A Globally Convergent Method for Finding the Number of Intrinsic Modes on Narrow-Banded Signals

TL;DR

This article proposed a novel variational method based on real field, focusing on evaluating the supporting baseline of a signal's spectrum, to further retrieve the intrinsic mode functions, and establishes the rigorous mathematical proof on the global convergence to the algorithm based on dual ascent in function space.

Abstract

Automatically determining the number of intrinsic mode functions (IMFs) and their center frequencies in Variational Mode Decomposition (VMD) still remains an open mathematical challenge, which often relies on heuristic prior settings, trial-and-error strategies or complex-field based methods that lack of theoretical guarantees on the convergence. In this article, we proposed a novel variational method based on real field, focusing on evaluating the supporting baseline of a signal's spectrum, to further retrieve the intrinsic mode functions. Our method treats automatic extraction of modes as a constrained optimization problem that adversarially maximizes the baseline integral while penalizing its curvature, and transform the problem into iteratively solving a fourth-order boundary value problem via Lagrangian duality. Furthermore, we establish the rigorous mathematical proof on the global convergence to our algorithm based on dual ascent in function space. Comprehensive numerical experiments on artificial and real-world signals including electrocardiogram (ECG) data show that our method can provide accurate estimates of IMFs and center frequencies, and comparison with methods like Successive VMD also shows our advantages in avoiding redundant modes while ensuring the recovery of necessary components, indicating that we have provided a robust, theoretically grounded initialization routine for VMD.

Figures (37)

• Figure 1: Illustration to the concept of supporting baseline: In (a)-(c) the solid line is $y=10\cos\pi x -10\cos 5\pi x + x^2$. The dashed line in (a) is a proper supporting baseline since it actually fits the lower bound of the solid line as well as ignoring high frequent details of the solid line. The dashed line in (b) has a gap to the bottom of the solid line so it is not suitable to be a good supporting baseline. In contrast, the dashed line in (c) fits too tightly to the lower bound of the solid line, capturing too much detail of the solid line so that it can be reckoned as the lower envelope rather than a proper supporting baseline.
• Figure 1: Iterations for finding the supporting baseline without ((a) and (b)) /with ((c) and (d)) smooth extrapolation to the spectrum when iterated to 500 ((a) and (c)) and 1500 ((b) and (d)) steps for the same spectrum. The solid line is the original spectrum and the dotted line is the evaluated supporting baseline. With smooth extrapolation to the original spectrum, the speed of convergence has significant improvement.
• Figure 1: The initial (a), final convergent state (b) and convergence curve in logarithmic coordinate system (c) with respect to the error for finding the optimum supporting baseline of FFT with respect to time series $y(t)=100\sin 20\pi t$. The iteration lasts for 5161 steps and costs 1.88 seconds.
• Figure 2: Illustration of the non-consistence of the error between the converged optimal supporting baseline and the original function. Fig. \ref{['fig:3a']} shows the converged $g(x)$ and Fig. \ref{['fig:3b']} shows $f(x)-g(x)$. Note that since $g(x)$ does have gap to the lower bound of $f(x)$ so that the bottom of $f(x)-g(x)$ is almost not on a horizontal line everywhere, which indicates there are some non-zero gaps. These non-zero gaps will have a significant impact on the stability of the subsequent modal number solution. \ref{['fig:3c']} shows the estimate kernel density pertaining the gap at the bottom. \ref{['fig:3d']} shows the estimated $f(x)-g(x)$ after removal of the gap at the bottom.
• Figure 2: The initial (a), final convergent state (b) and convergence curve in logarithmic coordinate system (c) for finding the supporting baseline of FFT with respect to FFT of $y(t)=10\cos 10\pi t+20\sin 20\pi t$. The iteration lasts for 5258 steps and costs 1.89 seconds.

Theorems & Definitions (11)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Proof 1
• Corollary 2.4
• Lemma 2.5
• Proof 2
• Theorem 2.6
• Proof 3
• Theorem 2.7