Singular spectral analysis‐based denoising without computing singular values via augmented Lagrange multiplier algorithm

Abstract

This study proposes an augmented Lagrange multiplier-based method to perform the singular spectral analysisbased denoising without computing the singular values. In particular, the one-dimensional (1D) signal is first mapped to a trajectory matrix using the window length L. Second, the trajectory matrix is represented as the sum of the signal dominant matrix and the noise-dominant matrix. The determination of these two matrices is formulated as an optimisation problem with the objective function being the sum of the rank of the signal dominant matrix and the l 0 norm of the noise-dominant matrix. This study employs the Schatten q-norm operator with q = 1/2 and the double nuclear-norm penalty for approximating the rank operator as well as the minimum concave penalty (MCP)-norm operator for approximating the l 0-norm operator. Third, some auxiliary variables are introduced and the augmented Lagrange multiplier algorithm is applied to find the optimal solution. Finally, the 1D denoised signal is obtained by applying the diagonal averaging method to the obtained signal dominant matrix. Computer numerical simulation results show that the authors' proposed method outperforms the existing methods.

Suppressing the Spikes in Electroencephalogram via an Iterative Joint Singular Spectrum Analysis and Low-Rank Decomposition Approach

“…Therefore, the singular spectrum analysis components corresponding to the small eigenvalues are summed up together to obtain an approximated despiked electroencephalogram. This is unlike the conventional singular spectrum analysis based denoising methods [15][16][17] x m . Here, it is required to determine m * via a thresholding method.…”

“…To suppress the spikes in the filtered electroencephalogram, the filtered electroencephalogram is required to decompose into various components and appropriate processing is applied to these components. Since the magnitudes of the spikes are large [4][5][6] and the singular spectrum analysis components are expressed as the magnitudes of the eigenvalues of the trajectory matrix [15][16][17], the singular spectrum analysis is an appropriate tool to decompose the filtered electroencephalogram into various components for suppressing the spikes.…”

“…2.1. Review on the Singular Spectrum Analysis [15][16][17] The singular spectrum analysis is to represent a signal as the sum of the singular spectrum analysis components. The procedures for performing the singular spectrum analysis are as follows.…”

“…On the other hand, the singular spectrum analysis components corresponding to the large eigenvalues are used for the reconstruction of the signal. Hence, they are summed up together to obtain the denoised signal [15][16][17]. However, as the spikes are with the large magnitudes and the singular spectrum analysis components corresponding to the spikes are usually corresponding to the large eigenvalues, the spikes are still corrupted in the denoised signal.…”

“…Since different approaches have their own advantages and disadvantages, this paper proposes an iterative joint singular spectrum analysis and low-rank decomposition approach for suppressing the spikes in an electroencephalogram. Unlike the conventional singular spectrum analysis based denoising methods [15][16][17], this paper sums up the singular spectrum analysis components corresponding to the small eigenvalues. Also, this paper does not completely discard the sparse component obtained by the low-rank decomposition.…”