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Regularization for high-dimensional covariance matrix

100%
Cui X., Li C., Zhao J., Zeng L., Zhang D., Pan J.
Special Matrices
|
2016
|
tom 4
|
nr 1
189-201
EN
In many applications, high-dimensional problem may occur often for various reasons, for example, when the number of variables under consideration is much bigger than the sample size, i.e., p >> n. For highdimensional data, the underlying structures of certain covariance matrix estimates are usually blurred due to substantial random noises, which is an obstacle to draw statistical inferences. In this paper, we propose a method to identify the underlying covariance structure by regularizing a given/estimated covariance matrix so that the noises can be filtered. By choosing an optimal structure from a class of candidate structures for the covariance matrix, the regularization is made in terms of minimizing Frobenius-norm discrepancy. The candidate class considered here includes the structures of order-1 moving average, compound symmetry, order-1 autoregressive and order-1 autoregressive moving average. Very intensive simulation studies are conducted to assess the performance of the proposed regularization method for very high-dimensional covariance problem. The simulation studies also show that the sample covariance matrix, although performs very badly in covariance estimation for high-dimensional data, can be used to correctly identify the underlying structure of the covariance matrix. The approach is also applied to real data analysis, which shows that the proposed regularization method works well in practice.
2
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Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions

100%
Gáspár C.
Open Mathematics
|
2013
|
tom 11
|
nr 8
1429-1440
EN
The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples.
3
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A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

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Besma K., Nadjib B., Faouzia R.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1649-1666
EN
In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.
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