Fitting covariance matrix models to simulations
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Data analysis in cosmology requires reliable covariance matrices. Covariance matrices derived from numerical simulations often require a very large number of realizations to be accurate. When a theoretical model for the covariance matrix exists, the parameters of the model can often be fit with many fewer simulations. We write a likelihood-based method for performing such a fit. We demonstrate how a model covariance matrix can be tested by examining the appropriate $χ^2$ distributions from simulations. We show that if model covariance has amplitude freedom, the expectation value of second moment of $χ^2$ distribution with a wrong covariance matrix will always be larger than one using the true covariance matrix. By combining these steps together, we provide a way of producing reliable covariances without ever requiring running a large number of simulations. We demonstrate our method on two examples. First, we measure the two-point correlation function of halos from a large set of $10000$ mock halo catalogs. We build a model covariance with $2$ free parameters, which we fit using our procedure. The resulting best-fit model covariance obtained from just $100$ simulation realizations proves to be as reliable as the numerical covariance matrix built from the full $10000$ set. We also test our method on a setup where the covariance matrix is large by measuring the halo bispectrum for thousands of triangles for the same set of mocks. We build a block diagonal model covariance with $2$ free parameters as an improvement over the diagonal Gaussian covariance. Our model covariance passes the $χ^2$ test only partially in this case, signaling that the model is insufficient even using free parameters, but significantly improves over the Gaussian one.Keywords:
Matrix (chemical analysis)
A simulation study comparing powers of the multivariate analysis (PROC GLM) with using a mixed model (PROC MIXED) using a variety of covariance structures for various treatment, time, and interaction affect sizes in a repeated measures design is conducted. Type 1 errors are estimated. Powers are estimated for a variety of covariance structures when the actual covariance structure is AR (1). It was found that the estimated powers for treatment effect were all very similar with PROC MIXED with the correct covariance structure having the largest estimated powers. When testing for time and interaction effect, it was found when in doubt that it was better to use a simpler covariance structure. The powers were generally higher in this case than with a more complex covariance structure.
Analysis of covariance
Repeated measures design
Covariance mapping
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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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With an experimental study,the application of covariance without interaction and with interaction in experiment with pre-measurement and post-measurement was discussed.And comparing with the of covariance for difference value designed by the authors,the two analysis methods had the similar results.
Analysis of covariance
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Covariance mapping
Matérn covariance function
General Covariance
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Covariance is a measure to characterize the joint variability of two complex uncertain variables. Since, calculating covariance is not easy based on uncertain measure, we present two formulas for covariance and pseudo covariance of complex uncertain variables. For calculating the covariance of comp lex uncertain variables, some theorems are proved and several formulas are provided by using the inverse uncertainty distribution. The main results are explained by using several examples.
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Covariance mapping
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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.
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A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.
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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.
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Covariance mapping
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Matérn covariance function
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A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.
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General Covariance
Covariance operator
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