Blind Unwrapping of Modulo Reduced Gaussian Vectors: Recovering MSBs from LSBs
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We consider the problem of recovering n i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo Δ, for some Δ > 0. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if an informed decoder that knows the covariance matrix can recover each sample with small error probability, and n is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions.Keywords:
Orthogonal complement
Complement
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It was described the investigation of statistic distributions used in tests of hypotheses about covariance matrix (in the cases when the matrix is known or the matrix was estimated) and the general assumption of correlation analysis is wrong. The obtained results allow determine, the limits of using of the correlation analysis methods for multivariate distributions modeling on the base of the exponential distribution family.
Statistic
Scatter matrix
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In this research article, k — Modulo Infinite Array and Complement of k — Modulo Infinite Array are introduced. The algebraic and Circulant properties for k — Modulo Infinite Array and Complement of k — Modulo Infinite Array are deliberated. The Norm and Spectrum of k — Modulo Infinite Array are derived and generalized for any dimension. The Generalized inverse is existed for k — Modulo Infinite Array and Complement of k — Modulo Infinite Array and also the Special inverses are elaborated. The subarrays of k — Modulo Infinite Array and Complement of k — Modulo Infinite Array have inherited the existence of Special inverses such as Quasi Commuting inverse, Drazin inverse and Group inverse, which are explored with suitable numerical illustrations.
Modulo operation
Complement
Multiplicative inverse
Orthogonal complement
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Abstract We consider testing the equality of vector means against a multivariate ordered alternative. An extension of the Abelson and Tukey test is first derived under the assumption that the covariance matrix is known. The asymptotic distribution is obtained when the covariance is estimated. We also extend Schaafsma and Smid's somewhere most powerful test to multivariate normal settings, under the assumption that the covariance matrix is known. An empirical power study shows that these tests perform better than their multi-sided χ2 test counterparts.
Scatter matrix
Matrix (chemical analysis)
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Extended Kalman filter (EKF) is prevailing for cooperative localization, where the cross-covariance (representing the correlation of estimated position) determines the benefit quantity from the local measurement. In this paper, the covariance factor set is adopted for cross-covariance maintaining in distributed architecture. During two exteroceptive measurements, the covariance factor set is propagated independently in each agent. When the updating information from the measuring agent is received by the other agents, a temporary relative master-slave relationship is determined between them. The updated correlation is retained in the receiver (slave) agent as a covariance factor. Meanwhile, the counterpart in the measuring (master) agent is set as identify matrix. The operation of matrix decomposition and the feedback for covariance update from slave to master is saved. Thus, the computational consumption and communication burden are reduced. It is significant for real-time cooperative localization.
Master/slave
Position (finance)
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A comparison among VMIX, VMAX and the adapted step-down Sullivan et al. (SE) tests for covariance matrix under bivariate normal assumption is presented. The type I error and power estimates were obtained by using Monte Carlo simulation under different scenarios with respect to covariance and correlation structures. In general, VMIX was more powerful than VMAX being SE more powerful than both, with few exceptions. SE test is more general since it can be used for normal and non-normal data, with no restriction with respect to the pattern of the covariance matrix shifts, and for larger dimension than the bivariate case.
Covariance mapping
Matrix (chemical analysis)
Matérn covariance function
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ABSTRACT The relative efficiency of maximum likelihood estimates is studied when taking advantage of underlying linear patterns in the covariances or correlations when estimating covariance matrices. We compare the variances of estimates of the covariance matrix obtained under two nested patterns with the assumption that the more restricted pattern is the true state. Formulas for the asymptotic variances are given which are exact for linear covariance patterns when explicit maximum likelihood estimates exist. Several specific examples are given using complete symmetry, circular symmetry and general covariance patterns as well as an example involving a covariance matrix with a linear pattern in the correlations.
General Covariance
Covariance mapping
Analysis of covariance
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This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from different Wishart ratios, that describe the process for the two consecutive time periods before and immediately after the change in the covariance structure took place. The product moments of these constructed random variables are highlighted and set the scene for a proposed measure to enable the practitioner to calculate the run-length probability to detect a shift immediately after a change in the covariance matrix occurs. Our results open a new approach and provides insight for detecting the change in the parameter structure as soon as possible once the underlying process, described by a multivariate normal process, encounters a permanent/sustained upward or downward shift.
Scatter matrix
Inverse-Wishart distribution
Matrix t-distribution
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The purpose of this article is to introdused the inference techniques for the mean vector μ, the correlation matrix π and the covariance matrix Σ of the multivariate normal sample and to apply these techniques using the software package STATISTICA 13.0 (StatSoft Inc, USA). The sample contains 50 weekly return observations (in percent) on each of ten stock portfolios constructed from stocks on the Toronto Stock Exchanges. Since the data are obtained as a random sample of multivariate normal distribution the Wishart distribution can be used to make inference about covariance matrix.
Inverse-Wishart distribution
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Scatter matrix
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The concept of moduloïd over a dioïd has been introduced in M. Gondran and M. Minoux In this paper we investigate two notions of dimension for moduloïds, one being weaker than the other. When D is a pseudo-ring both coïncide. Then we proove existence and unicity of weak bases for finitely generated moduloïds.
Modulo operation
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Covariance matrix estimation is a persistent challenge for cosmology. We focus on a class of model covariance matrices that can be generated with high accuracy and precision, using a tiny fraction of the computational resources that would be required to achieve comparably precise covariance matrices using mock catalogues. In previous work, the free parameters in these models were determined using sample covariance matrices computed using a large number of mocks, but we demonstrate that those parameters can be estimated consistently and with good precision by applying jackknife methods to a single survey volume. This enables model covariance matrices that are calibrated from data alone, with no reference to mocks.
Jackknife resampling
General Covariance
Covariance mapping
Matérn covariance function
Analysis of covariance
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