Sum Covariance Research Articles

A linear mixed model formulation for spatio-temporal random processes with computational advances for the product, sum, and product–sum covariance functions

To properly characterize a spatio-temporal random process, it is necessary to understand the process’ dependence structure. It is common to describe this dependence using a single random error having a complicated covariance. Instead of using the single random error approach, we describe spatio-temporal random processes using linear mixed models having several random errors; each random error describes a specific quality of the covariance. This linear mixed model formulation is general, intuitive, and contains many commonly used covariance functions as special cases. We focus on using the linear mixed model formulation to express three covariance functions: product (separable), sum (linear), and product–sum. We discuss benefits and drawbacks of each covariance function and propose novel algorithms using Stegle eigendecompositions, a recursive application of the Sherman–Morrison–Woodbury formula, and Helmert–Wolf blocking to efficiently invert their covariance matrices, even when every spatial location is not observed at every time point. Via a simulation study and an analysis of temperature data in Oregon, USA, we assess model performance and computational efficiency of these covariance functions when estimated using restricted maximum likelihood (likelihood-based) and Cressie’s weighted least squares (semivariogram-based). We end by offering guidelines for choosing among combinations of the covariance functions and estimation methods based on properties of observed data and the desired balance between model performance and computational efficiency.

Improving State Estimation Accuracy Through Incremental Meter Placement Using New Evolutionary Strategy

Power system network is an interconnected web of crucial electrical elements and system monitoring. It is imperative in ensuring system integrity and stability. In regard to this, state estimation is commonly employed to obtain the best estimate of the power system condition based on limited number of measurements. The placement of meters at appropriate locations is crucial in determining the accuracy of the state estimation. Hence, this paper presents a new optimal meter placement strategy for state estimation. A new evolutionary strategy for discrete optimization problem is proposed so that the location of additional meter placement will improve the accuracy of state estimation. The minimization of sum covariance error of state estimation is selected as the objective function to be minimized. Simulation results on the IEEE 30-bus system clearly shows that the proposed approach is able to outperform the conventional heuristic method in determining the optimal meter placement, which enhances the state estimation accuracy.

On Optimal Gaussian Signaling in MIMO Relay Channels With Partial Decode-and-Forward

For Gaussian multiple-input multiple-output (MIMO) relay channels with partial decode-and-forward, the optimal type of input distribution is still an open question in general. Recent research has revealed that in some other scenarios with unknown optimal input distributions (e.g., interference channels), improper (i.e., noncircular) Gaussian distributions can outperform proper (circular) Gaussian distributions. In this paper, we show that this is not the case for partial decode-and-forward in the Gaussian MIMO relay channel with Gaussian transmit signals, i.e., we show that a proper Gaussian input distribution is the optimal one among all Gaussian distributions. In order to prove this property, an innovation covariance matrix is introduced, and a decomposition is performed by considering the optimization over this matrix as an outer problem. A key point for showing optimality of proper signals then is a reformulation that reveals that one of the subproblems is equivalent to a sum rate maximization in a two-user MIMO broadcast channel under a sum covariance constraint, for which the optimality of proper signals can be shown.

Forcing by rare species and intransitive loops creates distinct bouts of extinction events conditioned by spatial pattern in competition communities

Using a spatially explicit multiple species com- petition framework, the pattern of extinction for an effect/ response competition cellular automata model is presented. It is found that, for a particular matrix with a negative row/ column sum covariance, (1) there are two phases of transient behavior, presented in the framework of graph partitioning, and (2) the ultimate behavior of the system in space can be understood as mainly the operation of three distinct intran- sitive competitive loops. Of particular interest is the effect that a set of initially excluded species have on the final outcome of spatial pattern formation. Even though initial exclusion happens quite rapidly, and it would seem that species excluded early could not have an effect on the eventual outcomes of competition, there is a clear statistical effect, contributing to the extant literature suggesting that rare or seemingly inconsequential species can effect the behavior of the common and obvious ones.

On strict positive definiteness of product and product–sum covariance models

Although positive definiteness is a sufficient condition for a function to be a covariance, the stronger strict positive definiteness is important for many applications, especially in spatial statistics, since it ensures that the kriging equations have a unique solution. In particular, spatial–temporal prediction has received a lot of attention, hence strictly positive definite spatial–temporal covariance models (or equivalently strictly conditionally negative definite variogram models) are needed. In this paper the necessary and sufficient condition for the product and the product–sum space–time covariance models to be strictly positive definite (or the variogram function to be strictly conditionally negative definite) is given. In addition it is shown that an example appeared in the recent literature which purports to show that product–sum covariance functions may be only semi-definite is itself invalid. Strict positive definiteness of the sum of products model is also discussed.

The Linear Coregionalization Model and the product-sum space-time variogram

The product covariance model, the product–sum covariance model, and the integrated product and integrated product–sum models have the advantage of being easily fitted by the use of “marginal” variograms. These models and the use of the marginals are described in a series of papers by De Iaco, Myers, and Posa. Such models allow not only estimating values at nondata locations but also prediction in future times, hence, they are useful for analyzing air pollution data, meteorological data, or ground water data. These three kinds of data are nearly always multivariate and because the processes determining the deposition or dynamics will affect all variates, a multivariate approach is desirable. It is shown that the use of marginal variograms for space–time modeling can be extended to the multivariate case and in particular to the use of the Linear Coregionalization Model (LCM) for cokriging in space–time. An application to an environmental data set is given.

Space–time analysis using a general product–sum model

A generalization of the product–sum covariance model introduced by De Cesare et al. (Statist. Probab. Lett. 51 (2001) 9) is given in this paper. This generalized model is non-separable and in general is non-integrable, hence, it cannot be obtained from the Cressie–Huang representation. Moreover, the product–sum model does not correspond to the use of a metric in space–time. It is shown that there are simple methods for estimating and modeling the covariance or variogram components of the product–sum model using data from realizations of spatial–temporal random fields.

Estimating and modeling space–time correlation structures

In this paper, a class of product–sum covariance models has been introduced for estimating and modeling space–time correlation structures. It is shown how the coefficients of this class of models are related to the global sill and “partial” spatial and temporal sills; moreover, some constraints on these sills have been given in order to assure positive definiteness of the product–sum covariance model. A brief comparative study with some other classes of spatial–temporal covariance models has been pointed out.

The consequences of using inadequate testers in the simplified triple test-cross

The genetical consequences of common alleles in the L1 and L2 testers of a simplified version of the triple test-cross which is applicable to populations of inbred lines are examined. The test for epistasis under these circumstances becomes ambiguous and can spuriously detect non-allelic interactions when they may not exist although it still provides a test for epistasis and the adequacy of the testers simultaneously. The tests of significance and the estimates of additive variation are biased to an extent related to the dominance and dominance x additive effects of the common loci while the significance and estimates of dominance variation are deflated because they reflect the dominance effects at the non-common loci only. The covariance of sums and differences is also underestimated for the same reasons. These expectations are illustrated by analysing the 190 simplified triple test-crosses that could be extracted from a 20 x 20 diallel set of crosses between pure-breeding lines of Nicotiana rustica.