Principal Component Analysis (PCA) is a commonly used tool for dimension reduction in analyzing high dimensional data; Multilinear Principal Component Analysis (MPCA) has the potential to serve the similar function for analyzing tensor structure data. MPCA and other tensor decomposition methods have been proved effective to reduce the dimensions for both real data analyses and simulation studies (Ye, 2005;Lu, Plataniotis and Venetsanopoulos, 2008;Kolda and Bader, 2009;Li, Kim and Altman, 2010). In this paper, we investigate MPCA's statistical properties and provide explanations for its advantages. Conventional PCA, vectorizing the tensor data, may lead to inefficient and unstable prediction due to its extremely large dimensionality. On the other hand, MPCA, trying to preserve the data structure, searches for low-dimensional multilinear projections and decreases the dimensionality efficiently. The asymptotic theories for order-two MPCA, including asymptotic distributions for principal components, associated projections and the explained variance, are developed. Finally, MPCA is shown to improve conventional PCA on analyzing the Olivetti Faces data set, by constructing more module oriented basis in reconstructing the test faces.
Cryo-electron microscopy (cryo-EM) has recently emerged as a powerful tool for obtaining three-dimensional (3D) structures of biological macromolecules in native states. A minimum cryo-EM image data set for deriving a meaningful reconstruction is comprised of thousands of randomly orientated projections of identical particles photographed with a small number of electrons. The computation of 3D structure from 2D projections requires clustering, which aims to enhance the signal to noise ratio in each view by grouping similarly oriented images. Nevertheless, the prevailing clustering techniques are often compromised by three characteristics of cryo-EM data: high noise content, high dimensionality and large number of clusters. Moreover, since clustering requires registering images of similar orientation into the same pixel coordinates by 2D alignment, it is desired that the clustering algorithm can label misaligned images as outliers. Herein, we introduce a clustering algorithm γ-SUP to model the data with a q-Gaussian mixture and adopt the minimum γ-divergence for estimation, and then use a self-updating procedure to obtain the numerical solution. We apply γ-SUP to the cryo-EM images of two benchmark macromolecules, RNA polymerase II and ribosome. In the former case, simulated images were chosen to decouple clustering from alignment to demonstrate γ-SUP is more robust to misalignment outliers than the existing clustering methods used in the cryo-EM community. In the latter case, the clustering of real cryo-EM data by our γ-SUP method eliminates noise in many views to reveal true structure features of ribosome at the projection level.
Generalized Bayes posterior distributions are formed by putting a fractional power on the likelihood before combining with the prior via Bayes's formula. This fractional power, which is often viewed as a remedy for potential model misspecification bias, is called the learning rate, and a number of data-driven learning rate selection methods have been proposed in the recent literature. Each of these proposals has a different focus, a different target they aim to achieve, which makes them difficult to compare. In this paper, we provide a direct head-to-head comparison of these learning rate selection methods in various misspecified model scenarios, in terms of several relevant metrics, in particular, coverage probability of the generalized Bayes credible regions. In some examples all the methods perform well, while in others the misspecification is too severe to be overcome, but we find that the so-called generalized posterior calibration algorithm tends to outperform the others in terms of credible region coverage probability.
Generalized Bayes posterior distributions are formed by putting a fractional power on the likelihood before combining with the prior via Bayes's formula. This fractional power, which is often viewed as a remedy for potential model misspecification bias, is called the learning rate, and a number of data-driven learning rate selection methods have been proposed in the recent literature. Each of these proposals has a different focus, a different target they aim to achieve, which makes them difficult to compare. In this paper, we provide a direct head-to-head empirical comparison of these learning rate selection methods in various misspecified model scenarios, in terms of several relevant metrics, in particular, coverage probability of the generalized Bayes credible regions. In some examples all the methods perform well, while in others the misspecification is too severe to be overcome, but we find that the so-called generalized posterior calibration algorithm tends to outperform the others in terms of credible region coverage probability.
SUMMARY We extend the notion of an influence or hat matrix to regression with functional responses and scalar predictors. For responses depending linearly on a set of predictors, our definition is shown to reduce to the conventional influence matrix for linear models. The pointwise degrees of freedom, the trace of the pointwise influence matrix, are shown to have an adaptivity property that motivates a two-step bivariate smoother for modeling nonlinear dependence on a single predictor. This procedure adapts to varying complexity of the nonlinear model at different locations along the function, and thereby achieves better performance than competing tensor product smoothers in an analysis of the development of white matter microstructure in the brain.
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