Covariance Components Research Articles

Influence of the mating design on the additive genetic variance in plant breeding populations

Key messageMating designs determine the realized additive genetic variance in a population sample. Deflated or inflated variances can lead to reduced or overly optimistic assessment of future selection gains.The additive genetic variance {V}_{A} inherent to a breeding population is a major determinant of short- and long-term genetic gain. When estimated from experimental data, it is not only the additive variances at individual loci (QTL) but also covariances between QTL pairs that contribute to estimates of {V}_{A}. Thus, estimates of {V}_{A} depend on the genetic structure of the data source and vary between population samples. Here, we provide a theoretical framework for calculating the expectation and variance of {V}_{A} from genotypic data of a given population sample. In addition, we simulated breeding populations derived from different numbers of parents (P = 2, 4, 8, 16) and crossed according to three different mating designs (disjoint, factorial and half-diallel crosses). We calculated the variance of {V}_{A} and of the parameter b reflecting the covariance component in {V}_{A}, standardized by the genic variance. Our results show that mating designs resulting in large biparental families derived from few disjoint crosses carry a high risk of generating progenies exhibiting strong covariances between QTL pairs on different chromosomes. We discuss the consequences of the resulting deflated or inflated {V}_{A} estimates for phenotypic and genome-based selection as well as for applying the usefulness criterion in selection. We show that already one round of recombination can effectively break negative and positive covariances between QTL pairs induced by the mating design. We suggest to obtain reliable estimates of {V}_{A} and its components in a population sample by applying statistical methods differing in their treatment of QTL covariances.

Dynamic mixed models with heterogeneous covariance components using multivariate GARCH innovations and the Dirichlet process mixture

This paper presents innovative strategies for correlated data analysis using dynamic mixed effects models to account for the autoregressive conditional-heteroscedasticity of innovations and cross-sectional dependence. In a multivariate setting, the traditional modeling scheme was extended to time-varying components of variances and covariances by initiating the heterogeneous GARCH model. We propose a Bayesian semi-parametric approach by utilizing the centered Dirichlet process as priors for the heterogeneity effects involved in the mean structure. We present a stochastic clustering strategy to distinguish similar patterns of covariates. The Hamiltonian Monte Carlo technique is used to ease the computation of time-series equations for variances and covariances with higher efficiency. We conduct comprehensive simulation studies to examine model features, especially in cluster-wise settings. We adopt the maximal overlap discrete wavelet transforms to isolate short- and long-run sample information for practical applications. The transformation facilitates analyzing the effect of high-frequency or trend variation in macroeconomic factors. We illustrate the advantage of our proposed modeling methodology in the empirical studies by investigating the short-run impacts of macroeconomic events on economic growth and its prediction accounting for the unexpected volatility in the financial market of some selected countries.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

New formulation of Galilean relativistic Maxwell theory

In this paper, we discuss Galilean relativistic Maxwell theory in detail. We first provide a set of mapping relations, derived systematically, that connect the covariant and contravariant vectors in the Lorentz relativistic and Galilean relativistic formulations. Exploiting this map, we construct the two limits of Galilean relativistic Maxwell theory from usual Maxwell's theory in the potential formalism for both contravariant and covariant vectors which are now distinct entities. Field equations are derived and their internal consistency is shown. The entire analysis is then performed in terms of electric and magnetic fields for both covariant and contravariant components. Duality transformations and their connection with boost symmetry are discussed which reveal a rich structure. The notion of twisted duality is introduced. Next we consider gauge symmetry, construct Noether currents and show their on-shell conservation. We also discuss shift symmetry under which the Lagrangian is invariant, where the corresponding currents are now on-shell conserved. At the end we analyse the theory by including sources for both contravariant and covariant sectors. We show that sources are now off-shell conserved

Electromagnetic field theory in superluminal spacetime

Recently, a number of experimental observations on the superluminal group velocities of pulses propagating in dispersive media have led to reconsidering electromagnetism theory in an unconventional framework. To consider faster-than-light phenomena, it is not necessary to replace the current relativistic theory, but it is sufficient to extend it to superluminal motions in a way that preserves the principle of causality. In the present paper, a new approach to study superluminal motions is proposed, which avoids introducing unphysical complex quantities and allows for the formulation of equations that are covariant according to a hyperbolic metric. In the framework of this formalism, Maxwell equations and the single-photon wave equation are obtained through superluminal transformations of ordinary equations. It is shown that the covariant and contravariant components of the superluminal electromagnetic field determine its magnitude and direction, respectively. Furthermore, the solutions of the transformed Maxwell equations are X-shaped ways propagating in the superluminal spacetime region that is delimited by the infinite light cone and the two-sheeted hyperboloid perpendicular to it. Instead, in quantum mechanics, the covariant and contravariant components of the electromagnetic field are proportional to the right- and left-handed helicities of the single photon, respectively.

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Evaluation of product bioequivalence when subject blood volume necessitates limited sampling strategies.

In a traditional blood level bioequivalence (BE) study, every subject provides drug concentrations at each blood sampling time. However, this approach is not suitable for animals whose blood volume limits or prohibits multiple sample collections. In our previous research, we presented an approach that can be applied to studies using a destructive sampling design where each animal provides only 1 blood sample that is then incorporated into a composite profile. Another situation we sometimes face is that of when the animals can contribute more than one sample but are still limited in the number of blood draws (e.g., 3) such that a complete profile per animal is not feasible. Unlike the destructive sampling situation, we cannot combine all blood samples into a single "composite" profile and ignore the correlation of values obtained from the same subject. To avoid the complexities associated with needing to include a covariance component among experimental units into the statistical model, we propose an approach whereby study subjects are randomly assigned to housing unit (e.g., cage or pen) and then randomly assigned to a sampling schedule within each housing unit. In doing so, housing unit rather than the individual subject serves as the experimental unit. This article provides an assessment of this alternative approach to assess product BE when only a limited number of samples can be obtained per study subject.

Electrodynamics and rational thermodynamics

AbstractThe electromagnetic fields (D, H) represent contravariant components of an antisymmetric 4‐tensor, while the fields (E, B) represent covariant components of the same 4‐tensor. Both sets are identical in Lorentz frames. The governing equations for them represent equations of balance for the flux of D and the flux B across open surfaces and they are invariant under arbitrary analytic transformation of space and time. This convenient property has motivated mechanicians to reformulate mechanics and thermodynamics in terms of the symmetric 4‐tensor of energy‐momentum so as to exhibit, perhaps, the same invariance as electromagnetism: general relativity. The interpretation of electromagnetism in terms of fields carries the theory a long way. But it breaks down when absorption and emission of radiation by bodies is concerned. In this case the proper view is that of a photon gas.Also the elegant union of electromagnetism and thermodynamics meets practical difficulties for bodies with internal structure, like electric and magnetic dipoles or for mixtures of charged constituents. The description of such bodies is limited to velocities much smaller than that of light. Still in those bodies the entropy inequality implies severe and interesting restrictions on the dependence of material properties on the electromagnetic fields. It will be evaluated by using the Lagrange multiplier method.In the early days of electromagnetism there was much confusion among physicists and engineers about the different formulation of the fields and about proper dimensions and units. This dilemma is also briefly explained in the article.

Met Office MOGREPS‐G initialisation using an ensemble of hybrid four‐dimensional ensemble variational (En‐4DEnVar) data assimilations

AbstractThe Met Office Global and Regional Ensemble Prediction System–Global (MOGREPS‐G) used an ensemble transform Kalman filter (ETKF) to perturb its initial conditions from its operational implementation in September 2008 until December 2019. In 2019, MOGREPS‐G became the first operational atmospheric ensemble to apply hybrid four‐dimensional ensemble variational data assimilation (En‐4DEnVar) to each of the 44 perturbed ensemble members. Other enhancements have also been added, including to the inflation used to improve ensemble spread. The combined impact of these changes on ensemble forecasts is overwhelmingly positive but initially more neutral for deterministic forecasts, which also use the ensemble to represent flow‐dependent forecast errors in their hybrid data assimilation updates. The latter result is not a surprise, because the deterministic forecast's hybrid data assimilation was initially weighted more strongly to the modelled stationary covariance component and not optimised to take full advantage of the upgraded ensemble. A subsequent operational upgrade in December 2020 has introduced shifting in addition to lagging to exploit the ensemble better in the deterministic forecast's hybrid data assimilation by including ensemble members from a previous cycle and also from adjacent forecast lead times to augment the ensemble without having to run additional forecasts. More weight has since been given to the ensemble in the deterministic forecast's hybrid data assimilation in May 2022. A key motive for adopting hybrid 4DEnVar in MOGREPS‐G is to reduce maintenance overheads by virtue of sharing much of the deterministic forecast system's data assimilation code. This also enables the ensemble to assimilate almost all observation types used by the deterministic forecast. The updated system also exploits parallelism better so as to be fast enough for operational use, despite assimilating more observations and being more computationally expensive than the Met Office's ETKF.

Genetic correlation between feed efficiency and carcass traits in Nellore cattle in Brazil

The aim was to estimate the genetic correlations between residual feed intake (RFI) and dry matter intake (DMI) with carcass finish (CF), rib eye area (REA), and marbling (MAR) of Nellore cattle. Data from 7,808 animals were considered. In addition, data from 2,261 females included in the complete database were also considered. Estimates of variance and covariance components, as well as heritabilities and genetic correlations were obtained by means of two-character analysis under animal model. Heritability estimates were found to be moderate for the RFI (0.22) and DMI (0.29) traits. It was observed that genetic correlation was close to zero for all traits, except between RFI and REA (-0.11). However, considering the female population, there was an increase in the estimated genetic correlation between RFI and DMI, although still a favorable genetic association of low magnitude (-0.30). There was also an increase in the genetic association of REA with RFI (-0.21). It can be concluded that the direct selection for RFI and DMI will not influence the CF, MAR, or REA of Nellore cattle. However, this selection may generate some favorable responses in MAR and REA in Nellore females.

Genetic Parameters for Functional Longevity, Type Traits, and Production in the Serbian Holstein.

In this study, the authors focused on the evaluation of the genetic parameters of longevity, milk yield traits, and type traits in dairy cattle populations in the Republic of Serbia. The total dataset used consisted of production records and pedigree data for 32,512 Holstein cows that calved from 1981 to 2015. The animal model was applied to determine the variance and covariance components and genetic parameters of the analyzed traits by applying the restricted maximum likelihood (REML) approach and using the programs VCE6 and PEST. The heritability of longevity traits was estimated using the Survival Kit V6.0 software package. Variance and covariance were estimated for five production traits: milk yield (MY), fat yield (FY), protein yield (PY), milk fat content (MC), and protein content (PC), and three longevity traits: length of productive life (LPL), lifetime milk yield (LMY), and the number of lactations achieved (NL) as well as for 18 standard type traits. Heritabilities for the milk, fat, and protein yield traits were 0.20 (MY), 0.15 (FY), and 0.19 (PY), respectively. The estimated coefficients of heritability for the longevity traits were higher when using the Weibull proportional hazards model compared to the traditional linear methods and ranged from 0.08 for NL to 0.10 for LPL. Heritability values for the type traits varied from a low of 0.10 (RLSsv-rear legs set-side view) to medium values of 0.32 (ST-stature). Genetic correlations were found between MY and the following longevity traits: LPL, LMY, and NL with values of -0.18, -0.11, and -0.09, respectively. Genetic correlations were found between MY and a number of linear type traits and varied from 0.02 (between MY and RUH-rear udder height) to 0.28 (between MY and FUA-fore udder attachment). Genetic correlations between the 18 investigated type traits ranged from -0.33 between TL (top line) and RTP (rear teats position) to 0.71 between AN (angularity) and RUH (rear udder height). Genetic correlations between most linear type traits and longevity traits (LPL, LMY, and NL) were generally negative and very low. The highest positive genetic correlation was found between UD and LPL (rxy = 0.38).

Modelling Factors Associated with Malnutrition and Anemia in Children under Five Years in Angola, Senegal, and Malawi by using a Joint Model

Background: In Sub-Saharan Africa, malnutrition and anemia contribute a higher percentage to infant morbidity. Malnutrition is known as the dearth of proper nutrition in the human body and it is an important risk factor for the burden of diseases. The lack of hemoglobin and red cells in the human body is known as anemia, and it is divided into three groups. This paper aims to identify the determinants of anemia and malnutrition in Angola, Senegal and Malawi. The novelty of this study includes creating a sample that will be jointly modelled to identify determinants of anemia and malnutrition. Methods: This paper used 2016 information from Angola, Senegal and Malawi Demographic and Health Survey to conduct a secondary data analysis. To create a pool sample for the analysis, the Angola, Malawi and Senegal Demographic and Health Survey data were combined. The joint model under the generalized linear mixed model was employed to identify the determinants of malnourishment and anemia among children under five years in Angola, Senegal, and Malawi. Results: The analysis of the data was performed in SAS 9.4. The results of the covariance components indicated a positive correlation between nutritional status and anemia status. Joint generalized linear mixed model results revealed that children under five years residing with a mother that has attained a primary level of education are 2.995 times more likely to be malnourished when compared to children under five years residing with a mother that have attained a higher level of education. Children under five years residing in the rural setting of Angola, Senegal, and Malawi are 1.473 times more likely to be malnourished when compared to children under five years residing in the urban setting of Angola, Senegal and Malawi. Conclusion: Based on the joint generalized linear mixed model results, type of residence, sex of the child, age of the child, mother's level of education, birth interval and wealth index are the correlates of malnourishment and anemia in Angola, Senegal and Malawi. There is a greater need for partnership and collaboration among the studied countries to achieve the SGD target.

A cross-sectional and longitudinal study of human brain development: The integration of cortical thickness, surface area, gyrification index, and cortical curvature into a unified analytical framework

Brain maturation studies typically examine relationships linking a single morphometric feature with cognition, behavior, age, or other demographic characteristics. However, the coordinated spatiotemporal arrangement of morphological features across development and their associations with behavior are unclear. Here, we examine covariation across multiple cortical features (cortical thickness [CT], surface area [SA], local gyrification index [GI], and mean curvature [MC]) using magnetic resonance images from the NIMH developmental cohort (ages 5–25). Neuroanatomical covariance was examined using non-negative matrix factorization (NMF), which decomposes covariance resulting in a parts-based representation. Cross-sectionally, we identified six components of covariation which demonstrate differential contributions of CT, GI, and SA in hetero- vs. unimodal areas. Using this technique to examine covariance in rates of change to identify longitudinal sources of covariance highlighted preserved SA in unimodal areas and changes in CT and GI in heteromodal areas. Using behavioral partial least squares (PLS), we identified a single latent variable (LV) that recapitulated patterns of reduced CT, GI, and SA related to older age, with limited contributions of IQ and SES. Longitudinally, PLS revealed three LVs that demonstrated a nuanced developmental pattern that highlighted a higher rate of maturational change in SA and CT in higher IQ and SES females. Finally, we situated the components in the changing architecture of cortical gradients. This novel characterization of brain maturation provides an important understanding of the interdependencies between morphological measures, their coordinated development, and their relationship to biological sex, cognitive ability, and the resources of the local environment.

Demodulation of non-stationary random signal using Hilbert transform

A narrow-band high frequency amplitude modulation as a model of vibration signal is considered. Use of Hilbert transform for the demodulation of periodically non-stationary random signal (PNRS) is discussed. Relations for spectral and covariance components of model signal, its Hilbert transform and cross-covariance components are obtained. Quadratures for modulation signal are extracted and analyzed. It is shown, that the Fourier coefficients of the auto-covariance functions of a signal and its Hilbert transform are the same and its cross-covariance functions differ only by a sign. The square of the modulus of the analytical signal is not a “squared envelope” in the known sense. A “squared envelope” in this case is a random process, whose mathematical expectation is equal to twice the variance of the raw signal. This results in an identity of cyclic spectrums of variances for analytic and raw signals. Thus, the Hilbert transform cannot be used directly as a demodulation procedure, and the “squared envelope” can be analyzed only as the implementation of a random process using PNVP methods. It is shown that band-pass filtering and the Hilbert transform can be used for extraction of modulating signal quadratures.

Genetic parameters for dry matter intake, energy balance, residual energy intake, and liability to diseases in German Holstein and Fleckvieh dairy cows.

Selection for feed efficiency (FE) is a hot topic in dairy cow breeding. Dry matter intake (DMI) and residual energy intake (REI) are mostly discussed as new selection traits. Selection for lower DMI or REI seems to increase FE if other traits, such as milk yield or health, are not affected negatively. However, genetic relationships with other traits have not been adequately investigated because of the difficulties in recording sufficient feed intake data for genetic evaluations. The aim of this study was to examine the genetic relationships between FE-related traits and liability to diseases throughout lactation. First, heritabilities for all traits are presented. Subsequently, genetic correlations between DMI, energy-corrected milk yield, energy balance (EB), and REI on the one hand and 3 disease categories (mastitis, claw and leg diseases, and all diseases) on the other throughout lactation in German Holstein (GH) dairy cows are illustrated. Production and health data from the projects optiKuh and eMissionCow were used. Data consisted of weekly observations recorded over a 325-wk period in 2,387 GH and over a 300-wk period in 632 Fleckvieh (FV) primiparous and multiparous dairy cows from 13 dairy research farms in Germany. Variance and covariance components were estimated univariately or bivariately with linear random regression models for production data and threshold random regression models for health data. Heritabilities for DMI, EB, and REI were on average 0.17 and 0.15, 0.14 and 0.15, as well as 0.11 and 0.14 in GH and FV, respectively. Heritabilities on the underlying scale for mastitis, claw and leg diseases, and all diseases were on average 0.17 and 0.16, 0.18 and 0.12, as well as 0.15 and 0.11 in GH and FV, respectively. In GH, almost all genetic correlations were negative, especially in early lactation. Within the first 50 d in milk, genetic correlations between DMI and REI on the one hand and disease categories on the other ranged from -0.25 to -0.14 for mastitis, from -0.31 to -0.13 for claw and leg diseases, and from -0.58 to -0.30 for all diseases. Consequently, selection for lower DMI or REI could lead to a higher liability to diseases, especially in early lactation. A possibility to mitigate these undesirable side effects could be lactation stage-specific selection for FE. For FV, further studies with more data are needed to assess genetic relationships.

Unmodeled-error-corrected stochastic assessment for a standalone GNSS receiver regardless of the number of tracked frequencies

Appropriate stochastic modeling is crucial to Global Navigation Satellite System (GNSS) applications. However, traditional methods based on the zero baseline and ultra-short baseline cannot accurately estimate the stochastic characteristics of GNSS measurements of a complete receiving system, i.e., receiver, antenna, low-noise amplifier, and cable. We propose the unmodeled-error-corrected stochastic assessment methods that can be used for a standalone receiver regardless of the number of tracked frequencies. First, the raw observation equations considering the unmodeled errors are deduced based on the geometry-free or geometry-based model. Second, some parameters are chosen as the datum for rank deficiency elimination. Third, the multi-epoch parameterization is used to correct the unmodeled errors in the reparametrized observation model. Finally, the covariance component estimation can be applied to determine the variance components. Three different and representative measurement campaigns were conducted, including the low-cost board, integrated receiver, and smart phone. The results show that the proposed method based on a standalone receiver can assess the stochastic characteristics of GNSS measurements. Since the stochastic characteristics of different low-cost receivers share different patterns, and the noise from the internal antenna cannot be easily ignored, the proposed method is essential.

Genetic Parameters of Reproductive Performances in Hungarian Large White, Landrace, and Their Crossbred F1 Pigs from 2010 to 2018

Genetic parameters, breeding values, and aggregate breeding values of number of piglets born alive (NBA), number of weaned piglets (NWE), and litter weight at weaning (LWWE) were predicted in the Hungarian Large White, Hungarian Landrace breeds, and in their cross (F1). Seven repeatability animal models were used. BLUP and REML methodology were used to estimate breeding values and variance–covariance components. PEST and VCE 6 software were used for estimating breeding values and variance components. Heritability for NBA and NWE was the same for all seven models. On the contrary, heritability estimates for LWWE were higher in comparison with NBA and NWE. The permanent environmental variance component was small for all traits. The large White breed had positive and significant genetic trends for all seven models and for all three traits. Landrace breed had significant trends for NBA, which was negative, and for NWE, the results were positive. The constructed indices result in one number (i.e., aggregate genetic merit); thus, the animals can be selected based on their overall performance considering the various aspects.

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component BΨP of the magnetic field vector with respect to the poloidal flux function ΨP. In an unperturbed plasma, BΨP can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of BΨP is small. Here, we are interested in the question whether Hamiltonian conservation laws are still satisfied when BΨP is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without BΨP yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with BΨP retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

Accurate Multilayered Shell Buckling Analysis via the Implicit-Mesh Discontinuous Galerkin Method

A novel formulation for the linear buckling analysis of multilayered shells is presented. High-order equivalent-single-layer shell theories based on the through-the-thickness expansion of the covariant components of the displacement field are employed. The novelty of the formulation regards the governing equations solution via implicit-mesh discontinuous Galerkin method. It is a high-order accurate numerical technique based on a discontinuous representation of the solution among the mesh elements and on the use of suitably defined boundary integrals to enforce the continuity of the solution at the inter-element interfaces as well as the boundary conditions. Owing to its discontinuous nature, it can be naturally employed with nonconventional meshes. In this work, it is combined with the implicitly defined mesh technique, whereby the mesh of the shell modeling domain is constructed by intersecting an easy-to-generate background grid and a level set function implicitly representing the cutouts. Several numerical examples are considered for the buckling loads of plates and shells modeled by different theories and characterized by various materials, geometry, boundary conditions, and cutouts. The obtained results are compared with literature and finite-element solutions, and they demonstrate the accuracy and the robustness of the proposed approach.

Optimization of undifferenced and uncombined PPP stochastic model based on covariance component estimation

Optimization of undifferenced and uncombined PPP stochastic model based on covariance component estimation