Estimations of singular functions of kernel cross-covariance operators

2021 ◽  
Vol 266 ◽  
pp. 105576
Author(s):  
Heng Chen ◽  
Di-Rong Chen ◽  
Yao Zhao
Keyword(s):  
Singular Functions ◽  
Cross Covariance ◽  
Covariance Operators

On Cross-Covariance Operators

1979 ◽  
Vol 37 (2) ◽  
pp. 325-329 ◽  
Author(s):  
A. F. Gualtierotti
Keyword(s):  
Cross Covariance ◽  
Covariance Operators

scholarly journals Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

2021 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Time Series ◽  
Hilbert Spaces ◽  
Upper Bounds ◽  
Estimation Errors ◽  
Cartesian Products ◽  
Major Task ◽  
Functional Time Series ◽  
Cross Covariance ◽  
Covariance Operators ◽  
Practical Tool

A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.

scholarly journals Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

2021 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Time Series ◽  
Hilbert Spaces ◽  
Upper Bounds ◽  
Estimation Errors ◽  
Cartesian Products ◽  
Major Task ◽  
Functional Time Series ◽  
Cross Covariance ◽  
Covariance Operators ◽  
Practical Tool

A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.

scholarly journals Replicating the complexity of natural surfaces: technique validation and applications for biomimetics, ecology and evolution

2018 ◽  
Vol 377 (2138) ◽  
pp. 20180265 ◽  
Author(s):  
Charchit Kumar ◽  
Alejandro Palacios ◽  
Venkata A. Surapaneni ◽  
Georg Bold ◽  
Marc Thielen ◽  
...  
Keyword(s):  
Biological Sciences ◽  
Industrial Applications ◽  
Model Parameters ◽  
Future Directions ◽  
Small Surface ◽  
Surface Information ◽  
Cross Covariance ◽  
Unique Method ◽  
Structural Aspects

The surfaces of animals, plants and abiotic structures are not only important for organismal survival, but they have also inspired countless biomimetic and industrial applications. Additionally, the surfaces of animals and plants exhibit an unprecedented level of diversity, and animals often move on the surface of plants. Replicating these surfaces offers a number of advantages, such as preserving a surface that is likely to degrade over time, controlling for non-structural aspects of surfaces, such as compliance and chemistry, and being able to produce large areas of a small surface. In this paper, we compare three replication techniques among a number of species of plants, a technical surface and a rock. We then use two model parameters (cross-covariance function ratio and relative topography difference) to develop a unique method for quantitatively evaluating the quality of the replication. Finally, we outline future directions that can employ highly accurate surface replications, including ecological and evolutionary studies, biomechanical experiments, industrial applications and improving haptic properties of bioinspired surfaces. The recent advances associated with surface replication and imaging technology have formed a foundation on which to incorporate surface information into biological sciences and to improve industrial and biomimetic applications. This article is part of the theme issue ‘Bioinspired materials and surfaces for green science and technology’.

An Efficient Dual-Resolution Approach for Ensemble Data Assimilation and Tests with Simulated Doppler Radar Data

2008 ◽  
Vol 136 (3) ◽  
pp. 945-963 ◽  
Author(s):  
Jidong Gao ◽  
Ming Xue
Keyword(s):  
High Resolution ◽  
Data Assimilation ◽  
Radial Velocity ◽  
Computational Cost ◽  
Radar Data ◽  
Horizontal Resolution ◽  
Velocity Data ◽  
Background Error ◽  
Cross Covariance ◽  
Radial Velocity Data

Abstract A new efficient dual-resolution (DR) data assimilation algorithm is developed based on the ensemble Kalman filter (EnKF) method and tested using simulated radar radial velocity data for a supercell storm. Radar observations are assimilated on both high-resolution and lower-resolution grids using the EnKF algorithm with flow-dependent background error covariances estimated from the lower-resolution ensemble. It is shown that the flow-dependent and dynamically evolved background error covariances thus estimated are effective in producing quality analyses on the high-resolution grid. The DR method has the advantage of being able to significantly reduce the computational cost of the EnKF analysis. In the system, the lower-resolution ensemble provides the flow-dependent background error covariance, while the single-high-resolution forecast and analysis provides the benefit of higher resolution, which is important for resolving the internal structures of thunderstorms. The relative smoothness of the covariance obtained from the lower 4-km-resolution ensemble does not appear to significantly degrade the quality of analysis. This is because the cross covariance among different variables is of first-order importance for “retrieving” unobserved variables from the radar radial velocity data. For the DR analysis, an ensemble size of 40 appears to be a reasonable choice with the use of a 4-km horizontal resolution in the ensemble and a 1-km resolution in the high-resolution analysis. Several sensitivity tests show that the DR EnKF system is quite robust to different observation errors. A 4-km thinned data resolution is a compromise that is acceptable under the constraint of real-time applications. A data density of 8 km leads to a significant degradation in the analysis.

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