Статті в журналах: "Covariance matrice"

1

Cappuccio, Nunzio, and Diego Lubian. "Ordering of Covariance Matrice." Econometric Theory 12, no. 4 (October 1996): 746–48. http://dx.doi.org/10.1017/s0266466600007106.

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2

Sigaud, Olivier, and Freek Stulp. "Adaptation de la matrice de covariance pour l’apprentissage par renforcement direct." Revue d'intelligence artificielle 27, no. 2 (April 30, 2013): 243–63. http://dx.doi.org/10.3166/ria.27.243-263.

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3

Fourdrinier, Dominique, William E. Strawderman, and Martin T. Wells. "Estimation robuste pour des lois à symétrie elliptique à matrice de covariance inconnue." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 326, no. 9 (May 1998): 1135–40. http://dx.doi.org/10.1016/s0764-4442(98)80076-1.

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4

Appourchaux, T., and L. Gizon. "The Art of Fitting P-Mode Spectra." Symposium - International Astronomical Union 185 (1998): 43–44. http://dx.doi.org/10.1017/s0074180900238230.

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For deriving p-mode parameters from m, v diagrammes, one has to treat correctly the statistics of the observation. The correct statistical treatment of these diagrammes was first achieved by Schou (1992) (PhD thesis, Aarhus University). Fitting p-mode spectra requires 4 major steps: 1.Compute the mode leakage matrices2.Compute mode covariance matrices from the previous matrices3.Compute the noise covariance matrices4.Compute and maximize the likelihood of the observation

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5

Khoder, Wassim. "Recalage de la navigation inertielle hybride par le filtrage de Kalman sans parfum paramétré à quaternions." MATEC Web of Conferences 261 (2019): 06003. http://dx.doi.org/10.1051/matecconf/201926106003.

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Dans ce papier, nous avons développé un algorithme d’hybridation (recalage) de la navigation inertielle, noté Q-SUKF, qui combine le filtre de Kalman sans parfum à paramètre (SUKF) et l’utilisation des propriétés de rotation et d’unicité des quaternions (Q) pour représenter l’attitude. L’utilisation des quaternions unités dans le calcul de la matrice de covariance d’erreurs prédite empêche les problèmes de singularité et la dérive des informations d’attitude. L’augmentation de l’incertitude dans les angles d’attitude, est modélisé par un vecteur de rotation pour garantir que la normalisation du quaternion est toujours maintenue dans le filtre. Le Q-SUKF proposé est bien adapté pour estimer récursivement les états de la navigation, quelque soient les valeurs initiales sur les angles d’attitude ou la dynamique des mouvements du mobile, à l’aide des capteurs externes qui sont complémentaires et/ou redondants.

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6

Meyer, Karin, and Mark Kirkpatrick. "Up hill, down dale: quantitative genetics of curvaceous traits." Philosophical Transactions of the Royal Society B: Biological Sciences 360, no. 1459 (July 7, 2005): 1443–55. http://dx.doi.org/10.1098/rstb.2005.1681.

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‘Repeated’ measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as ‘function-valued’ (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances. Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multi-dimensional analyses are discussed.

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7

Alekseychik, Pavel, Gabriel Katul, Ilkka Korpela, and Samuli Launiainen. "Eddies in motion: visualizing boundary-layer turbulence above an open boreal peatland using UAS thermal videos." Atmospheric Measurement Techniques 14, no. 5 (May 18, 2021): 3501–21. http://dx.doi.org/10.5194/amt-14-3501-2021.

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Abstract. High-resolution thermal infrared (TIR) imaging is opening up new vistas in biosphere–atmosphere heat exchange studies. The rapidly developing unmanned aerial systems (UASs) and specially designed cameras offer opportunities for TIR survey with increasingly high resolution, reduced geometric and radiometric noise, and prolonged flight times. A state-of-the-art science platform is assembled using a Matrice 210 V2 drone equipped with a Zenmuse XT2 thermal camera and deployed over a pristine boreal peatland with the aim of testing its performance in a heterogeneous sedge-fen ecosystem. The study utilizes the capability of the UAS platform to hover for prolonged times (about 20 min) at a height of 500 m a.g.l. while recording high frame rate (30 Hz) TIR videos of an area of ca. 430 × 340 m. A methodology is developed to derive thermal signatures of near-ground coherent turbulent structures impinging on the land surface, surface temperature spectra, and heat fluxes from the retrieved videos. The size, orientation, and movement of the coherent structures are computed from the surface temperature maps, and their dependency on atmospheric conditions is examined. A range of spectral and wavelet-based approaches are used to infer the properties of the dominant turbulent scene structures. A ground-based eddy-covariance system and an in situ meteorological setup are used for reference.

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8

Gonzalez-Ondina, Jose M., Lewis Sampson, and Georgy I. Shapiro. "A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling." Journal of Marine Science and Engineering 9, no. 12 (December 20, 2021): 1461. http://dx.doi.org/10.3390/jmse9121461.

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Data assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). Since the true values are never known in practice, the error covariance matrices containing values of the covariance functions at different locations, are estimated approximately. Several methods have been devised to compute these matrices, one of the most widely used is the one developed by Hollingsworth and Lönnberg (H-L). This method requires to bin (combine) the data points separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method uses functional analysis which allows to overcome some shortcomings of the H-L method, for example, the assumption of statistical isotropy. It also eliminates the intermediate steps used in the H-L method such as binning the innovations (differences between observations and the model), and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also demonstrate that in many cases the new method allows to use the separable convolution mathematical algorithm to increase the computational speed significantly, up to an order of magnitude. The Projection Method (PROM) also allows computing 2D and 3D covariance functions in addition to the standard 1D case.

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9

Zhang, Peng, Wen Juan Qi, and Zi Li Deng. "Covariance Intersection Fusion Kalman Estimator for Multi-Sensor System with Measurements Delays." Applied Mechanics and Materials 475-476 (December 2013): 460–65. http://dx.doi.org/10.4028/www.scientific.net/amm.475-476.460.

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To handle the state estimation fusion problem between local estimation errors for the system with unknown cross-covariances and to avoid a large computation complexity of cross-covariances, for a multi-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the measurement transformation method, an equivalent system without measurement delays is obtained, and then using the covariance intersection (CI) fusion method, the covariance intersection fusion steady-state Kalman estimator is presented. It is proved that its accuracy is higher than that of each local estimator, and is lower than that of optimal Kalman fuser weighted by matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relations, hence it has good performances.

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10

Fang (方啸), Xiao, Tim Eifler, and Elisabeth Krause. "2D-FFTLog: efficient computation of real-space covariance matrices for galaxy clustering and weak lensing." Monthly Notices of the Royal Astronomical Society 497, no. 3 (June 17, 2020): 2699–714. http://dx.doi.org/10.1093/mnras/staa1726.

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ABSTRACT Accurate covariance matrices for two-point functions are critical for inferring cosmological parameters in likelihood analyses of large-scale structure surveys. Among various approaches to obtaining the covariance, analytic computation is much faster and less noisy than estimation from data or simulations. However, the transform of covariances from Fourier space to real space involves integrals with two Bessel integrals, which are numerically slow and easily affected by numerical uncertainties. Inaccurate covariances may lead to significant errors in the inference of the cosmological parameters. In this paper, we introduce a 2D-FFTLog algorithm for efficient, accurate, and numerically stable computation of non-Gaussian real-space covariances for both 3D and projected statistics. The 2D-FFTLog algorithm is easily extended to perform real-space bin-averaging. We apply the algorithm to the covariances for galaxy clustering and weak lensing for a Dark Energy Survey Year 3-like and a Rubin Observatory’s Legacy Survey of Space and Time Year 1-like survey, and demonstrate that for both surveys, our algorithm can produce numerically stable angular bin-averaged covariances with the flat sky approximation, which are sufficiently accurate for inferring cosmological parameters. The code CosmoCov for computing the real-space covariances with or without the flat-sky approximation is released along with this paper.

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11

Aboutaleb, Youssef M., Mazen Danaf, Yifei Xie, and Moshe E. Ben-Akiva. "Sparse covariance estimation in logit mixture models." Econometrics Journal 24, no. 3 (March 19, 2021): 377–98. http://dx.doi.org/10.1093/ectj/utab008.

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Summary This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC (mixed integer sparse covariance), that uses a mixed-integer optimization (MIO) program to find an optimal block diagonal structure specification for the covariance matrix, corresponding to subsets of correlated coefficients, for any desired sparsity level using Markov Chain Monte Carlo (MCMC) posterior draws from the unrestricted full covariance matrix. The optimal sparsity level of the covariance matrix is determined using out-of-sample validation. We demonstrate the ability of MISC to correctly recover the true covariance structure from synthetic data. In an empirical illustration using a stated preference survey on modes of transportation, we use MISC to obtain a sparse covariance matrix indicating how preferences for attributes are related to one another.

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12

Silverstein, Jack W., and Z. D. Bai. "Covariance Matrices." Annals of Probability 27, no. 3 (July 1999): 1536–55. http://dx.doi.org/10.1214/aop/1022677458.

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13

Klypin, Anatoly, Francisco Prada, and Joyce Byun. "Suppressing cosmic variance with paired-and-fixed cosmological simulations: average properties and covariances of dark matter clustering statistics." Monthly Notices of the Royal Astronomical Society 496, no. 3 (April 2, 2020): 3862–69. http://dx.doi.org/10.1093/mnras/staa734.

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ABSTRACT Making cosmological inferences from the observed galaxy clustering requires accurate predictions for the mean clustering statistics and their covariances. Those are affected by cosmic variance – the statistical noise due to the finite number of harmonics. The cosmic variance can be suppressed by fixing the amplitudes of the harmonics instead of drawing them from a Gaussian distribution predicted by the inflation models. Initial realizations also can be generated in pairs with 180○ flipped phases to further reduce the variance. Here, we compare the consequences of using paired-and-fixed versus Gaussian initial conditions on the average dark matter clustering and covariance matrices predicted from N-body simulations. As in previous studies, we find no measurable differences between paired-and-fixed and Gaussian simulations for the average density distribution function, power spectrum, and bispectrum. Yet, the covariances from paired-and-fixed simulations are suppressed in a complicated scale- and redshift-dependent way. The situation is particularly problematic on the scales of Baryon acoustic oscillations where the covariance matrix of the power spectrum is lower by only $\sim 20{{\ \rm per\ cent}}$ compared to the Gaussian realizations, implying that there is not much of a reduction of the cosmic variance. The non-trivial suppression, combined with the fact that paired-and-fixed covariances are noisier than from Gaussian simulations, suggests that there is no path towards obtaining accurate covariance matrices from paired-and-fixed simulations – result, that is theoretically expected and accepted in the field. Because the covariances are crucial for the observational estimates of galaxy clustering statistics and cosmological parameters, paired-and-fixed simulations, though useful for some applications, cannot be used for the production of mock galaxy catalogues.

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14

Philcox, Oliver H. E., Daniel J. Eisenstein, Ross O’Connell, and Alexander Wiegand. "rascalc: a jackknife approach to estimating single- and multitracer galaxy covariance matrices." Monthly Notices of the Royal Astronomical Society 491, no. 3 (November 18, 2019): 3290–317. http://dx.doi.org/10.1093/mnras/stz3218.

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ABSTRACT To make use of clustering statistics from large cosmological surveys, accurate and precise covariance matrices are needed. We present a new code to estimate large-scale galaxy two-point correlation function (2PCF) covariances in arbitrary survey geometries that, due to new sampling techniques, runs ∼104 times faster than previous codes, computing finely binned covariance matrices with negligible noise in less than 100 CPU-hours. As in previous works, non-Gaussianity is approximated via a small rescaling of shot noise in the theoretical model, calibrated by comparing jackknife survey covariances to an associated jackknife model. The flexible code, rascalc, has been publicly released, and automatically takes care of all necessary pre- and post-processing, requiring only a single input data set (without a prior 2PCF model). Deviations between large-scale model covariances from a mock survey and those from a large suite of mocks are found to be indistinguishable from noise. In addition, the choice of input mock is shown to be irrelevant for desired noise levels below ∼105 mocks. Coupled with its generalization to multitracer data sets, this shows the algorithm to be an excellent tool for analysis, reducing the need for large numbers of mock simulations to be computed.

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15

Bongiorno, C., D. Challet, and G. Loeper. "Filtering time-dependent covariance matrices using time-independent eigenvalues." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 2 (February 1, 2023): 023402. http://dx.doi.org/10.1088/1742-5468/acb7ed.

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Abstract We propose a data-driven, model-free, way to reduce the noise of covariance matrices of time-varying systems. If the true covariance matrix is time-invariant, non-linear shrinkage of the eigenvalues is known to yield the optimal estimator for large matrices. Such a method outputs eigenvalues that are highly dependent on the inputs, as common sense suggests. When the covariance matrix is time-dependent, we show that it is generally better to use the set of eigenvalues that encode the average influence of the future on present eigenvalues resulting in a set of time-independent average eigenvalues. This situation is widespread in nature, one example being financial markets, where non-linear shrinkage remains the gold-standard filtering method. Our approach outperforms non-linear shrinkage both for the Frobenius norm distance, which is the typical loss function used for covariance filtering and for financial portfolio variance minimization, which makes our method generically relevant to many problems of multivariate inference. Further analysis of financial data suggests that the expected overlap between past eigenvectors and future ones is systematically overestimated by methods designed for constant covariances matrices. Our method takes a simple empirical average of the eigenvector overlap matrix, which is enough to outperform non-linear shrinkage.

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16

Smith, Kimberly, Courtenay Strong, and Firas Rassoul-Agha. "Multisite Generalization of the SHArP Weather Generator." Journal of Applied Meteorology and Climatology 57, no. 9 (September 2018): 2113–27. http://dx.doi.org/10.1175/jamc-d-17-0236.1.

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AbstractGeneralization of point-scale stochastic weather generators to simultaneously produce output at multiple sites provides more powerful support for hydrology and climate change impact studies. Generalization preserves the statistical properties of each individual site while maintaining proper spatial correlation over the domain. Here, generalization of the daily precipitation and temperature components of the stochastic harmonic autoregressive parametric (SHArP) weather generator is presented. The generalization process for temperature involves conversion of vector time series to matrix time series that capture between-site covariances of maximum and minimum daily temperature. Between-site temperature covariances depend on spatial precipitation-occurrence patterns (POPs), of which there are up to 2M for M sites. To dramatically reduce the number of covariance matrices that drive temperature, multisite SHArP uses empirical orthogonal function analysis to categorize the POPs and harmonic smoothing to reduce the number of parameters describing the temporal evolution (annual cycle) of the elements in the covariance matrices. By modeling precipitation-regime-specific residuals, the model is shown to capture statistically significant spatial and temporal contrasts in observed temperature covariance. For precipitation simulation, we extend existing techniques by adding a trend term to the occurrence and amount parameters. Multisite generalization of the framework is illustrated by simulating stochastic historical and future temperature and precipitation across complex terrain over northern Utah on the basis of historical station observations and historical and future statistically downscaled climate model output.

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17

Koch, K. R., H. Kuhlmann, and W. D. Schuh. "Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances." Journal of Geodesy 84, no. 6 (March 19, 2010): 383–97. http://dx.doi.org/10.1007/s00190-010-0375-5.

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18

Fumagalli, Alessandra, Matteo Biagetti, Alex Saro, Emiliano Sefusatti, Anže Slosar, Pierluigi Monaco, and Alfonso Veropalumbo. "Fitting covariance matrix models to simulations." Journal of Cosmology and Astroparticle Physics 2022, no. 12 (December 1, 2022): 022. http://dx.doi.org/10.1088/1475-7516/2022/12/022.

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Abstract 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.

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19

Blot, Linda, Martin Crocce, Emiliano Sefusatti, Martha Lippich, Ariel G. Sánchez, Manuel Colavincenzo, Pierluigi Monaco, et al. "Comparing approximate methods for mock catalogues and covariance matrices II: power spectrum multipoles." Monthly Notices of the Royal Astronomical Society 485, no. 2 (February 19, 2019): 2806–24. http://dx.doi.org/10.1093/mnras/stz507.

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ABSTRACT We study the accuracy of several approximate methods for gravitational dynamics in terms of halo power spectrum multipoles and their estimated covariance matrix. We propagate the differences in covariances into parameter constraints related to growth rate of structure, Alcock–Paczynski distortions, and biasing. We consider seven methods in three broad categories: algorithms that solve for halo density evolution deterministically using Lagrangian trajectories (ICE–COLA, pinocchio, and peakpatch), methods that rely on halo assignment schemes on to dark matter overdensities calibrated with a target N-body run (halogen, patchy), and two standard assumptions about the full density probability distribution function (Gaussian and lognormal). We benchmark their performance against a set of three hundred N-body simulations, running similar sets of approximate simulations with matched initial conditions, for each method. We find that most methods reproduce the monopole to within $5{{\ \rm per\ cent}}$, while residuals for the quadrupole are sometimes larger and scale dependent. The variance of the multipoles is typically reproduced within $10{{\ \rm per\ cent}}$. Overall, we find that covariances built from approximate simulations yield errors on model parameters within $10{{\ \rm per\ cent}}$ of those from the N-body-based covariance.

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20

Stępniak, Czesław. "Inverting covariance matrices." Discussiones Mathematicae Probability and Statistics 26, no. 2 (2006): 163. http://dx.doi.org/10.7151/dmps.1080.

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21

Dorvlo, Atsu S. S. "Generating covariance matrices." International Journal of Mathematical Education in Science and Technology 31, sup2 (March 2000): 287–89. http://dx.doi.org/10.1080/00207390050032261.

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22

Loh, Wei-Liem. "Estimating Covariance Matrices." Annals of Statistics 19, no. 1 (March 1991): 283–96. http://dx.doi.org/10.1214/aos/1176347982.

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23

Tian, Yongge. "Matrix rank and inertia formulas in the analysis of general linear models." Open Mathematics 15, no. 1 (February 27, 2017): 126–50. http://dx.doi.org/10.1515/math-2017-0013.

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Abstract Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs), and how to use the formulas in statistical analysis of GLMs. We first derive analytical expressions of best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem, and present some well-known results on ordinary least-squares predictors/ordinary least-squares estimators (OLSPs/OLSEs). We then establish some fundamental rank and inertia formulas for covariance matrices related to BLUPs/BLUEs and OLSPs/OLSEs, and use the formulas to characterize a variety of equalities and inequalities for covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. As applications, we use these equalities and inequalities in the comparison of the covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. The work on the formulations of BLUPs/BLUEs and OLSPs/OLSEs, and their covariance matrices under GLMs provides direct access, as a standard example, to a very simple algebraic treatment of predictors and estimators in linear regression analysis, which leads a deep insight into the linear nature of GLMs and gives an efficient way of summarizing the results.

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24

Carta, Lynn, and David Carta. "Nematode specific gravity profiles and applications to flotation extraction and taxonomy." Nematology 2, no. 2 (2000): 201–10. http://dx.doi.org/10.1163/156854100508935.

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AbstractA technique is described that refines the standard sugar flotation procedure used to isolate nematodes from their surroundings. By centrifuging nematodes in a number of increasing specific gravity solutions and plotting the fraction floating, the cumulative probability distribution of the population’s specific gravity is generated. By assuming normality, the population mean, μ, and standard deviation, σ, are found by a nonlinear least squares procedure. These density parameters along with their error covariance matrix may be used as a taxonomic physical character. A chi-squared test is derived for comparing populations. Mean and standard deviation pairs (μ, σ) were found for the specific gravities of the adult stage of the plant parasites Pratylenchus agilis (1.068, 0.017), P. scribneri (1.073, 0.028), P. penetrans (1.058, 0.008) and the bacterial-feeder Caenorhabditis elegans (1.091, 0.016). La technique exposée affine le procédure standard par flottation au sucre utilisée pour séparer les nématodes de leur milieu. La centrifugation des nématodes dans une série de solutions de densités spécifiques et la mise en diagramme de la valeur de la fraction surnageante permettent de connaître le répartition de la probabilité cumulée de la densité spécifique de la population en cause. La normalité étant supposée, la moyenne de la population, μ, et la déviation standard, σ, sont calculées par la méthode des moindres carrés non linéaires. Ces paramètres relatifs à la densité ainsi que leur matrice de co-variance d’erreur peuvent être utilisés en taxinomie comme caractère physique. Un test chi2 en est dérivé pour comparer les populations entre elles. Des données en paires — moyenne (μ) et écart-type (σ) — ont été définies pour les densités des adultes des espèces phytoparasites Pratylenchus agilis (1,068; 0,017), P. scribneri (1,073; 0,028), P. penetrans (1,058; 0,008), ainsi que pour l’espèce bactérivore Caenorhabditis elegans (1,091; 0,016).

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25

Bishop, Craig H., Bo Huang, and Xuguang Wang. "A Nonvariational Consistent Hybrid Ensemble Filter." Monthly Weather Review 143, no. 12 (December 1, 2015): 5073–90. http://dx.doi.org/10.1175/mwr-d-14-00391.1.

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Abstract A consistent hybrid ensemble filter (CHEF) for using hybrid forecast error covariance matrices that linearly combine aspects of both climatological and flow-dependent matrices within a nonvariational ensemble data assimilation scheme is described. The CHEF accommodates the ensemble data assimilation enhancements of (i) model space ensemble covariance localization for satellite data assimilation and (ii) Hodyss’s method for improving accuracy using ensemble skewness. Like the local ensemble transform Kalman filter (LETKF), the CHEF is computationally scalable because it updates local patches of the atmosphere independently of others. Like the sequential ensemble Kalman filter (EnKF), it serially assimilates batches of observations and uses perturbed observations to create ensembles of analyses. It differs from the deterministic (no perturbed observations) ensemble square root filter (ESRF) and the EnKF in that (i) its analysis correction is unaffected by the order in which observations are assimilated even when localization is required, (ii) it uses accurate high-rank solutions for the posterior error covariance matrix to serially assimilate observations, and (iii) it accommodates high-rank hybrid error covariance models. Experiments were performed to assess the effect on CHEF and ESRF analysis accuracy of these differences. In the case where both the CHEF and the ESRF used tuned localized ensemble covariances for the forecast error covariance model, the CHEF’s advantage over the ESRF increased with observational density. In the case where the CHEF used a hybrid error covariance model but the ESRF did not, the CHEF had a substantial advantage for all observational densities.

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26

Zhang, Peng, Wen Juan Qi, and Zi Li Deng. "Parallel Covariance Intersection Fusion Optimal Kalman Filter." Applied Mechanics and Materials 475-476 (December 2013): 436–41. http://dx.doi.org/10.4028/www.scientific.net/amm.475-476.436.

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For multisensor network systems with unknown cross-covariances, a novel multi-level parallel covariance intersection (PCI) fusion Kalman filter is presented in this paper, which is realized by the multi-level parallel two-sensor covariance intersection (CI) fusers, so it only requires to solve the optimization problems of several one-dimensional nonlinear cost functions in parallel with loss computation burden. It can significantly reduce the computation time and increase data processing rate when the number of sensors is very large. It is proved that the PCI fuser is consistent, and its accuracy is higher than that of each local filter and is lower than that of the optimal Kalman fuser weighted by matrices. The geometric interpretation of accuracy relations based on the covariance ellipses is given. A simulation example for tracking systems verifies the accuracy relations.

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Yuan, Sihan, and Daniel J. Eisenstein. "Decorrelating the errors of the galaxy correlation function with compact transformation matrices." Monthly Notices of the Royal Astronomical Society 486, no. 1 (March 27, 2019): 708–24. http://dx.doi.org/10.1093/mnras/stz899.

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Abstract Covariance matrix estimation is a persistent challenge for cosmology, often requiring a large number of synthetic mock catalogues. The off-diagonal components of the covariance matrix also make it difficult to show representative error bars on the 2-point correlation function (2PCF) since errors computed from the diagonal values of the covariance matrix greatly underestimate the uncertainties. We develop a routine for decorrelating the projected and anisotropic 2PCF with simple and scale-compact transformations on the 2PCF. These transformation matrices are modelled after the Cholesky decomposition and the symmetric square root of the Fisher matrix. Using mock catalogues, we show that the transformed projected and anisotropic 2PCF recover the same structure as the original 2PCF while producing largely decorrelated error bars. Specifically, we propose simple Cholesky-based transformation matrices that suppress the off-diagonal covariances on the projected 2PCF by ${\sim } 95{{\ \rm per\ cent}}$ and that on the anisotropic 2PCF by ${\sim } 87{{\ \rm per\ cent}}$. These transformations also serve as highly regularized models of the Fisher matrix, compressing the degrees of freedom so that one can fit for the Fisher matrix with a much smaller number of mocks.

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28

Tieplova, D. "Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples." Zurnal matematiceskoj fiziki, analiza, geometrii 13, no. 1 (March 25, 2017): 82–98. http://dx.doi.org/10.15407/mag13.01.082.

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29

Engle, Robert F., Olivier Ledoit, and Michael Wolf. "Large Dynamic Covariance Matrices." Journal of Business & Economic Statistics 37, no. 2 (December 22, 2017): 363–75. http://dx.doi.org/10.1080/07350015.2017.1345683.

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30

Trenkler, Götz. "Ordering of Covariance Matrices." Econometric Theory 11, no. 4 (August 1995): 796. http://dx.doi.org/10.1017/s0266466600009750.

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31

Loh, Wei-Liem. "Estimating covariance matrices II." Journal of Multivariate Analysis 36, no. 2 (February 1991): 163–74. http://dx.doi.org/10.1016/0047-259x(91)90055-7.

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32

Eriksen, P. Svante. "Proportionality of Covariance Matrices." Annals of Statistics 15, no. 2 (June 1987): 732–48. http://dx.doi.org/10.1214/aos/1176350372.

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Carroll, T. L., and J. M. Byers. "Dimension from covariance matrices." Chaos: An Interdisciplinary Journal of Nonlinear Science 27, no. 2 (February 2017): 023101. http://dx.doi.org/10.1063/1.4975063.

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34

Pillai, Natesh S., and Jun Yin. "Universality of covariance matrices." Annals of Applied Probability 24, no. 3 (June 2014): 935–1001. http://dx.doi.org/10.1214/13-aap939.

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35

Ludwig, Monika. "Covariance matrices and valuations." Advances in Applied Mathematics 51, no. 3 (August 2013): 359–66. http://dx.doi.org/10.1016/j.aam.2012.12.003.

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36

Gunawan, B., and JW James. "The use of 'bending' in multiple trait selection of Border Leicester - Merino synthetic populations." Australian Journal of Agricultural Research 37, no. 5 (1986): 539. http://dx.doi.org/10.1071/ar9860539.

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The consistency of phenotypic and genetic parameters estimated for various body weight and wool characters in Border Leicester-Merino synthetic populations was investigated by calculating the eigenvalues of matrices of phenotypic covariances (P), genetic covariances (G), and the product of the inverse of the phenotypic with the genetic covariance matrix (P-1G). If these estimates were found to be inconsistent (non-positive definite), the bending technique was applied before genetic selection indices were calculated. In general, the P were positive definite, but the G or P-1G were always non-positive definite. The results suggest that P and G should always be checked carefully before genetic selection indices are calculated. The bending technique was quite effective in giving reasonable results for accuracy of selection.

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37

Blais, J. A. Rod. "Optimal Modeling and Filtering of Stochastic Time Series for Geoscience Applications." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/895061.

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Sequences of observations or measurements are often modeled as realizations of stochastic processes with some stationary properties in the first and second moments. However in practice, the noise biases and variances are likely to be different for different epochs in time or regions in space, and hence such stationarity assumptions are often questionable. In the case of strict stationarity with equally spaced data, the Wiener-Hopf equations can readily be solved with fast Fourier transforms (FFTs) with optimal computational efficiency. In more general contexts, covariance matrices can also be diagonalized using the Karhunen-Loève transforms (KLTs), or more generally using empirical orthogonal and biorthogonal expansions, which are unfortunately much more demanding in terms of computational efforts. In cases with increment stationarity, the mathematical modeling can be modified and generalized covariances can be used with some computational advantages. The general nonlinear solution methodology is also briefly overviewed with the practical limitations. These different formulations are discussed with special emphasis on the spectral properties of covariance matrices and illustrated with some numerical examples. General recommendations are included for practical geoscience applications.

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38

Qi, Wen Juan, Peng Zhang, Zi Li Deng, and Yuan Gao. "Covariance Intersection Fusion Smoothers for Multichannel ARMA Signal with Colored Measurement Noises." Applied Mechanics and Materials 373-375 (August 2013): 716–22. http://dx.doi.org/10.4028/www.scientific.net/amm.373-375.716.

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For multichannel autoregressive moving average (ARMA) signal with colored measurement noises, based on classical Kalman filtering theory, a covariance intersection (CI) fusion smoother without cross-covariances is presented by the augmented state space model. It has the advantage that the computation of cross-covariances is avoid, so it can significantly reduce the computational burden, and it can solve the fusion problem for multi-sensor systems with unknown cross-covariances. Under the unbiased linear minimum variance (ULMV) criterion, three optimal weighted fusion smoothers with matrix weights, scalar weights and diagonal weights are also presented respectively. The accuracy comparison of the CI fuser with the other three weighted fusers is given. It is shown that its accuracy is higher than that of each local smoother, and is lower than or close to that of the optimal fuser weighted by matrices. So the presented fusion smoother is better in performance.

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39

Trucíos, Carlos, Mauricio Zevallos, Luiz K. Hotta, and André A. P. Santos. "Covariance Prediction in Large Portfolio Allocation." Econometrics 7, no. 2 (May 9, 2019): 19. http://dx.doi.org/10.3390/econometrics7020019.

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Many financial decisions, such as portfolio allocation, risk management, option pricing and hedge strategies, are based on forecasts of the conditional variances, covariances and correlations of financial returns. The paper shows an empirical comparison of several methods to predict one-step-ahead conditional covariance matrices. These matrices are used as inputs to obtain out-of-sample minimum variance portfolios based on stocks belonging to the S&P500 index from 2000 to 2017 and sub-periods. The analysis is done through several metrics, including standard deviation, turnover, net average return, information ratio and Sortino’s ratio. We find that no method is the best in all scenarios and the performance depends on the criterion, the period of analysis and the rebalancing strategy.

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40

Penny, Stephen G. "The Hybrid Local Ensemble Transform Kalman Filter." Monthly Weather Review 142, no. 6 (May 28, 2014): 2139–49. http://dx.doi.org/10.1175/mwr-d-13-00131.1.

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Abstract Hybrid data assimilation methods combine elements of ensemble Kalman filters (EnKF) and variational methods. While most approaches have focused on augmenting an operational variational system with dynamic error covariance information from an ensemble, this study takes the opposite perspective of augmenting an operational EnKF with information from a simple 3D variational data assimilation (3D-Var) method. A class of hybrid methods is introduced that combines the gain matrices of the ensemble and variational methods, rather than linearly combining the respective background error covariances. A hybrid local ensemble transform Kalman filter (Hybrid-LETKF) is presented in two forms: 1) a traditionally motivated Hybrid/Covariance-LETKF that combines the background error covariance matrices of LETKF and 3D-Var, and 2) a simple-to-implement algorithm called the Hybrid/Mean-LETKF that falls into the new class of hybrid gain methods. Both forms improve analysis errors when using small ensemble sizes and low observation coverage versus either LETKF or 3D-Var used alone. The results imply that for small ensemble sizes, allowing a solution to be found outside of the space spanned by ensemble members provides robustness in both hybrid methods compared to LETKF alone. Finally, the simplicity of the Hybrid/Mean-LETKF design implies that this algorithm can be applied operationally while requiring only minor modifications to an existing operational 3D-Var system.

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41

Philcox, Oliver H. E., and Daniel J. Eisenstein. "Estimating covariance matrices for two- and three-point correlation function moments in Arbitrary Survey Geometries." Monthly Notices of the Royal Astronomical Society 490, no. 4 (October 16, 2019): 5931–51. http://dx.doi.org/10.1093/mnras/stz2896.

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ABSTRACT We present configuration-space estimators for the auto- and cross-covariance of two- and three-point correlation functions (2PCF and 3PCF) in general survey geometries. These are derived in the Gaussian limit (setting higher order correlation functions to zero), but for arbitrary non-linear 2PCFs (which may be estimated from the survey itself), with a shot-noise rescaling parameter included to capture non-Gaussianity. We generalize previous approaches to include Legendre moments via a geometry-correction function calibrated from measured pair and triple counts. Making use of importance sampling and random particle catalogues, we can estimate model covariances in fractions of the time required to do so with mocks, obtaining estimates with negligible sampling noise in ∼10 (∼100) CPU-hours for the 2PCF (3PCF) autocovariance. We compare results to sample covariances from a suite of BOSS DR12 mocks and find the matrices to be in good agreement, assuming a shot-noise rescaling parameter of 1.03 (1.20) for the 2PCF (3PCF). To obtain strongest constraints on cosmological parameters, we must use multiple statistics in concert; having robust methods to measure their covariances at low computational cost is thus of great relevance to upcoming surveys.

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42

Nye, Tom M. W., Brad J. C. Baxter, and Walter R. Gilks. "A Covariance Matrix Inversion Problem arising from the Construction of Phylogenetic Trees." LMS Journal of Computation and Mathematics 10 (2007): 119–31. http://dx.doi.org/10.1112/s1461157000001327.

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AbstractWe describe an efficient algorithm for the inversion of covariance matrices that arise in the context of phylogenetic tree construction. Phylogenetic trees describe the evolutionary relationships between species, and their construction is computationally demanding. Many approaches involve the symmetric matrix of evolutionary distances between species. Regarding these distances as random variables, the corresponding set of variances and covariances form a rank-4 tensor, and the inner-product defined by its inverse can be used to assign statistical scores to candidate trees. We describe a natural set of assumptions for the phylogenetic tree under construction, and show how under these assumptions the covariance tensor for a tree with n leaves can be inverted in O(n2) operations. In addition to presenting the inversion algorithm, we hope this article will open algebraic and computational problems from the field of phylogeny to a wider audience.

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43

Chang, Wei-Yu, Jothiram Vivekanandan, and Tai-Chi Chen Wang. "Estimation of X-Band Polarimetric Radar Attenuation and Measurement Uncertainty Using a Variational Method." Journal of Applied Meteorology and Climatology 53, no. 4 (April 2014): 1099–119. http://dx.doi.org/10.1175/jamc-d-13-0191.1.

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AbstractA variational algorithm for estimating measurement error covariance and the attenuation of X-band polarimetric radar measurements is described. It concurrently uses both the differential reflectivity ZDR and propagation phase ΦDP. The majority of the current attenuation estimation techniques use only ΦDP. A few of the ΦDP-based methods use ZDR as a constraint for verifying estimated attenuation. In this paper, a detailed observing system simulation experiment was used for evaluating the performance of the variational algorithm. The results were compared with a single-coefficient ΦDP-based method. Retrieved attenuation from the variational method is more accurate than the results from a single coefficient ΦDP-based method. Moreover, the variational method is less sensitive to measurement noise in radar observations. The variational method requires an accurate description of error covariance matrices. Relative weights between measurements and background values (i.e., mean value based on long-term DSD measurements in the variational method) are determined by their respective error covariances. Instead of using ad hoc values, error covariance matrices of background and radar measurement are statistically estimated and their spatial characteristics are studied. The estimated error covariance shows higher values in convective regions than in stratiform regions, as expected. The practical utility of the variational attenuation correction method is demonstrated using radar field measurements from the Taiwan Experimental Atmospheric Mobile-Radar (TEAM-R) during 2008’s Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX). The accuracy of attenuation-corrected X-band radar measurements is evaluated by comparing them with collocated S-band radar measurements.

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44

de Santi, Natalí S. M., and L. Raul Abramo. "Improving cosmological covariance matrices with machine learning." Journal of Cosmology and Astroparticle Physics 2022, no. 09 (September 1, 2022): 013. http://dx.doi.org/10.1088/1475-7516/2022/09/013.

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Abstract Cosmological covariance matrices are fundamental for parameter inference, since they are responsible for propagating uncertainties from the data down to the model parameters. However, when data vectors are large, in order to estimate accurate and precise covariance matrices we need huge numbers of observations, or rather costly simulations - neither of which may be viable. In this work we propose a machine learning approach to alleviate this problem in the context of the covariance matrices used in the study of large-scale structure. With only a small amount of data (matrices built with samples of 50-200 halo power spectra) we are able to provide significantly improved covariance matrices, which are almost indistinguishable from the ones built from much larger samples (thousands of spectra). In order to perform this task we trained convolutional neural networks to denoise the covariance matrices, using in the training process a data set made up entirely of spectra extracted from simple, inexpensive halo simulations (mocks). We then show that the method not only removes the noise in the covariance matrices of the cheap simulation, but it is also able to successfully denoise the covariance matrices of halo power spectra from N-body simulations. We compare the denoised matrices with the noisy sample covariance matrices using several metrics, and in all of them the denoised matrices score significantly better, without any signs of spurious artifacts. With the help of the Wishart distribution we show that the end product of the denoiser can be compared with an effective sample augmentation in the input matrices. Finally, we show that, by using the denoised covariance matrices, the cosmological parameters can be recovered with nearly the same accuracy as when using covariance matrices built with a sample of 30,000 spectra in the case of the cheap simulations, and with 15,000 spectra in the case of the N-body simulations. Of particular interest is the bias in the Hubble parameter H 0, which was significantly reduced after applying the denoiser.

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45

SERVICE, PHILIP M. "The genetic structure of female life history in D. melanogaster: comparisons among populations." Genetical Research 75, no. 2 (April 2000): 153–66. http://dx.doi.org/10.1017/s0016672399004322.

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Two questions were addressed: (1) What is the genetic variance–covariance structure of a suite of four female life history traits in D. melanogaster? and (2) Does the genetic architecture of these traits differ among populations? Three populations of D. melanogaster were studied. Genetic variances and covariances were estimated by sib analysis three times for each population: immediately upon establishment of populations in the laboratory, and subsequently after approximately 6 months and 2 years of laboratory culture. Entire genetic variance–covariance matrices, as well as their individual components, were compared between populations by means of likelihood ratio tests. All traits studied were significantly heritable in at least one-half of estimates. Despite large sample sizes, additive genetic covariances were for the most part not statistically significant, and only two significant negative covariance estimates were obtained throughout the experiments. Therefore, these experiments provide little support for evolutionary life history theories that are based on negative genetic correlations among life history components. Neither do they support the idea that genetic variance for fitness components is maintained by trade-offs. Evidence suggests that the G matrix of one population was initially different from those of the other two populations. Those differences disappeared after 2 years of laboratory culture. At the level of individual (co)variance components, there were relatively few differences among populations, and the overall impression was that the three populations had generally similar genetic architectures for the traits studied.

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46

Lacasa, Fabien. "The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys." Astronomy & Astrophysics 634 (February 2020): A74. http://dx.doi.org/10.1051/0004-6361/201936683.

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As galaxy surveys improve their precision thanks to lower levels of noise and the push toward small, non-linear scales, the need for accurate covariances beyond the classical Gaussian formula becomes more acute. Here I investigate the analytical implementation and impact of non-Gaussian covariance terms that I had previously derived for the galaxy angular power spectrum. Braiding covariance is such an interesting class of such terms and it gets contributions both from in-survey and super-survey modes, the latter proving difficult to calibrate through simulations. I present an approximation for braiding covariance which speeds up the process of numerical computation. I show that including braiding covariance is a necessary condition for including other non-Gaussian terms, namely the in-survey 2-, 3-, and 4-halo covariance. Indeed these terms yield incorrect covariance matrices with negative eigenvalues if considered on their own. I then move to quantify the impact on parameter constraints, with forecasts for a survey with Euclid-like galaxy density and angular scales. Compared with the Gaussian case, braiding and in-survey covariances significantly increase the error bars on cosmological parameters, in particular by 50% for the dark energy equation of state w. The error bars on the halo occupation distribution (HOD) parameters are also affected between 12% and 39%. Accounting for super-sample covariance (SSC) also increases parameter errors, by 90% for w and between 7% and 64% for HOD. In total, non-Gaussianity increases the error bar on w by 120% (between 15% and 80% for other cosmological parameters) and the error bars on HOD parameters between 17% and 85%. Accounting for the 1-halo trispectrum term on top of SSC, as has been done in some current analyses, is not sufficient for capturing the full non-Gaussian impact: braiding and the rest of in-survey covariance have to be accounted for. Finally, I discuss why the inclusion of non-Gaussianity generally eases up parameter degeneracies, making cosmological constraints more robust for astrophysical uncertainties. I released publicly the data and a Python notebook reproducing the results and plots of the article.

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47

Uribe-Murcia, Karen, Jorge A. Ortega-Contreras, Eli G. Pale-Ramon, Miguel Vazquez-Olguin, and Yuriy S. Shmaliy. "Comparison of the Kalman Filter and the Unbiased FIR Filter for Network Systems with Multiples Output Delays and Lost Data." WSEAS TRANSACTIONS ON CIRCUITS AND SYSTEMS 21 (August 2, 2022): 176–81. http://dx.doi.org/10.37394/23201.2022.21.19.

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In this article, a comparison of the UFIR and Kalman filter to estimate a tracking vehicle system variables is developed considering two possible observation output models. The time stamp approach and the predictive compensation are used to analyze the problem from multiple perturbations, which produces random delayed data and losses during transmissions. For the estimation, a transformation model and a decorrelation covariance matrices are developed with the aim of assure optimal conditions and minimizing the estimation error. Finally, several real situations, miss modeling, uncertain noise covariances, and uncertain probabilities are proposed to demonstrate the effectiveness and robustness of the filter proposed.

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48

O’Connell, Ross, and Daniel J. Eisenstein. "Large covariance matrices: accurate models without mocks." Monthly Notices of the Royal Astronomical Society 487, no. 2 (May 21, 2019): 2701–17. http://dx.doi.org/10.1093/mnras/stz1359.

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Abstract 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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49

Mc Kay, R. J. "Simultaneous procedures for covariance matrices." Communications in Statistics - Theory and Methods 18, no. 2 (January 1989): 429–43. http://dx.doi.org/10.1080/03610928908829909.

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50

Clifford, David. "Computation of Spatial Covariance Matrices." Journal of Computational and Graphical Statistics 14, no. 1 (March 2005): 155–67. http://dx.doi.org/10.1198/106186005x27626.

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