Because of the relatively high levels of genetic relationships among potential bull sires and bull dams, innovative selection tools should consider both genetic gain and genetic relationships in a long-term perspective. Optimum genetic contribution theory using official estimated breeding values for a moderately heritable trait (production index, Index-PROD), and a lowly heritable functional trait (index for somatic cell score, Index-SCS) was applied to find optimal allocations of bull dams and bull sires. In contrast to previous practical applications using optimizations based on Lagrange multipliers, we focused on semi-definite programming (SDP). The SDP methodology was combined with either pedigree (a(ij)) or genomic relationships (f(ij)) among selection candidates. Selection candidates were 484 genotyped bulls, and 499 preselected genotyped bull dams completing a central test on station. In different scenarios separately for PROD and SCS, constraints on the average pedigree relationships among future progeny were varied from a(ij)=0.08 to a(ij)=0.20 in increments of 0.01. Corresponding constraints for single nucleotide polymorphism-based kinship coefficients were derived from regression analysis. Applying the coefficient of 0.52 with an intercept of 0.14 estimated for the regression pedigree relationship on genomic relationship, the corresponding range to alter genomic relationships varied from f(ij) = 0.18 to f(ij) = 0.24. Despite differences for some bulls in genomic and pedigree relationships, the same trends were observed for constraints on pedigree and corresponding genomic relationships regarding results in genetic gain and achieved coefficients of relationships. Generally, allowing higher values for relationships resulted in an increase of genetic gain for Index-PROD and Index-SCS and in a reduction in the number of selected sires. Interestingly, more sires were selected for all scenarios when restricting genomic relationships compared with restricting pedigree relationships. For example, at constraint of f(ij)=0.185 and selection on Index-PROD, the number of selected sires was 35. In contrast, only 21 sires were selected at the comparable constraint on additive genetic relationship of a(ij)=0.09. A further reduction in relationships is possible when using SDP output (i.e., suggested genetic contributions of selected parents) and applying a simulated annealing algorithm to define specific mating plans. However, the advantage of this strategy is limited to a short-term perspective and probably not successful in the period of genomic selection allowing a substantial reduction of generation intervals.
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.
The iteratively reweighted least-squares approach to self-tuning robust adjustment of parameters in linear regression models with autoregressive (AR) and t-distributed random errors, previously established in Kargoll et al. (in J Geod 92(3):271–297, 2018. 10.1007/s00190-017-1062-6), is extended to multivariate approaches. Multivariate models are used to describe the behavior of multiple observables measured contemporaneously. The proposed approaches allow for the modeling of both auto- and cross-correlations through a vector-autoregressive (VAR) process, where the components of the white-noise input vector are modeled at every time instance either as stochastically independent t-distributed (herein called “stochastic model A”) or as multivariate t-distributed random variables (herein called “stochastic model B”). Both stochastic models are complementary in the sense that the former allows for group-specific degrees of freedom (df) of the t-distributions (thus, sensor-component-specific tail or outlier characteristics) but not for correlations within each white-noise vector, whereas the latter allows for such correlations but not for different dfs. Within the observation equations, nonlinear (differentiable) regression models are generally allowed for. Two different generalized expectation maximization (GEM) algorithms are derived to estimate the regression model parameters jointly with the VAR coefficients, the variance components (in case of stochastic model A) or the cofactor matrix (for stochastic model B), and the df(s). To enable the validation of the fitted VAR model and the selection of the best model order, the multivariate portmanteau test and Akaike’s information criterion are applied. The performance of the algorithms and of the white noise test is evaluated by means of Monte Carlo simulations. Furthermore, the suitability of one of the proposed models and the corresponding GEM algorithm is investigated within a case study involving the multivariate modeling and adjustment of time-series data at four GPS stations in the EUREF Permanent Network (EPN).
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
In time series analyses, covariance modeling is an essential part of stochastic methods such as prediction or filtering. For practical use, general families of covariance functions with large flexibilities are necessary to model complex correlations structures such as negative correlations. Thus, families of covariance functions should be as versatile as possible by including a high variety of basis functions. Another drawback of some common covariance models is that they can be parameterized in a way such that they do not allow all parameters to vary. In this work, we elaborate on the affiliation of several established covariance functions such as exponential, Matérn-type, and damped oscillating functions to the general class of covariance functions defined by autoregressive moving average (ARMA) processes. Furthermore, we present advanced limit cases that also belong to this class and enable a higher variability of the shape parameters and, consequently, the representable covariance functions. For prediction tasks in applications with spatial data, the covariance function must be positive semi-definite in the respective domain. We provide conditions for the shape parameters that need to be fulfilled for positive semi-definiteness of the covariance function in higher input dimensions.
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