Self-tuning variational mode decomposition

Highlights

• A self-tuning variational mode decomposition (SVMD) is proposed.

• SVMD can adaptively update the parameters $K$ and $\alpha$.

• Several properties of SVMD are studied and compared with those of other methods.

• Comparative studies of real-world applications show the advantages of SVMD.

Abstract

Variational mode decomposition (VMD) has attracted a lot of attention recently owing to its robustness to sampling frequency and its high-frequency resolution. However, its performance highly depends on two key preset parameters (the mode number $K$ and the penalty parameter $\alpha$), both of which tightly limit its adaptability and applications. In this study, a self-tuning VMD (SVMD) is proposed to tackle this problem. Within the proposed method, $K$ and $\alpha$ update themselves respectively and adaptively via the energy ratio and orthogonality between modes in the frequency domain. The proposed SVMD is similar to a matching pursuit method and it shows a VMD-like equivalent filter bank structure but with much less mode-mixing probability. We show that SVMD is more robust to both changes in $\alpha$ and noise level than the original VMD; also, it has better convergence and reduces mode-mixing and end-effect. The experiments on SVMD indicate that SVMD outmatches several classic signal decomposition algorithms. In the end, real-world applications in three fields, namely, length of day variation analysis in geophysics, climate cycle study in meteorology, and oscillation detection in process control, are provided to demonstrate the effectiveness and advantages of the proposed SVMD.

Introduction

In various fields, signal and data processing, which is a crucial and necessary task in both scientific research and practical applications, has spurred significant interest. Because the information one cares about is often contaminated by noise, nonlinearity, or nonstationary, etc., the primary goal of signal processing is to reveal underlying information and structures, such as the frequency component in the signal [1].

Fourier transform, short-time Fourier transform, Wigner-Ville distribution, and Wavelet transform are based on basis function expansion. These methods have the advantages of simplicity, uniqueness and symmetry, but they lack flexibility and they are not amenable enough for complex signal analysis. Most of the complexity of natural signals can be attributed to the activities of different time scales, so these signals can be considered as a combination of several simple sub-signals. During the past decades, researchers have proposed several adaptive signal decomposition techniques to decompose signals into their components. These methods are truly data-driven and posteriori. In 1998, Huang et al. [2] first proposed the empirical mode decomposition (EMD), which was the most well-known decomposition method applied to various fields. However, this algorithm lacks a mathematical foundation and it is sensitive to noise and sampling [3]. Later, inspired by EMD, Smith [4] developed local mean decomposition (LMD) by separating amplitude-modulation (AM) and frequency-modulation (FM), but this method also suffers from mode-mixing and end-effect problems. To remedy these issues, Frei and Osorio [5] put forward intrinsic time-scale decomposition (ITD). Although ITD has low computational complexity, its time resolution is poor. Unlike the above EMD-based decomposition framework, Gilles [6] devised the empirical wavelet transform (EWT) guided by the wavelet theory. Although EWT alleviates the mode-mixing problem with less computation than EMD, its frequency band division and filter selection are still unsolved.

More recently, Dragomiretskiy and Zosso [7] proposed the variational mode decomposition (VMD) algorithm that is based on convex optimization theory. It has an attractive performance in several aspects [8], such as robustness and anti-mode-mixing. However, its effectiveness highly depends on the mode number $K$ and the penalty coefficient $\alpha$. Some works try to iteratively optimize $K$ by the exhaustive method. For example, Li et al. [9] used the criterion of approximate complete reconstruction to determine an appropriate mode number. Lian et al. [10] selected the $K$ value based on the characteristic of the intrinsic mode obtained from VMD. Cai et al. [11] used the FFT spectrum to choose the mode number parameter. Although the methods mentioned above are simple, they are risky because the performance of VMD is regulated by both $K$ and $\alpha$, not just one. The other approaches optimize both $K$ and $\alpha$ simultaneously. Most of them define specific fitness functions for different applications and then use some intelligent algorithms to search parameters, such as AFSA-based (artificial fish swarm algorithm) [12], GOA-based (grasshopper optimization algorithm) [13], and MOPSO-based (multi-objective particle swarm optimization) [14], etc. Although these intelligent search methods consider both parameters, they do not address the fundamental problem of VMD and their effectiveness should depend on the selection of optimization algorithms and fitness functions.

In this paper, a self-tuning version of the original VMD (SVMD) is developed to tackle the tuning problems of $K$ and $\alpha$. Motived by the recent related methods [15,16], this paper proposes a novel objective function that is solved by updating the mode and the center frequency alternatively in the frequency domain. In the meantime, an adaptive updating law of the penalty coefficient is also merged during the iterations of SVMD. Also, the initialization method is developed to improve the convergence capability of SVMD. The development of the proposed SVMD scheme will be detailed in the section as follows. Section 2 provides an overview of the original VMD. The proposed SVMD is described elaborately in Section 3. The study on the properties of SVMD is given in Section 4, including the filter bank structure, tones versus sampling, tones separation, robustness, and convergence. Experiments are also used to show that SVMD outmatches several classic signal decomposition algorithms. In Section 5, real-world applications in three fields, including the length of day variation analysis in geophysics, the climate cycle study in meteorology, and oscillation detection in process control, are provided to validate the effectiveness and advantages of the proposed methodology. The concluding remarks are given in Section 6.