scholarly journals Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution

2017 ◽  
Vol 2017 ◽  
pp. 1-24 ◽  
Author(s):  
Filippo Palombi ◽  
Simona Toti ◽  
Romina Filippini
Keyword(s):  
Covariance Matrix ◽  
Numerical Technique ◽  
Euclidean Ball ◽  
Chi Square ◽  
Iterative Schemes ◽  
Numerical Reconstruction ◽  
Second Moments ◽  
The Matrix ◽  
Truncated Normal ◽  
Full Covariance Matrix

We relate the matrixSBof the second moments of a spherically truncated normal multivariate to its full covariance matrixΣand present an algorithm to invert the relation and reconstructΣfromSB. While the eigenvectors ofΣare left invariant by the truncation, its eigenvalues are nonuniformly damped. We show that the eigenvalues ofΣcan be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over an Euclidean ball, for which we extend a numerical technique, originally proposed by Ruben in 1962, based on a series expansion in chi-square distributions. In order to study the feasibility of our approach, we examine the convergence rate of some iterative schemes on suitably chosen ensembles of Wishart matrices. We finally discuss the practical difficulties arising in sample space and outline a regularization of the problem based on perturbation theory.

Impact of AMSU-A Radiances in a Column Ensemble Kalman Filter

2018 ◽  
Vol 146 (12) ◽  
pp. 3949-3976 ◽  
Author(s):  
Herschel L. Mitchell ◽  
P. L. Houtekamer ◽  
Sylvain Heilliette
Keyword(s):  
Covariance Matrix ◽  
Background Error ◽  
Error Covariance Matrix ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Microwave Sounding Unit ◽  
The Matrix ◽  
Microwave Sounding ◽  
Background Temperature ◽  
The Difference

Abstract A column EnKF, based on the Canadian global EnKF and using the RTTOV radiative transfer (RT) model, is employed to investigate issues relating to the EnKF assimilation of Advanced Microwave Sounding Unit-A (AMSU-A) radiance measurements. Experiments are performed with large and small ensembles, with and without localization. Three different descriptions of background temperature error are considered: 1) using analytical vertical modes and hypothetical spectra, 2) using the vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling, and 3) using the vertical modes and spectrum of the static background error covariance matrix employed to initiate a global data assimilation cycle. It is found that the EnKF performs well in some of the experiments with background error description 1, and yields modest error reductions with background error description 3. However, the EnKF is virtually unable to reduce the background error (even when using a large ensemble) with background error description 2. To analyze these results, the different background error descriptions are viewed through the prism of the RT model by comparing the trace of the matrix , where is the RT model and is the background error covariance matrix. Indeed, this comparison is found to explain the difference in the results obtained, which relates to the degree to which deep modes are, or are not, present in the different background error covariances. The results suggest that, after 2 weeks of cycling, the global EnKF has virtually eliminated all background error structures that can be “seen” by the AMSU-A radiances.

ГАУССОВ ПУЧОК КАК МОДЕЛЬ ИСТОЧНИКА ИОНОВ

2021 ◽  
Author(s):  
Н.В. Коненков ◽  
А.И. Иванов ◽  
В.А. Степанов
Keyword(s):  
Gaussian Beam ◽  
Probability Density Functions ◽  
Ion Sources ◽  
Density Functions ◽  
Ion Trajectory ◽  
Second Moments ◽  
Matching Efficiency ◽  
The Matrix ◽  
Trajectory Method ◽  
Ellipse Area

Для расчета статистического аксептанса КФМ использовался траекторный метод. Функция плотности вероятности захваченных фазовых точек предназначена для определения матриц вторых моментов. Элементы этих матриц описывают эллипсы захвата на X и Y фазовых плоскостях. Мерой согласования Гауссова пучка и аксептанса квадруполя служат площади эллипсов. При постоянных параметрах Гауссова пучка ионов эффективность согласования слабо уменьшается с увеличением разрешающей способности. Полученные данные будут полезны при проектировании современных источников ионов. To calculate the statistical QMF acceptance, an ion trajectory method has been used. The probability density functions of accepted points allow fitting the matrix of the second moments. The elements of these matrices describe the acceptance ellipses on phase X and Y planes. The measure of the coupling Gaussian beam and quadrupole acceptance is ellipse area. Colored distributions of the input and output coordinates and velocities are presented, in which the initial phases are marked with different colors. It was found that with increasing resolution, the statistical acceptance ellipses are nested into each other. At constant parameters of the input Gaussian beam, the matching efficiency weakly decreases with resolution. The obtained data will be useful for creation a new modern ion sources.

scholarly journals Estimation accuracy of covariance matrices when their eigenvalues are almost duplicated

2018 ◽  
Vol 7 ◽  
Author(s):  
Kantaro Shimomura ◽  
Kazushi Ikeda
Keyword(s):  
Covariance Matrix ◽  
Estimation Method ◽  
Estimation Accuracy ◽  
Essential Information ◽  
The Matrix ◽  
The Difference ◽  
Processing Techniques ◽  
Monotonic Property ◽  
Signal Processing Techniques ◽  
Special Case

The covariance matrix of signals is one of the most essential information in multivariate analysis and other signal processing techniques. The estimation accuracy of a covariance matrix is degraded when some eigenvalues of the matrix are almost duplicated. Although the degradation is theoretically analyzed in the asymptotic case of infinite variables and observations, the degradation in finite cases are still open. This paper tackles the problem using the Bayesian approach, where the learning coefficient represents the generalization error. The learning coefficient is derived in a special case, i.e., the covariance matrix is spiked (all eigenvalues take the same value except one) and a shrinkage estimation method is employed. Our theoretical analysis shows a non-monotonic property that the learning coefficient increases as the difference of eigenvalues increases until a critical point and then decreases from the point and converged to the distinct case. The result is validated by numerical experiments.

scholarly journals Multivariate Ensemble Sensitivity Analysis for Super Typhoon Haiyan (2013)

2019 ◽  
Vol 147 (9) ◽  
pp. 3467-3480 ◽  
Author(s):  
Sijing Ren ◽  
Lili Lei ◽  
Zhe-Min Tan ◽  
Yi Zhang
Keyword(s):  
Sensitivity Analysis ◽  
High Resolution ◽  
Covariance Matrix ◽  
Multivariate Regression ◽  
Ensemble Forecast ◽  
Initial Condition ◽  
Maximum Wind ◽  
Diagonal Approximation ◽  
Low Levels ◽  
Full Covariance Matrix

Abstract Ensemble sensitivity is often a diagonal approximation to the multivariate regression, leading to a simple univariate regression. Comparatively, the multivariate ensemble sensitivity retains the full covariance matrix when computing the multivariate regression. The performances of both univariate and multivariate ensemble sensitivities in multiscale flows have not been thoroughly examined, and the demonstration of the latter in realistic applications has been sparse. A high-resolution ensemble forecast of Typhoon Haiyan (2013) is used to examine the performances of the two ensemble sensitivities. Compared to the multivariate sensitivity, the univariate sensitivity overestimates the forecast metric, especially at higher levels. To increase the predicted Haiyan’s intensity, multivariate ensemble sensitivity gives initial perturbations characterized by a warming area around the center of the storm, an increased moisture area around the eyewall, a stronger primary circulation around the radius of maximum wind, and stronger inflow at low levels and stronger outflow at high levels. Perturbed initial condition experiments verify that the predicted response from the multivariate sensitivity is more accurate than that from the univariate sensitivity. Therefore, the ability of multivariate sensitivity to provide more accurate predicted responses than the univariate sensitivity has been demonstrated in a realistic multiscale flow application.

scholarly journals CONVEX REPLICA SYMMETRY BREAKING FROM POSITIVITY AND THERMODYNAMIC LIMIT

2004 ◽  
Vol 18 (04n05) ◽  
pp. 585-591 ◽  
Author(s):  
PIERLUIGI CONTUCCI ◽  
SANDRO GRAFFI
Keyword(s):  
Symmetry Breaking ◽  
Covariance Matrix ◽  
Thermodynamic Limit ◽  
Matrix Elements ◽  
Replica Symmetry ◽  
Replica Symmetry Breaking ◽  
Random Energy Model ◽  
Random Energy ◽  
The Matrix ◽  
Convexity Conditions

Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity conditions on the matrix elements.

scholarly journals MVDR Algorithm Based on Estimated Diagonal Loading for Beamforming

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Yuteng Xiao ◽  
Jihang Yin ◽  
Honggang Qi ◽  
Hongsheng Yin ◽  
Gang Hua
Keyword(s):  
Signal Processing ◽  
Covariance Matrix ◽  
Optimization Model ◽  
Matrix Theory ◽  
Minimum Variance ◽  
Signal Model ◽  
Diagonal Loading ◽  
Minimum Variance Distortionless Response ◽  
The Matrix ◽  
Beamforming Algorithm

Beamforming algorithm is widely used in many signal processing fields. At present, the typical beamforming algorithm is MVDR (Minimum Variance Distortionless Response). However, the performance of MVDR algorithm relies on the accurate covariance matrix. The MVDR algorithm declines dramatically with the inaccurate covariance matrix. To solve the problem, studying the beamforming array signal model and beamforming MVDR algorithm, we improve MVDR algorithm based on estimated diagonal loading for beamforming. MVDR optimization model based on diagonal loading compensation is established and the interval of the diagonal loading compensation value is deduced on the basis of the matrix theory. The optimal diagonal loading value in the interval is also determined through the experimental method. The experimental results show that the algorithm compared with existing algorithms is practical and effective.

Impurity precipitation on dislocations— a theory of strain ageing

1962 ◽  
Vol 266 (1325) ◽  
pp. 209-221 ◽  
Keyword(s):  
Numerical Technique ◽  
Dislocation Core ◽  
Elastic Interaction ◽  
Low Carbon ◽  
Transfer Velocity ◽  
Strain Ageing ◽  
Carbon And Nitrogen ◽  
The Matrix ◽  
Continuous Rod ◽  
Kinetics Of

A direct numerical technique has been used to investigate the kinetics of impurity precipitation on dislocations making full allowance for diffusion and with a strong elastic interaction between the solute atoms and the dislocations. A distinction is made between the growth of discrete precipitate particles and continuous rod-like precipitates on the dislocations. The kinetics for the former mode of precipitation are obtained for various values of a constant transfer velocity across the precipitate matrix interface; it is found that the fraction of solute remaining in free solution decreases exponentially with time. The continuous rod-like mode of precipitation is a consequence of a relatively high binding energy in the dislocation core, and it is shown that such a situation should lead to a transient variation of the transfer velocity across the matrix-core interface. Under these conditions, the kinetics of precipitation closely resemble the experimentally determined strain ageing kinetics in some low -carbon and nitrogen steels.

scholarly journals On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices

2018 ◽  
Vol 33 ◽  
pp. 24-40 ◽  
Author(s):  
Jolanta Pielaszkiewicz ◽  
Dietrich Von Rosen ◽  
Martin Singull
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Asymptotic Distribution ◽  
Joint Distribution ◽  
Recursive Formula ◽  
Multivariate Normal ◽  
Wishart Matrices ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

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