Validity of spatial covariance matrices over time and frequency

2003 ◽  
Author(s):  
I. Viering ◽  
H. Hofstetter ◽  
W. Utschick
Keyword(s):  
Covariance Matrices ◽  
Spatial Covariance ◽  
Over Time

Classification Based on Multivariate Repeated Measures with Time Effect on Mean Vector and an AR (1) Correlation Structure on the Repeated Measures

2005 ◽  
Vol 57 (1-2) ◽  
pp. 49-66 ◽  
Author(s):  
Anuradba Roy ◽  
Ravindra Khattree
Keyword(s):  
Maximum Likelihood ◽  
Repeated Measures ◽  
Likelihood Estimation ◽  
Covariance Matrices ◽  
Time Effect ◽  
Population Parameters ◽  
Time Effects ◽  
Mean Vectors ◽  
Mean Vector ◽  
Over Time

In repeated measures studies how observations change over time is often of prime interest. Modelling this time effect in the context of discrimination, is the objective of this article. We study the problem of classification with multiple q-variate observations with time effect on each individual. The covariance matrices as well as mean vectors are mordelled respectively to accommodate the correlation between the successive repeated measures and to describe the time effects. Computation schemes for maximum likelihood estimation of required population parameters are provided.

scholarly journals Embedding Tangent Space Extreme Learning Machine for EEG Decoding in Brain Computer Interface Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mingwei Zhang ◽  
Yao Hou ◽  
Rongnian Tang ◽  
Youjun Li
Keyword(s):  
Extreme Learning Machine ◽  
Motor Imagery ◽  
Tangent Space ◽  
Brain Computer Interface ◽  
Covariance Matrices ◽  
Computer Interface ◽  
Spatial Covariance ◽  
Learning Machine ◽  
Low Dimensional ◽  
Bci System

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.

An Advanced Formulation of Kalman Filter Time Series Reference Frame Realization for Geophysical Applications

2020 ◽  
Author(s):  
Xiaoping Wu ◽  
Bruce Haines ◽  
Michael Heflin ◽  
Felix Landerer
Keyword(s):  
Time Series ◽  
Kalman Filter ◽  
Reference Frame ◽  
State Vector ◽  
Terrestrial Reference Frame ◽  
Covariance Matrices ◽  
Time Variable ◽  
The State ◽  
Geocenter Motion ◽  
Over Time

<p>A Kalman filter and time series approach to the International Terrestrial Reference Frame (ITRF) realization (KALREF) has been developed and used in JPL. KALREF combines weekly or daily SLR, VLBI, GNSS and DORIS data and realizes a terrestrial reference frame in the form of time-variable geocentric station coordinate time series. The origin is defined at nearly instantaneous Center-of-Mass of the Earth system (CM) sensed by weekly SLR data and the scale is implicitly defined by the weighted averages of those of weekly SLR and daily VLBI data. The standard KALREF formulation describes the state vector in terms of time variable station coordinates and other constant parameters. Such a formulation is fine for station positions and their uncertainties or covariance matrices at individual epochs. However, coordinate errors are strongly correlated over time given KALREF’s unique nature of combining different technique data with various frame strengths through local tie measurements and co-motion constraints and its use of random walk processes. For long time series and large space geodetic networks in the ITRF, KALREF cannot keep track of such correlations over time. If they are ignored when forming geocentric displacements for geophysical inverse or network shift geocenter motion studies, the covariance matrices of coordinate differences cannot adequately represent those of displacements. Consequently, significant non-uniqueness and inaccuracies would occur in the results of studies using such matrices. To overcome this difficulty, an advanced KALREF formulation is implemented that features explicit displacement parameters in the state vector that would allow the Kalman filter and smoother to compute and return covariance matrices of displacements. The use of displacement covariance matrices reduces the impact of time correlated errors and completely solves the non-uniqueness problem. However, errors in the displacements are still correlated in time. Further calibrations are needed to accurately assess covariance matrices of derivative quantities such as averages, velocities and accelerations during various time periods. We will present KALREF results of the new formulation and their use along with newly reprocessed RL06 GRACE gravity data in a new unified inversion for geocenter motion.</p>

Uncertainties of TWS Time Series for Arbitrary Regions - Modelled vs. Formal Covariances

2021 ◽  
Author(s):  
Eva Boergens ◽  
Andreas Kvas ◽  
Henryk Dobslaw ◽  
Annette Eicker ◽  
Christoph Dahle ◽  
...  
Keyword(s):  
Time Series ◽  
Covariance Matrices ◽  
Gobi Desert ◽  
Covariance Model ◽  
Spatial Covariance ◽  
Formal Approach ◽  
Data Set ◽  
Terrestrial Water ◽  
Spherical Harmonic Synthesis ◽  
Processing Steps

<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>

Modeling dynamic processes with panel data: An application of continuous time models to prevention research

2020 ◽  
Vol 45 (1) ◽  
pp. 28-39
Author(s):  
Pascal R. Deboeck ◽  
David A. Cole ◽  
Kristopher J. Preacher ◽  
Rex Forehand ◽  
Bruce E. Compas
Keyword(s):  
Panel Data ◽  
Continuous Time ◽  
Covariance Matrices ◽  
Post Intervention ◽  
Repeated Observations ◽  
Effective Interventions ◽  
Continuous Time Processes ◽  
Underlying Processes ◽  
Over Time ◽  
Continuous Time Models

Many interventions are characterized by repeated observations on the same individuals (e.g., baseline, mid-intervention, two to three post-intervention observations), which offer the opportunity to consider differences in how individuals vary over time. Effective interventions may not be limited to changing means, but instead may also include changes to how variables affect each other over time. Continuous time models offer the opportunity to specify differing underlying processes for how individuals change from one time to the next, such as whether it is the level or change in a variable that is related to changes in an outcome of interest. After introducing continuous time models, we show how different processes can produce different expected covariance matrices. Thus, models representing differing underlying processes can be compared, even with a relatively small number of repeated observations. A substantive example comparing models that imply different underlying continuous time processes will be fit using panel data, with parameters reflecting differences in dynamics between control and intervention groups.

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