretic condition for the weak recovery of both clusters is also identified.
Sparse principal component analysis (PCA) is a popular tool for dimensional reduction of high-dimensional data. Despite its massive popularity, there is still a lack of theoretically justifiable Bayesian sparse PCA that is computationally scalable. A major challenge is choosing a suitable prior for the loadings matrix, as principal components are mutually orthogonal. We propose a spike and slab prior that meets this orthogonality constraint and show that the posterior enjoys both theoretical and computational advantages. Two computational algorithms, the PX-CAVI and the PX-EM algorithms, are developed. Both algorithms use parameter expansion to deal with the orthogonality constraint and to accelerate their convergence speeds. We found that the PX-CAVI algorithm has superior empirical performance than the PX-EM algorithm and two other penalty methods for sparse PCA. The PX-CAVI algorithm is then applied to study a lung cancer gene expression dataset. $mathsf{R}$ package $mathsf{VBsparsePCA}$ with an implementation of the algorithm is available on The Comprehensive R Archive Network.
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