224 research outputs found
On the Early History of the Singular Value Decomposition
This paper surveys the contributions of five
mathematicians\,---\,Eugenio Beltrami (1835--1899), Camille Jordan
(1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt
(1876--1959), and Hermann Weyl (1885--1955)\,---\,who were responsible
for establishing the existence of the singular value decomposition
and developing its theory.
(Also cross-referenced as UMIACS-TR-92-31
Shrinkage Function And Its Applications In Matrix Approximation
The shrinkage function is widely used in matrix low-rank approximation,
compressive sensing, and statistical estimation. In this article, an elementary
derivation of the shrinkage function is given. In addition, applications of the
shrinkage function are demonstrated in solving several well-known problems,
together with a new result in matrix approximation
Joint Covariance Estimation with Mutual Linear Structure
We consider the problem of joint estimation of structured covariance
matrices. Assuming the structure is unknown, estimation is achieved using
heterogeneous training sets. Namely, given groups of measurements coming from
centered populations with different covariances, our aim is to determine the
mutual structure of these covariance matrices and estimate them. Supposing that
the covariances span a low dimensional affine subspace in the space of
symmetric matrices, we develop a new efficient algorithm discovering the
structure and using it to improve the estimation. Our technique is based on the
application of principal component analysis in the matrix space. We also derive
an upper performance bound of the proposed algorithm in the Gaussian scenario
and compare it with the Cramer-Rao lower bound. Numerical simulations are
presented to illustrate the performance benefits of the proposed method
Assessing Protein Conformational Sampling Methods Based on Bivariate Lag-Distributions of Backbone Angles
Despite considerable progress in the past decades, protein structure prediction remains one of the major unsolved problems in computational biology. Angular-sampling-based methods have been extensively studied recently due to their ability to capture the continuous conformational space of protein structures. The literature has focused on using a variety of parametric models of the sequential dependencies between angle pairs along the protein chains. In this article, we present a thorough review of angular-sampling-based methods by assessing three main questions: What is the best distribution type to model the protein angles? What is a reasonable number of components in a mixture model that should be considered to accurately parameterize the joint distribution of the angles? and What is the order of the local sequence–structure dependency that should be considered by a prediction method? We assess the model fits for different methods using bivariate lag-distributions of the dihedral/planar angles. Moreover, the main information across the lags can be extracted using a technique called Lag singular value decomposition (LagSVD), which considers the joint distribution of the dihedral/planar angles over different lags using a nonparametric approach and monitors the behavior of the lag-distribution of the angles using singular value decomposition. As a result, we developed graphical tools and numerical measurements to compare and evaluate the performance of different model fits. Furthermore, we developed a web-tool (http://www.stat.tamu.edu/∼madoliat/LagSVD) that can be used to produce informative animations
Integrating Data Transformation in Principal Components Analysis
Principal component analysis (PCA) is a popular dimension-reduction method to reduce the complexity and obtain the informative aspects of high-dimensional datasets. When the data distribution is skewed, data transformation is commonly used prior to applying PCA. Such transformation is usually obtained from previous studies, prior knowledge, or trial-and-error. In this work, we develop a model-based method that integrates data transformation in PCA and finds an appropriate data transformation using the maximum profile likelihood. Extensions of the method to handle functional data and missing values are also developed. Several numerical algorithms are provided for efficient computation. The proposed method is illustrated using simulated and real-world data examples. Supplementary materials for this article are available online
Modified Linear Projection for Large Spatial Data Sets
Recent developments in engineering techniques for spatial data collection
such as geographic information systems have resulted in an increasing need for
methods to analyze large spatial data sets. These sorts of data sets can be
found in various fields of the natural and social sciences. However, model
fitting and spatial prediction using these large spatial data sets are
impractically time-consuming, because of the necessary matrix inversions.
Various methods have been developed to deal with this problem, including a
reduced rank approach and a sparse matrix approximation. In this paper, we
propose a modification to an existing reduced rank approach to capture both the
large- and small-scale spatial variations effectively. We have used simulated
examples and an empirical data analysis to demonstrate that our proposed
approach consistently performs well when compared with other methods. In
particular, the performance of our new method does not depend on the dependence
properties of the spatial covariance functions.Comment: 29 pages, 5 figures, 4 table
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