scholarly journals Correction to: Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices

2019 ◽  
Vol 51 (6) ◽  
pp. 843-843
Author(s):  
Benjamin J. K. Davis ◽  
Frank C. Curriero
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mingwei Zhang ◽  
Yao Hou ◽  
Rongnian Tang ◽  
Youjun Li

In motor imagery brain computer interface system, the spatial covariance matrices of EEG signals which carried important discriminative information have been well used to improve the decoding performance of motor imagery. However, the covariance matrices often suffer from the problem of high dimensionality, which leads to a high computational cost and overfitting. These problems directly limit the application ability and work efficiency of the BCI system. To improve these problems and enhance the performance of the BCI system, in this study, we propose a novel semisupervised locality-preserving graph embedding model to learn a low-dimensional embedding. This approach enables a low-dimensional embedding to capture more discriminant information for classification by efficiently incorporating information from testing and training data into a Riemannian graph. Furthermore, we obtain an efficient classification algorithm using an extreme learning machine (ELM) classifier developed on the tangent space of a learned embedding. Experimental results show that our proposed approach achieves higher classification performance than benchmark methods on various datasets, including the BCI Competition IIa dataset and in-house BCI datasets.


2021 ◽  
Author(s):  
Mirko Mälicke

Abstract. Geostatistical methods are widely used in almost all geoscientific disciplines, i.e. for interpolation, re-scaling, data assimilation or modelling. At its core geostatistics aims to detect, quantify, describe, analyze and model spatial covariance of observations. The variogram, a tool to describe this spatial covariance in a formalized way, is at the heart of every such method. Unfortunately, many applications of geostatistics rather focus on the interpolation method or the result, than the quality of the estimated variogram. Not least because estimating a variogram is commonly left as a task for computers and some software implementations do not even show a variogram to the user. This is a miss, because the quality of the variogram largely determines, whether the application of geostatistics makes sense at all. Furthermore, the Python programming language was missing a mature, well-established and tested package for variogram estimation a couple of years ago. Here I present SciKit-GStat, an open source Python package for variogram estimation, that fits well into established frameworks for scientific computing and puts the focus on the variogram before more sophisticated methods are about to be applied. SciKit-GStat is written in a mutable, object-oriented way that mimics the typical geostatistical analysis workflow. Its main strength is the ease of usage and interactivity and it is therefore usable with only a little or even no knowledge in Python. During the last few years, other libraries covering geostatistics for Python developed along with SciKit-GStat. Today, the most important ones can be interfaced by SciKit-GStat. Additionally, established data structures for scientific computing are reused internally, to keep the user from learning complex data models, just for using SciKit-GStat. Common data structures along with powerful interfaces enable the user to use SciKit-GStat along with other packages in established workflows, rather than forcing the user to stick to the authors programming paradigms. SciKit-GStat ships with a large number of predefined procedures, algorithms and models, such as variogram estimators, theoretical spatial models or binning algorithms. Common approaches to estimate variograms are covered and can be used out of the box. At the same time, the base class is very flexible and can be adjusted to less common problems, as well. Last but not least, it was made sure, that a user is aided at implementing new procedures, or even extending the core functionality as much as possible, to extend SciKit-GStat to uncovered use-cases. With broad documentation, user guide, tutorials and good unit-test coverage, SciKit-GStat enables the user to focus on variogram estimation, rather than implementation details.


2021 ◽  
Author(s):  
Eva Boergens ◽  
Andreas Kvas ◽  
Henryk Dobslaw ◽  
Annette Eicker ◽  
Christoph Dahle ◽  
...  

<p>Knowledge of the variances and covariances of gridded terrestrial water storage anomalies (TWS) as observed with GRACE and GRACE-FO is crucial for many applications thereof. For example, data assimilation into different models, trend estimations, or combinations with other data set require reliable estimations of the variances and covariances. Today, the Level-2 Stokes coefficients are provided with formal variance-covariance matrices which can yield variance-covariance matrices of the gridded data after a labourious variance propagation through all post-processing steps, including filtering and spherical harmonic synthesis. Unfortunately, this is beyond the capabilities of many, if not most, users.</p><p><br>This is why, we developed a spatial covariance model for gridded TWS data. The covariance model results in non-homogeneous, non-stationary, and anisotropic covariances. This model also accommodates a wave-like behaviour in latitudinal-directed correlations caused by residual striping errors. The model is applied to both VDK3 filtered GFZ RL06 and ITSG-Grace2018 TWS data. </p><p><br>With thus derived covariances it is possible to estimate the uncertainties of mean TWS time series for any arbitrary region such as river basins. On the other hand, such time series uncertainties can also be derived from the afore mentioned formal covariance matrices. Here, only the formal covariance matrices of ITSG-Grace2018 are used which are also filtered with the VDK3 filter. All together, we are able to compare globally the time series uncertainties of both the modelled and formal approach. Further, the modelled uncertainties are compared to empirical standard deviations in arid regions in the Arabian, Sahara, and Gobi desert where residual hydrological signal can be neglected. Both in the temporal and spatial domain they show a very satisfying agreement proving the usefulness of the covariance model for the users. </p>


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