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Register a new userA Comparison of Functional Principal Component Analysis Methods with Accelerometry Applications
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Added by
Bohan Wu
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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The association between a persons physical activity and various health outcomes is an area of active research. The National Health and Nutrition Examination Survey (NHANES) data provide a valuable resource for studying these associations. NHANES accelerometry data has been used by many to measure individuals activity levels. A common approach for analyzing accelerometry data is functional principal component analysis (FPCA). The first part of the paper uses Poisson FPCA (PFPCA), Gaussian FPCA (GFPCA), and nonnegative and regularized function decomposition (NARFD) to extract features from the count-valued NHANES accelerometry data. The second part of the paper compares logistic regression, random forests, and AdaBoost models based on GFPCA, NARFD, or PFPCA scores in the context of mortality prediction. The results show that Poisson FPCA is the best FPCA model for the inference of accelerometry data, and the AdaBoost model based on Poisson FPCA scores gives the best mortality prediction results.
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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.
Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional manifold. For this purpose a regularization approach is adopted, introducing a smoothing penalty coherent with the geodesic distance over the manifold. The model introduced can be applied to any manifold topology, can naturally handle missing data and functional samples evaluated in different grids of points. We approach the discretization task by means of finite element analysis and propose an efficient iterative algorithm for its resolution. We compare the performances of the proposed algorithm with other approaches classically adopted in literature. We finally apply the proposed method to resting state functional magnetic resonance imaging data from the Human Connectome Project, where the method shows substantial differential variations between brain regions that were not apparent with other approaches.
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvement over the naive method of combining PCA with regression. To achieve our results, we first observe that ridge regression can be used to obtain a smooth projection onto the top principal components. We then sharpen this approximation to true projection using a low-degree polynomial approximation to the matrix step function. Step function approximation is a topic of long-term interest in scientific computing. We extend prior theory by constructing polynomials with simple iterative structure and rigorously analyzing their behavior under limited precision.
Functional principal component analysis (FPCA) could become invalid when data involve non-Gaussian features. Therefore, we aim to develop a general FPCA method to adapt to such non-Gaussian cases. A Kenalls $tau$ function, which possesses identical eigenfunctions as covariance function, is constructed. The particular formulation of Kendalls $tau$ function makes it less insensitive to data distribution. We further apply it to the estimation of FPCA and study the corresponding asymptotic consistency. Moreover, the effectiveness of the proposed method is demonstrated through a comprehensive simulation study and an application to the physical activity data collected by a wearable accelerometer monitor.
Functional binary datasets occur frequently in real practice, whereas discrete characteristics of the data can bring challenges to model estimation. In this paper, we propose a sparse logistic functional principal component analysis (SLFPCA) method to handle the functional binary data. The SLFPCA looks for local sparsity of the eigenfunctions to obtain convenience in interpretation. We formulate the problem through a penalized Bernoulli likelihood with both roughness penalty and sparseness penalty terms. An efficient algorithm is developed for the optimization of the penalized likelihood using majorization-minimization (MM) algorithm. The theoretical results indicate both consistency and sparsistency of the proposed method. We conduct a thorough numerical experiment to demonstrate the advantages of the SLFPCA approach. Our method is further applied to a physical activity dataset.
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