Journal articles: 'Covariance'

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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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Gao, Boran, Can Yang, Jin Liu, and Xiang Zhou. "Accurate genetic and environmental covariance estimation with composite likelihood in genome-wide association studies." PLOS Genetics 17, no. 1 (January 4, 2021): e1009293. http://dx.doi.org/10.1371/journal.pgen.1009293.

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Genetic and environmental covariances between pairs of complex traits are important quantitative measurements that characterize their shared genetic and environmental architectures. Accurate estimation of genetic and environmental covariances in genome-wide association studies (GWASs) can help us identify common genetic and environmental factors associated with both traits and facilitate the investigation of their causal relationship. Genetic and environmental covariances are often modeled through multivariate linear mixed models. Existing algorithms for covariance estimation include the traditional restricted maximum likelihood (REML) method and the recent method of moments (MoM). Compared to REML, MoM approaches are computationally efficient and require only GWAS summary statistics. However, MoM approaches can be statistically inefficient, often yielding inaccurate covariance estimates. In addition, existing MoM approaches have so far focused on estimating genetic covariance and have largely ignored environmental covariance estimation. Here we introduce a new computational method, GECKO, for estimating both genetic and environmental covariances, that improves the estimation accuracy of MoM while keeping computation in check. GECKO is based on composite likelihood, relies on only summary statistics for scalable computation, provides accurate genetic and environmental covariance estimates across a range of scenarios, and can accommodate SNP annotation stratified covariance estimation. We illustrate the benefits of GECKO through simulations and applications on analyzing 22 traits from five large-scale GWASs. In the real data applications, GECKO identified 50 significant genetic covariances among analyzed trait pairs, resulting in a twofold power gain compared to the previous MoM method LDSC. In addition, GECKO identified 20 significant environmental covariances. The ability of GECKO to estimate environmental covariance in addition to genetic covariance helps us reveal strong positive correlation between the genetic and environmental covariance estimates across trait pairs, suggesting that common pathways may underlie the shared genetic and environmental architectures between traits.

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Gneiting, Tilmann, Zoltán Sasvári, and Martin Schlather. "Analogies and correspondences between variograms and covariance functions." Advances in Applied Probability 33, no. 3 (September 2001): 617–30. http://dx.doi.org/10.1239/aap/1005091356.

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Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

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Gneiting, Tilmann, Zoltán Sasvári, and Martin Schlather. "Analogies and correspondences between variograms and covariance functions." Advances in Applied Probability 33, no. 03 (September 2001): 617–30. http://dx.doi.org/10.1017/s0001867800011034.

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Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

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Yuan, Ke-Hai, and Peter M. Bentler. "9. Structural Equation Modeling with Robust Covariances." Sociological Methodology 28, no. 1 (August 1998): 363–96. http://dx.doi.org/10.1111/0081-1750.00052.

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Existing methods for structural equation modeling involve fitting the ordinary sample covariance matrix by a proposed structural model. Since a sample covariance is easily influenced by a few outlying cases, the standard practice of modeling sample covariances can lead to inefficient estimates as well as inflated fit indices. By giving a proper weight to each individual case, a robust covariance will have a bounded influence function as well as a nonzero breakdown point. These robust properties of the covariance estimators will be carried over to the parameter estimators in the structural model if a technically appropriate procedure is used. We study such a procedure in which robust covariances replace ordinary sample covariances in the context of the Wishart likelihood function. This procedure is easy to implement in practice. Statistical properties of this procedure are investigated. A fit index is given based on sampling from an elliptical distribution. An estimating equation approach is used to develop a variety of robust covariances, and consistent covariances of these robust estimators, needed for standard errors and test statistics, follow from this approach. Examples illustrate the inflated statistics and distorted parameter estimates obtained by using sample covariances when compared with those obtained by using robust covariances. The merits of each method and its relevance to specific types of data are discussed.

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Shamsipour, Pejman, Denis Marcotte, Michel Chouteau, Martine Rivest, and Abderrezak Bouchedda. "3D stochastic gravity inversion using nonstationary covariances." GEOPHYSICS 78, no. 2 (March 1, 2013): G15—G24. http://dx.doi.org/10.1190/geo2012-0122.1.

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The flexibility of geostatistical inversions in geophysics is limited by the use of stationary covariances, which, implicitly and mostly for mathematical convenience, assumes statistical homogeneity of the studied field. For fields showing sharp contrasts due, for example, to faults or folds, an approach based on the use of nonstationary covariances for cokriging inversion was developed. The approach was tested on two synthetic cases and one real data set. Inversion results based on the nonstationary covariance were compared to the results from the stationary covariance for two synthetic models. The nonstationary covariance better recovered the known synthetic models. With the real data set, the nonstationary assumption resulted in a better match with the known surface geology.

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NIKOLIĆ, HRVOJE. "QUANTUM DETERMINISM FROM QUANTUM GENERAL COVARIANCE." International Journal of Modern Physics D 15, no. 12 (December 2006): 2171–75. http://dx.doi.org/10.1142/s0218271806009595.

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The requirement of general covariance of quantum field theory (QFT) naturally leads to quantization based on the manifestly covariant De Donder–Weyl formalism. To recover the standard noncovariant formalism without violating covariance, fields need to depend on time in a specific deterministic manner. This deterministic evolution of quantum fields is recognized as a covariant version of the Bohmian hidden-variable interpretation of QFT.

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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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Wang, Xuguang, Chris Snyder, and Thomas M. Hamill. "On the Theoretical Equivalence of Differently Proposed Ensemble–3DVAR Hybrid Analysis Schemes." Monthly Weather Review 135, no. 1 (January 1, 2007): 222–27. http://dx.doi.org/10.1175/mwr3282.1.

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Abstract Hybrid ensemble–three-dimensional variational analysis schemes incorporate flow-dependent, ensemble-estimated background-error covariances into the three-dimensional variational data assimilation (3DVAR) framework. Typically the 3DVAR background-error covariance estimate is assumed to be stationary, nearly homogeneous, and isotropic. A hybrid scheme can be achieved by 1) directly replacing the background-error covariance term in the cost function by a linear combination of the original background-error covariance with the ensemble covariance or 2) through augmenting the state vector with another set of control variables preconditioned upon the square root of the ensemble covariance. These differently proposed hybrid schemes are proven to be equivalent. The latter framework may be a simpler way to incorporate ensemble information into operational 3DVAR schemes, where the preconditioning is performed with respect to the background term.

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Yaremchuk, Max, Dmitri Nechaev, and Chudong Pan. "A Hybrid Background Error Covariance Model for Assimilating Glider Data into a Coastal Ocean Model." Monthly Weather Review 139, no. 6 (June 1, 2011): 1879–90. http://dx.doi.org/10.1175/2011mwr3510.1.

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Abstract A hybrid background error covariance (BEC) model for three-dimensional variational data assimilation of glider data into the Navy Coastal Ocean Model (NCOM) is introduced. Similar to existing atmospheric hybrid BEC models, the proposed model combines low-rank ensemble covariances with the heuristic Gaussian-shaped covariances to estimate forecast error statistics. The distinctive features of the proposed BEC model are the following: (i) formulation in terms of inverse error covariances, (ii) adaptive determination of the rank m of with information criterion based on the innovation error statistics, (iii) restriction of the heuristic covariance operator to the null space of , and (iv) definition of the BEC magnitudes through separate analyses of the innovation error statistics in the state space and the null space of . The BEC model is validated by assimilation experiments with simulated and real data obtained during a glider survey of the Monterey Bay in August 2003. It is shown that the proposed hybrid scheme substantially improves the forecast skill of the heuristic covariance model.

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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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Ramouz, Sabah, Yosra Afrasteh, Mirko Reguzzoni, and Abdolreza Safari. "Assessment of local covariance estimation through Least Squares Collocation over Iran." Advances in Geosciences 50 (March 4, 2020): 65–75. http://dx.doi.org/10.5194/adgeo-50-65-2020.

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Abstract. Covariance determination as the heart of Least Squares Collocation gravity field modeling is based on fitting an analytical covariance to the empirical covariance, which is stemmed from gravimetric data. The main objective of this study is to process different local covariance strategies over four regions with different topography and spatial data distribution in Iran. For this purpose, Least Squares Collocation based on Remove – Compute – Restore technique is implemented. In the Remove step, gravity reduction in regions with a denser distribution and a rougher topography is more effective. In the Compute step, the assessment of the Collocation estimates on the gravity anomaly control points illustrates that data density is more relevant than topography roughness to have a good covariance determination. Moreover, among the different attempts of localizing the covariance estimation, a recursive approach correcting the covariance parameters based on the agreement between Least Squares Collocation estimates and control points shows better performance. Furthermore, we could see that covariance localization in a region with sparse or bad distributed observations is a challenging task and may not necessarily improve the Collocation gravity modeling. Indeed, the geometrical fitness of the empirical and analytical covariances – which is usually a qualitative test to verify the precision of the covariance determination – is not always an adequate criterion.

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Hu, Zhentao, Linlin Yang, Yong Jin, Han Wang, and Shibo Yang. "Strong Tracking PHD Filter Based on Variational Bayesian with Inaccurate Process and Measurement Noise Covariance." Sensors 21, no. 4 (February 5, 2021): 1126. http://dx.doi.org/10.3390/s21041126.

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Assuming that the measurement and process noise covariances are known, the probability hypothesis density (PHD) filter is effective in real-time multi-target tracking; however, noise covariance is often unknown and time-varying for an actual scene. To solve this problem, a strong tracking PHD filter based on Variational Bayes (VB) approximation is proposed in this paper. The measurement noise covariance is described in the linear system by the inverse Wishart (IW) distribution. Then, the fading factor in the strong tracking principle uses the optimal measurement noise covariance at the previous moment to control the state prediction covariance in real-time. The Gaussian IW (GIW) joint distribution adopts the VB approximation to jointly return the measurement noise covariance and the target state covariance. The simulation results show that, compared with the traditional Gaussian mixture PHD (GM-PHD) and the VB-adaptive PHD, the proposed algorithm has higher tracking accuracy and stronger robustness in a more reasonable calculation time.

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Hubbert, J. C., and V. N. Bringi. "Studies of the Polarimetric Covariance Matrix. Part II: Modeling and Polarization Errors." Journal of Atmospheric and Oceanic Technology 20, no. 7 (July 1, 2003): 1011–22. http://dx.doi.org/10.1175/1456.1.

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Abstract A polarimetric radar covariance matrix model is described to study the behavior of the co-to-cross covariances in precipitation. The 2 × 2 propagation matrix with attenuation, differential attenuation, and differential phase is coupled to the backscatter matrix leading to a propagation-modified covariance matrix model. System polarization errors are included in this model as well. This model is used to study the behavior of the magnitude and phase of the co-to-cross covariances and the linear depolarization ratio (LDR) in rainfall. It is shown that the model predictions are consistent with data collected with the Colorado State University (CSU)–University of Chicago–Illinois State Water Survey (CHILL) radar in intense rainfall. A method is also given for estimating the system polarization errors from covariance matrix data collected in intense rainfall.

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McRoberts, Ronald E., Erik Næsset, Terje Gobakken, Gherardo Chirici, Sonia Condés, Zhengyang Hou, Svetlana Saarela, Qi Chen, Göran Ståhl, and Brian F. Walters. "Assessing components of the model-based mean square error estimator for remote sensing assisted forest applications." Canadian Journal of Forest Research 48, no. 6 (June 2018): 642–49. http://dx.doi.org/10.1139/cjfr-2017-0396.

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Model-based inference is an alternative to probability-based inference for small areas or remote areas for which probability sampling is difficult. Model-based mean square error estimators incorporate three components: prediction covariance, residual variance, and residual covariance. The latter two components are often considered negligible, particularly for large areas, but no thresholds that justify ignoring them have been reported. The objectives of the study were threefold: (i) to compare analytical and bootstrap estimators of model parameter covariances as the primary factors affecting prediction covariance; (ii) to estimate the contribution of residual variance to overall variance; and (iii) to estimate thresholds for residual spatial correlation that justify ignoring this component. Five datasets were used, three from Europe, one from Africa, and one from North America. The dependent variable was either forest volume or biomass and the independent variables were either Landsat satellite image bands or airborne laser scanning metrics. Three conclusions were noteworthy: (i) analytical estimators of the model parameter covariances tended to be biased; (ii) the effects of residual variance were mostly negligible; and (iii) the effects of spatial correlation on residual covariance vary by multiple factors but decrease with increasing study area size. For study areas greater than 75 km2 in size, residual covariance could generally be ignored.

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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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García-Berthou, Emili, and Ramon Moreno-Amich. "Multivariate Analysis of Covariance in Morphometric Studies of the Reproductive Cycle." Canadian Journal of Fisheries and Aquatic Sciences 50, no. 7 (July 1, 1993): 1394–99. http://dx.doi.org/10.1139/f93-159.

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A new approach for analysing morphometric data of reproductive cycles is proposed, involving multivariate analysis of covariance of the directly measured variables (e.g., total weight and gonadal weight), with body length being the covariate. Multivariate, univariate, and between-group tests can be used progressively if significant differences have been found previously. Seasonal variation and other factors of interest can be described with predicted means of the model, adjusted for covariate, rendering the use of indices such as condition factor and gonadosomatic index unnecessary. A special design of multivariate analysis of covariance, with a pooled covariate by factor interaction, can be used to test the fundamental assumption of homogeneous slopes (parallelism hypothesis) in the standard multivariate analysis of covariance. Data for an Iberian brackish water cyprinodontid fish are used to demonstrate the proposed method.

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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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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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Schubert, Till, Johannes Korte, Jan Martin Brockmann, and Wolf-Dieter Schuh. "A Generic Approach to Covariance Function Estimation Using ARMA-Models." Mathematics 8, no. 4 (April 15, 2020): 591. http://dx.doi.org/10.3390/math8040591.

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Covariance function modeling is an essential part of stochastic methodology. Many processes in geodetic applications have rather complex, often oscillating covariance functions, where it is difficult to find corresponding analytical functions for modeling. This paper aims to give the methodological foundations for an advanced covariance modeling and elaborates a set of generic base functions which can be used for flexible covariance modeling. In particular, we provide a straightforward procedure and guidelines for a generic approach to the fitting of oscillating covariance functions to an empirical sequence of covariances. The underlying methodology is developed based on the well known properties of autoregressive processes in time series. The surprising simplicity of the proposed covariance model is that it corresponds to a finite sum of covariance functions of second-order Gauss–Markov (SOGM) processes. Furthermore, the great benefit is that the method is automated to a great extent and directly results in the appropriate model. A manual decision for a set of components is not required. Notably, the numerical method can be easily extended to ARMA-processes, which results in the same linear system of equations. Although the underlying mathematical methodology is extensively complex, the results can be obtained from a simple and straightforward numerical method.

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Quinn, Courtney, Terence J. O'Kane, and Vassili Kitsios. "Application of a local attractor dimension to reduced space strongly coupled data assimilation for chaotic multiscale systems." Nonlinear Processes in Geophysics 27, no. 1 (February 19, 2020): 51–74. http://dx.doi.org/10.5194/npg-27-51-2020.

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Abstract. The basis and challenge of strongly coupled data assimilation (CDA) is the accurate representation of cross-domain covariances between various coupled subsystems with disparate spatio-temporal scales, where often one or more subsystems are unobserved. In this study, we explore strong CDA using ensemble Kalman filtering methods applied to a conceptual multiscale chaotic model consisting of three coupled Lorenz attractors. We introduce the use of the local attractor dimension (i.e. the Kaplan–Yorke dimension, dimKY) to prescribe the rank of the background covariance matrix which we construct using a variable number of weighted covariant Lyapunov vectors (CLVs). Specifically, we consider the ability to track the nonlinear trajectory of each of the subsystems with different variants of sparse observations, relying only on the cross-domain covariance to determine an accurate analysis for tracking the trajectory of the unobserved subdomain. We find that spanning the global unstable and neutral subspaces is not sufficient at times where the nonlinear dynamics and intermittent linear error growth along a stable direction combine. At such times a subset of the local stable subspace is also needed to be represented in the ensemble. In this regard the local dimKY provides an accurate estimate of the required rank. Additionally, we show that spanning the full space does not improve performance significantly relative to spanning only the subspace determined by the local dimension. Where weak coupling between subsystems leads to covariance collapse in one or more of the unobserved subsystems, we apply a novel modified Kalman gain where the background covariances are scaled by their Frobenius norm. This modified gain increases the magnitude of the innovations and the effective dimension of the unobserved domains relative to the strength of the coupling and timescale separation. We conclude with a discussion on the implications for higher-dimensional systems.

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Genton, Marc G., and Olivier Perrin. "On a time deformation reducing nonstationary stochastic processes to local stationarity." Journal of Applied Probability 41, no. 1 (March 2004): 236–49. http://dx.doi.org/10.1239/jap/1077134681.

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A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.

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Genton, Marc G., and Olivier Perrin. "On a time deformation reducing nonstationary stochastic processes to local stationarity." Journal of Applied Probability 41, no. 01 (March 2004): 236–49. http://dx.doi.org/10.1017/s0021900200014170.

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A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.

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Bédard, Joël, Jean-François Caron, Mark Buehner, Seung-Jong Baek, and Luc Fillion. "Hybrid Background Error Covariances for a Limited-Area Deterministic Weather Prediction System." Weather and Forecasting 35, no. 3 (May 6, 2020): 1051–66. http://dx.doi.org/10.1175/waf-d-19-0069.1.

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Abstract This study introduces an experimental regional assimilation configuration for a 4D ensemble–variational (4D-EnVar) deterministic weather prediction system. A total of 16 assimilation experiments covering July 2014 are presented to assess both experimental regional climatological background error covariances and updates in the treatment of flow-dependent error covariances. The regional climatological background error covariances are estimated using statistical correlations between variables instead of using balance operators. These error covariance estimates allow the analyses to fit more closely with the assimilated observations than when using the lower-resolution global background error covariances (due to shorter correlation scales), and the ensuing forecasts are significantly improved. The use of ensemble-based background error covariances is also improved by reducing vertical and horizontal localization length scales for the flow-dependent background error covariance component. Also, reducing the number of ensemble members employed in the deterministic analysis (from 256 to 128) reduced computational costs by half without degrading the accuracy of analyses and forecasts. The impact of the relative contributions of the climatological and flow-dependent background error covariance components is also examined. Results show that the experimental regional system benefits from giving a lower (higher) weight to climatological (flow-dependent) error covariances. When compared with the operational assimilation configuration of the continental prediction system, the proposed modifications to the background error covariances improve both surface and upper-air RMSE scores by nearly 1%. Still, the use of a higher-resolution ensemble to estimate flow-dependent background error covariances does not yet provide added value, although it is expected to allow for a better use of dense observations in the future.

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Buffalo, Vince, and Graham Coop. "Estimating the genome-wide contribution of selection to temporal allele frequency change." Proceedings of the National Academy of Sciences 117, no. 34 (August 12, 2020): 20672–80. http://dx.doi.org/10.1073/pnas.1919039117.

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Rapid phenotypic adaptation is often observed in natural populations and selection experiments. However, detecting the genome-wide impact of this selection is difficult since adaptation often proceeds from standing variation and selection on polygenic traits, both of which may leave faint genomic signals indistinguishable from a noisy background of genetic drift. One promising signal comes from the genome-wide covariance between allele frequency changes observable from temporal genomic data (e.g., evolve-and-resequence studies). These temporal covariances reflect how heritable fitness variation in the population leads changes in allele frequencies at one time point to be predictive of the changes at later time points, as alleles are indirectly selected due to remaining associations with selected alleles. Since genetic drift does not lead to temporal covariance, we can use these covariances to estimate what fraction of the variation in allele frequency change through time is driven by linked selection. Here, we reanalyze three selection experiments to quantify the effects of linked selection over short timescales using covariance among time points and across replicates. We estimate that at least 17 to 37% of allele frequency change is driven by selection in these experiments. Against this background of positive genome-wide temporal covariances, we also identify signals of negative temporal covariance corresponding to reversals in the direction of selection for a reasonable proportion of loci over the time course of a selection experiment. Overall, we find that in the three studies we analyzed, linked selection has a large impact on short-term allele frequency dynamics that is readily distinguishable from genetic drift.

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Chandramohan, Jagadeesh, Robert D. Foley, and Ralph L. Disney. "Thinning of point processes—covariance analyses." Advances in Applied Probability 17, no. 1 (March 1985): 127–46. http://dx.doi.org/10.2307/1427056.

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Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

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Chandramohan, Jagadeesh, Robert D. Foley, and Ralph L. Disney. "Thinning of point processes—covariance analyses." Advances in Applied Probability 17, no. 01 (March 1985): 127–46. http://dx.doi.org/10.1017/s0001867800014695.

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Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

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Yang, Chao, Dongbin Xiu, and Robert Michael Kirby. "VISUALIZATION OF COVARIANCE AND CROSS-COVARIANCE FIELDS." International Journal for Uncertainty Quantification 3, no. 1 (2013): 25–38. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.2011003369.

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Khoury, Justin, Godfrey E. J. Miller, and Andrew J. Tolley. "How general relativity and Lorentz covariance arise from the spatially-covariant effective field theory of the transverse, traceless graviton." International Journal of Modern Physics D 23, no. 12 (October 2014): 1442012. http://dx.doi.org/10.1142/s0218271814420127.

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Traditional derivations of general relativity (GR) from the graviton degrees of freedom assume spacetime Lorentz covariance as an axiom. In this paper, we survey recent evidence that GR is the unique spatially-covariant effective field theory of the transverse, traceless graviton degrees of freedom. The Lorentz covariance of GR, having not been assumed in our analysis, is thus plausibly interpreted as an accidental or emergent symmetry of the gravitational sector. From this point of view, Lorentz covariance is a necessary feature of low-energy graviton dynamics, not a property of spacetime. This result has revolutionary implications for fundamental physics.

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Ge, Baoshuang, Hai Zhang, Liuyang Jiang, Zheng Li, and Maaz Butt. "Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance." Sensors 19, no. 6 (March 19, 2019): 1371. http://dx.doi.org/10.3390/s19061371.

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The unscented Kalman filter (UKF) is widely used to address the nonlinear problems in target tracking. However, this standard UKF shows unstable performance whenever the noise covariance mismatches. Furthermore, in consideration of the deficiencies of the current adaptive UKF algorithm, this paper proposes a new adaptive UKF scheme for the time-varying noise covariance problems. First of all, the cross-correlation between the innovation and residual sequences is given and proven. On this basis, a linear matrix equation deduced from the innovation and residual sequences is applied to resolve the process noise covariance in real time. Using the redundant measurements, an improved measurement-based adaptive Kalman filtering algorithm is applied to estimate the measurement noise covariance, which is entirely immune to the state estimation. The results of the simulation indicate that under the condition of time-varying noise covariances, the proposed adaptive UKF outperforms the standard UKF and the current adaptive UKF algorithm, hence improving tracking accuracy and stability.

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de Araujo, Marcelo R. A., and B. E. Coulman. "Parent-offspring regression in meadow bromegrass (Bromus riparius Rehm.): Evaluation of two methodologies on heritability estimates." Canadian Journal of Plant Science 84, no. 1 (January 1, 2004): 125–27. http://dx.doi.org/10.4141/p02-119.

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To determine the nature and extent of inflation of estimates of heritabilities by parent-offspring regression methods, 40 clones of meadow bromegrass (Bromus riparius Rehm.) and their half-sib progenies were studied in completely randomized block design trials, with six replications in Saskatoon and Melfort, Canada. Clones and progenies were evaluated for dry matter yield, seed yield, plant height, fertility index and harvest index. The results of the analysis showed a consistent inflation of heritability estimates derived from the simple parent-offspring regression, when compared to the regression estimate by variance-covariance analysis. The two methods successfully removed the environmental covariances from the estimates. However, in the simple regression analysis, error covariance was not removed from the numerat or; therefore, heritabilities estimated by this methodology were higher than those estimated by the variance-covariance method. It was concluded that estimates derived from variance-covariance analysis provide less biased estimates of heritability. Key words: Regression analysis, heritability, meadow bromegrass

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Zinde-Walsh, Victoria. "Some Exact Formulae for Autoregressive Moving Average Processes." Econometric Theory 4, no. 3 (December 1988): 384–402. http://dx.doi.org/10.1017/s0266466600013360.

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This paper demonstrates that for a finite stationary autoregressive moving average process the inverse of the covariance matrix differs from the matrix of the covariances of the inverse process by a matrix of low rank. The formula for the exact inverse of the covariance matrix of the scalar or multivariate process is provided. We obtain approximations based on this formula and evaluate some of the approximate results in the existing literature. Applications to computational algorithms and to the distributions of two-step estimators are discussed. In addition the paper contains the formula for the determinant of the covariance matrix which is useful in exact maximum likelihood estimation; it also lists the expressions for the autocovariances of scalar autoregressive moving average processes.

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Guo, Xiaoyue, Li Zhang, and Yuxiang Xing. "Analytical covariance estimation for iterative CT reconstruction methods." Biomedical Physics & Engineering Express 8, no. 3 (March 11, 2022): 035007. http://dx.doi.org/10.1088/2057-1976/ac58bf.

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Abstract Covariance of reconstruction images are useful to analyze the magnitude and correlation of noise in the evaluation of systems and reconstruction algorithms. The covariance estimation requires a big number of image samples that are hard to acquire in reality. A covariance propagation method from projection by a few noisy realizations is studied in this work. Based on the property of convergent points of cost funtions, the proposed method is composed of three steps, (1) construct a relationship between the covariance of projection and corresponding reconstruction from cost functions at its convergent point, (2) simplify the covariance relationship constructed in (1) by introducing an approximate gradient of penalties, and (3) obtain an analytical covariance estimation according to the simplified relationship in (2). Three approximation methods for step (2) are studied: the linear approximation of the gradient of penalties (LAM), the Taylor apprximation (TAM), and the mixture of LAM and TAM (MAM). TV and qGGMRF penalized weighted least square methods are experimented on. Results from statistical methods are used as reference. Under the condition of unstable 2nd derivative of penalties such as TV, the covariance image estimated by LAM accords to reference well but of smaller values, while the covarianc estimation by TAM is quite off. Under the conditon of relatively stable 2nd derivative of penalties such as qGGMRF, TAM performs well and LAM is again with a negative bias in magnitude. MAM gives a best performance under both conditions by combining LAM and TAM. Results also show that only one noise realization is enough to obtain reasonable covariance estimation analytically, which is important for practical usage. This work suggests the necessity and a new way to estimate the covariance for non-quadratically penalized reconstructions. Currently, the proposed method is computationally expensive for large size reconstructions.Computational efficiency is our future work to focus.

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Jensen, Soren Tolver. "Covariance Hypotheses Which are Linear in Both the Covariance and the Inverse Covariance." Annals of Statistics 16, no. 1 (March 1988): 302–22. http://dx.doi.org/10.1214/aos/1176350707.

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GRIGORE, D. R. "ON MANIFEST COVARIANCE CONDITION IN THE LAGRANGIAN FORMALISM." International Journal of Modern Physics A 07, no. 17 (July 10, 1992): 4073–89. http://dx.doi.org/10.1142/s0217751x92001824.

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The manifest covariance condition for a Poincaré covariant theory is analyzed in the framework of the Lagrangian formalism in a systematic way. We prove that this result is of a kinematic nature and also show that in some cases one needs an appropriate generalization of the Lagrangian formalism. Then the manifest covariance condition is analyzed in a Galilei invariant theory, obtaining the most general type of interaction allowed by this condition.

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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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Wermuth, Nanny, D. R. Cox, and Giovanni M. Marchetti. "Covariance chains." Bernoulli 12, no. 5 (October 2006): 841–62. http://dx.doi.org/10.3150/bj/1161614949.

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Naudts, Jan, and Maciej Kuna. "Covariance systems." Journal of Physics A: Mathematical and General 34, no. 43 (October 19, 2001): 9265–80. http://dx.doi.org/10.1088/0305-4470/34/43/311.

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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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O. Kunjunni, Sajana, and Sajesh T. Abraham. "Sn covariance." Communications in Statistics - Theory and Methods 49, no. 24 (June 16, 2019): 6133–38. http://dx.doi.org/10.1080/03610926.2019.1628275.

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Gao, Rong, Hamed Ahmadzade, and Mojtaba Esfahani. "Covariance and Pseudo-Covariance of Complex Uncertain Variables." Journal of Intelligent & Fuzzy Systems 36, no. 1 (February 16, 2019): 241–51. http://dx.doi.org/10.3233/jifs-181233.

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Card, D. A., E. S. Wisniewski, D. E. Folmer, and A. W. Castleman. "The relationship between covariance and anti-covariance mapping." International Journal of Mass Spectrometry 223-224 (January 2003): 355–63. http://dx.doi.org/10.1016/s1387-3806(02)00928-4.

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Ringh, Axel, Johan Karlsson, and Anders Lindquist. "Multidimensional Rational Covariance Extension with Approximate Covariance Matching." SIAM Journal on Control and Optimization 56, no. 2 (January 2018): 913–44. http://dx.doi.org/10.1137/17m1127922.

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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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Han, Bingyan, and Hoi Ying Wong. "Optimal investment and consumption problems under correlation ambiguity." IMA Journal of Management Mathematics 31, no. 1 (February 18, 2019): 69–89. http://dx.doi.org/10.1093/imaman/dpz002.

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Abstract Consider an economy with $d$ stochastic factors that have an ambiguous variance–covariance matrix. An ambiguity- and risk-averse agent seeks to determine the optimal investment and consumption strategy that is robust to the uncertainty in the covariances. We formulate the robust decision rule as an expected utility maximization over the worst-case scenario with respect to all possible covariances. As this variance–covariance ambiguity leads to robust optimal decisions over a set of non-equivalent probability measures, the $G$-expectation framework is adopted to characterize the problem as a maximin optimization. Our problem formulation can be applied to finite and infinite horizon investment–consumption problems with or without a subsistence consumption constraint. We demonstrate our models using two examples including the defined contribution pension problem and lifetime optimal investment–consumption problems.

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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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Santos, Daniel, Alessandro Dal'Col Lúcio, Alberto Cargnelutti Filho, Lindolfo Storck, Leandro Homrich Lorentz, and Denison Esequiel Schabarum. "Effect of neighborhood and plot size on experiments with multiple-harvest oleraceous crops." Pesquisa Agropecuária Brasileira 49, no. 4 (April 2014): 257–64. http://dx.doi.org/10.1590/s0100-204x2014000400003.

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The objective of this work was to determine the efficiency of the Papadakis method on the quality evaluation of experiments with multiple-harvest oleraceous crops, and on the estimate of the covariate and the ideal plot size. Data from nine uniformity trials (five with bean pod, two with zucchini, and two with sweet pepper) and from one experiment with treatments (with sweet pepper) were used. Through the uniformity trials, the best way to calculate the covariate was defined and the optimal plot size was calculated. In the experiment with treatments, analyses of variance and covariance were performed, in which the covariate was calculated by the Papadakis method, and experimental precision was evaluated based on four statistics. The use of analysis of covariance with the covariate obtained by the Papadakis method increases the quality of experiments with multiple-harvest oleraceous crops and allows the use of smaller plot sizes. The best covariate is the one that considers a neighboring plot of each side of the reference plot.

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Kong, Rong, Ming Xue, and Chengsi Liu. "Development of a Hybrid En3DVar Data Assimilation System and Comparisons with 3DVar and EnKF for Radar Data Assimilation with Observing System Simulation Experiments." Monthly Weather Review 146, no. 1 (January 1, 2018): 175–98. http://dx.doi.org/10.1175/mwr-d-17-0164.1.

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Abstract A hybrid ensemble–3DVar (En3DVar) system is developed and compared with 3DVar, EnKF, “deterministic forecast” EnKF (DfEnKF), and pure En3DVar for assimilating radar data through perfect-model observing system simulation experiments (OSSEs). DfEnKF uses a deterministic forecast as the background and is therefore parallel to pure En3DVar. Different results are found between DfEnKF and pure En3DVar: 1) the serial versus global nature and 2) the variational minimization versus direct filter updating nature of the two algorithms are identified as the main causes for the differences. For 3DVar (EnKF/DfEnKF and En3DVar), optimal decorrelation scales (localization radii) for static (ensemble) background error covariances are obtained and used in hybrid En3DVar. The sensitivity of hybrid En3DVar to covariance weights and ensemble size is examined. On average, when ensemble size is 20 or larger, a 5%–10% static covariance gives the best results, while for smaller ensembles, more static covariance is beneficial. Using an ensemble size of 40, EnKF and DfEnKF perform similarly, and both are better than pure and hybrid En3DVar overall. Using 5% static error covariance, hybrid En3DVar outperforms pure En3DVar for most state variables but underperforms for hydrometeor variables, and the improvement (degradation) is most notable for water vapor mixing ratio qυ (snow mixing ratio qs). Overall, EnKF/DfEnKF performs the best, 3DVar performs the worst, and static covariance only helps slightly via hybrid En3DVar.

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Gao, Jidong, and David J. Stensrud. "Some Observing System Simulation Experiments with a Hybrid 3DEnVAR System for Storm-Scale Radar Data Assimilation." Monthly Weather Review 142, no. 9 (September 2014): 3326–46. http://dx.doi.org/10.1175/mwr-d-14-00025.1.

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A hybrid three-dimensional ensemble–variational data assimilation (3DEnVAR) algorithm is developed based on the 3D variational data assimilation (3DVAR) and ensemble Kalman filter (EnKF) programs with the Advanced Regional Prediction System (ARPS). The method uses the extended control variable approach to combine the static and ensemble-derived flow-dependent forecast error covariances. The method is applied to the assimilation of simulated data from two radars for a supercell storm. Some sensitivity experiments are performed to answer questions about how flow-dependent covariance estimated from the forecast ensemble can be best used in the hybrid 3DEnVAR scheme. When the ensemble size is relatively small (with 5 or 10 ensemble members), it is found that experiments with a weaker weighting value for the ensemble covariance leads to better analysis results. Even when severe sampling errors exist, introducing ensemble-estimated covariances into the variational method still benefits the analysis. For reasonably large ensemble sizes (50–100 members), a stronger relative weighting (>0.8) for the ensemble covariance leads to better analyses from the hybrid 3DEnVAR. In addition, the sensitivity experiments also indicate that the best results are obtained when the number of the augmented control variables is a function of three spatial dimensions and ensemble members, and is the same for all analysis variables.

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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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Journal articles on the topic 'Covariance'

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