A Kronecker-based Covariance Specification for Spatially Continuous Multivariate Data

We propose a covariance specification for modeling spatially continuous multivariate data. This model is based on a reformulation of Kronecker’s product of covariance matrices for Gaussian random fields. We illustrate the case with the Matérn function used for specifying marginal covariances. The structure holds for other choices of covariance functions with parameters varying in their usual domains, which makes the estimation process more accessible. The reduced computational time and flexible generalization for increasing number of variables, make it an attractive alternative for modelling spatially continuous data. Theoretical results for the likelihood function and the derivatives of the covariance matrix are presented. The proposed model is fitted to the literature’s soil250 dataset, and adequacy measures, forecast errors and estimation times are compared with the ones obtained based on classical models. Furthermore, the model is fitted to the classic meuse dataset to illustrate the model’s flexibility in a four-variate analysis. A simulation study is performed considering different parametric scenarios to evaluate the asymptotic properties of the maximum likelihood estimators. The satisfactory results, its simpler structure and the reduced estimation time make the proposed model a candidate approach for multivariate analysis of spatial data.

A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling

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.

Spherical harmonic covariance and magnitude function encodings for beamformer design

Microphone and speaker array designs have increasingly diverged from simple topologies due to diversity of physical host geometries and use cases. Effective beamformer design must now account for variation in the array’s acoustic radiation pattern, spatial distribution of target and noise sources, and intended beampattern directivity. Relevant tasks such as representing complex pressure fields, specifying spatial priors, and composing beampatterns can be efficiently synthesized using spherical harmonic (SH) basis functions. This paper extends the expansion of common stationary covariance functions onto the SHs and proposes models for encoding magnitude functions on a sphere. Conventional beamformer designs are reformulated in terms of magnitude density functions and beampatterns along SH bases. Applications to speaker far-field response fitting, cross-talk cancelation design, and microphone beampattern fitting are presented.

Covariance functions under B-spline polynomials to model Polled Nellore cattle growth

B-spline functions have been used in random regression models (RRM) to model animal weight from birth to adulthood because they are less vulnerable to common difficulties of other methods. However, its application to model growth traits of Polled Nellore cattle has been little studied. Therefore, this study aimed to evaluate polynomial functions of different orders and segment numbers to model effects associated with the Polled Nellore cattle growth curve. For this purpose, we used 15,148 weight records of 3,115 animals aged between 1 and 660 days and reared in northern Brazil and born between 1995 and 2010. Random effects were modeled using B-spline polynomials. As random effects, we considered the direct and maternal genetic additives, as well as direct and maternal permanent environments. As fixed effects were included contemporary group, cow age at calving (linear and quadratic) and fourth-order Legendre polynomials to represent average growth curve. The residue was modeled by considering seven age classes. The bestfitted model was the one that considered cubic B-spline functions with four knots for direct additive genetic effects and three knots for maternal genetic, animal permanent environment, and maternal permanent environment effects (C6555). Therefore, covariance functions under B-spline polynomials are efficient and can be used to model the growth curve of Polled Nellore cattle from birth to 660 days of age.

pyaneti II: A multi-dimensional Gaussian process approach to analysing spectroscopic time-series

The two most successful methods for exoplanet detection rely on the detection of planetary signals in photometric and radial velocity time-series. This depends on numerical techniques that exploit the synergy between data and theory to estimate planetary, orbital, and/or stellar parameters. In this work we present a new version of the exoplanet modelling code pyaneti. This new release has a special emphasis on the modelling of stellar signals in radial velocity time-series. The code has a built-in multi-dimensional Gaussian process approach to modelling radial velocity and activity indicator time-series with different underlying covariance functions. This new version of the code also allows multi-band and single transit modelling; it runs on Python 3, and features overall improvements in performance. We describe the new implementation and provide tests to validate the new routines that have direct application to exoplanet detection and characterisation. We have made the code public and freely available at https://github.com/oscaribv/pyaneti. We also present the codes citlalicue and citlalatonac that allow one to create synthetic photometric and spectroscopic time-series, respectively, with planetary and stellar-like signals.

The CUSUM statistic of change point under NA sequences

In this paper, we investigate the CUSUM statistic of change point under the negatively associated (NA) sequences. By establishing the consistency estimators for mean and covariance functions respectively, the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge, which extends the results obtained under the case of an independent normal sample and the moving average processes. Finally, the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis.