Adjoints and low-rank covariance representation
نویسندگان
چکیده
Quantitative measures of the uncertainty of Earth system estimates can be as important as the estimates themselves. Direct calculation of second moments of estimation errors, as described by the covariance matrix, is impractical when the number of degrees of freedom of the system state is large and the sources of uncertainty are not completely known. Theoretical analysis of covariance equations can help guide the formulation of low-rank covariance approximations, such as those used in ensemble and reducedstate approaches for prediction and data assimilation. We use the singular value decomposition and recently developed positive map techniques to analyze a family of covariance equations that includes stochastically forced linear systems. We obtain covariance estimates given imperfect knowledge of the sources of uncertainty and we obtain necessary conditions for low-rank approximations to be appropriate. The results are illustrated in a stochastically forced system with time-invariant linear dynamics.
منابع مشابه
Deep Unsupervised Domain Adaptation for Image Classification via Low Rank Representation Learning
Domain adaptation is a powerful technique given a wide amount of labeled data from similar attributes in different domains. In real-world applications, there is a huge number of data but almost more of them are unlabeled. It is effective in image classification where it is expensive and time-consuming to obtain adequate label data. We propose a novel method named DALRRL, which consists of deep ...
متن کاملImage segmentation with superpixel-based covariance descriptors in low-rank representation
This paper investigates the problem of image segmentation using superpixels. We propose two approaches to enhance the discriminative ability of the superpixel’s covariance descriptors. In the first one, we employ the Log-Euclidean distance as the metric on the covariance manifolds, and then use the RBF kernel to measure the similarities between covariance descriptors. The second method is focus...
متن کاملTrek Separation for Gaussian Graphical Models
Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that inclu...
متن کاملTile Low Rank Cholesky Factorization for Climate/Weather Modeling Applications on Manycore Architectures
Covariance matrices are ubiquitous in computational science and engineering. In particular, large covariance matrices arise from multivariate spatial data sets, for instance, in climate/weather modeling applications to improve prediction using statistical methods and spatial data. One of the most time-consuming computational steps consists in calculating the Cholesky factorization of the symmet...
متن کاملMultichannel signal processing using spatial rank covariance matrices
This paper addresses the problem of estimating the covariance matrix reliably when the assumptions, such as Gaussianity, on the probabilistic nature of multichannel data do not necessarily hold. Multivariate spatial sign and rank functions, which are generalizations of univariate sign and centered rank, are introduced. Furthermore, spatial rank covariance matrix and spatial Kendall’s tau covari...
متن کامل