1,721,046 research outputs found
Polynomial-exponential 2D data models, Hankel-block-Hankel matrices and zero-dimensional ideals
No full textSubspace-based methods are popular for analysis of two-dimensional data that can be modeled by sums of polynomially modulated exponential (or “polynomial-exponential”) functions. In this paper we touch some problems concerning rank properties of Hankel-block-Hankel matrices, which are used in subspace-based methods. We review the correspondence between polynomial-exponential functions and zero-dimensional ideals. Then we demonstrate the usefulness of this correspondence for the problems being considered
Improved initial approximation for errors-in-variables system identification
No full textErrors-in-variables system identification can be posed and solved as a Hankel structured low-rank approximation problem. In this paper different estimates based on suboptimal low-rank approximations are considered. The estimates are shown to have almost the same efficiency and lead to the same minimum when supplied as an initial approximation to local optimization solver of the structured low-rank approximation problem. In this paper it is shown that increasing Hankel matrix window length improves suboptimal estimates for autonomous systems and does not improve them for systems with inputs
An Algebraic View on Finite Rank in 2D-SSA
No full textThe 2D-SSA method provides a decomposition of a 2D-array (a function of two variables, e.g. digital image) into a sum of identifiable components. For the decomposition to be proper, these components should be close to 2D-arrays of finite rank. This paper is devoted to study of arrays of finite rank by means of polynomial ideals generated by arrays. The 2D-arrays are considered as functionals of polynomials. A general form of arrays of finite rank is obtained. The structure of finite-rank arrays and their trajectory spaces is investigated
Structured low-rank approximation with missing data
No full textThe approach of SIAM J. Matrix Anal. Appl., 26(4):1083--1099 for solving structured total least squares problems is generalized to weighted structured low-rank approximation with missing data. The method proposed is based on elimination of the correction matrix and solution of the resulting nonlinear least squares problem by local optimization methods. The elimination step is a singular linear least-norm problem, which admits an analytic solution. Two approaches are proposed for the nonlinear least-squares minimization: minimization subject to equality constraints and unconstrained minimization with regularized cost function. The method is generalized to weighted low-rank approximation with singular weight matrix and is illustrated on matrix completion, system identification, and data-driven simulation problems. An extended version of the paper is a literate program, implementing the method and reproducing the presented results
Structured Low-rank Approximation as a Rational Function Minimization
Full text linkMany problems of system identification, model reduction and signal processing can be posed and solved as a structured low-rank approximation problem (SLRA). In this paper a reformulation of SLRA as minimization of a multivariate rational function is considered. Using two different parametrizations, we show that the problem reduces to optimization over a compact manifold or to a set of optimization problems over bounded domains of Euclidean space. We make a review of methods of polynomial algebra for global optimization of the rational cost function
On signal and extraneous roots in Singular Spectrum Analysis
Full text linkIn the present paper we study properties of roots of characteristic polynomials for the linear recurrent formulae (LRF) that govern time series. We also investigate how the values of these roots affect Singular Spectrum Analysis implications, in what concerns separation of components, SSA forecasting and related signal parameter estimation methods. The roots of the characteristic polynomial for an LRF comprise the signal roots, which determine the structure of the time series, and extraneous roots. We show how the separability of two time series can be characterized in terms of their signal roots. All possible cases of exact separability are enumerated. We also examine properties of extraneous roots of the LRF used in SSA forecasting algorithms, which is equivalent to the Min-Norm vector in subspace-based estimation methods. We apply recent theoretical results for orthogonal polynomials on the unit circle, which enable us to precisely describe the asymptotic distribution of extraneous roots relative to the position of the signal roots
Filtering of digital terrain models by 2D Singular Spectrum Analysis
No full textSingular Spectrum Analysis (SSA) has been approved as a model-free technique to analyse time series. SSA can solve different problems such as decomposition into a sum of trend, periodicities, and noise, smoothing, and others. In this paper, we validate abilities of 2D-SSA (the extension of SSA to analyse two-dimensional scalar fields) to treat digital terrain models (DTMs). The study is exemplified by a 30-arc-second digital elevation model of a part of South America derived from GTOPO30. Results demonstrate that 2D-SSA is an efficient method to denoise and generalise DTMs. It can be also used to decompose a topographic surface into components of continental, regional, and local scales
Symptom and syndrome analysis of categorial series, logical principles and forms of logic
No full textThe calculation of variables of one metering type by the variables of others metering types as the process leads to the forms of logic which are described by means of the collineation group of the projective geometry. Missing metering types are replaced with the appropriate logic principles. The logic of sufficiency (preferences) on the basis of orders and the logic principle of duality is considered in detail on an RNA connectivity analysis example. The minimal sum of diagonal elements in the cross-tabulation of two finitely-linear combinations (symptoms) of fragments which differ by a shift on the given quantity of symbols is chosen as a relation between two dichotomizing series. If this relation is zero then correct classification takes place. In this paper the probability of random classification with given number of errors is estimated. Logic principle of duality allows to distinguish between weak and strong statistically significant relations
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