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Ridge detection for nonstationary multicomponent signals with time-varying wave-shape functions and its applications

Yan-Wei Su, Gi-Ren Liu, Yuan-Chung Sheu, Hau-Tieng Wu

TL;DR

A novel ridge detection algorithm for time-frequency analysis, particularly tailored for intricate nonstationary time series encompassing multiple non-sinusoidal oscillatory components, is introduced, shape-adaptive mode decomposition-based multiple harmonic ridge detection (SAMD-MHRD).

Abstract

We introduce a novel ridge detection algorithm for time-frequency (TF) analysis, particularly tailored for intricate nonstationary time series encompassing multiple non-sinusoidal oscillatory components. The algorithm is rooted in the distinctive geometric patterns that emerge in the TF domain due to such non-sinusoidal oscillations. We term this method \textit{shape-adaptive mode decomposition-based multiple harmonic ridge detection} (\textsf{SAMD-MHRD}). A swift implementation is available when supplementary information is at hand. We demonstrate the practical utility of \textsf{SAMD-MHRD} through its application to a real-world challenge. We employ it to devise a cutting-edge walking activity detection algorithm, leveraging accelerometer signals from an inertial measurement unit across diverse body locations of a moving subject.

Ridge detection for nonstationary multicomponent signals with time-varying wave-shape functions and its applications

TL;DR

A novel ridge detection algorithm for time-frequency analysis, particularly tailored for intricate nonstationary time series encompassing multiple non-sinusoidal oscillatory components, is introduced, shape-adaptive mode decomposition-based multiple harmonic ridge detection (SAMD-MHRD).

Abstract

We introduce a novel ridge detection algorithm for time-frequency (TF) analysis, particularly tailored for intricate nonstationary time series encompassing multiple non-sinusoidal oscillatory components. The algorithm is rooted in the distinctive geometric patterns that emerge in the TF domain due to such non-sinusoidal oscillations. We term this method \textit{shape-adaptive mode decomposition-based multiple harmonic ridge detection} (\textsf{SAMD-MHRD}). A swift implementation is available when supplementary information is at hand. We demonstrate the practical utility of \textsf{SAMD-MHRD} through its application to a real-world challenge. We employ it to devise a cutting-edge walking activity detection algorithm, leveraging accelerometer signals from an inertial measurement unit across diverse body locations of a moving subject.
Paper Structure (27 sections, 1 theorem, 42 equations, 12 figures, 2 algorithms)

This paper contains 27 sections, 1 theorem, 42 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical models and analysis tools
  3. Mathematical model
  4. Analysis by TF analysis tools---a quick summary
  5. Ridge detection as a key step---existing algorithms
  6. Challenges of existing approaches
  7. Proposed ridge detection algorithm
  8. Step 1: fit multiple ridges for harmonics (MHRD)
  9. Step 2: Peel off ridges for harmonics by SAMD
  10. Iteration
  11. Speed up SAMD-MHRD with auxiliary information
  12. Parameters selection
  13. Numerical simulation
  14. Single IMT function ($L=1$)
  15. Multiple IMT functions ($L>1$)
  16. ...and 12 more sections

Key Result

Proposition 3.1

Fix $K\geq 2$. The curve $\mathbf{h}_k:[N]\rightarrow[M]$ satisfying where $1\leq k\leq K-1$, is one of the rows of $\mathbf{c}^*$ in (multiCurveExt:with Fundamental).

Figures (12)

  • Figure 1: The overall flowchart of analyzing nonstationary oscillatory time series and the challenge of RD. (a) A portion of the simulated noisy signal $Y(t)$ with the signal-to-noise level of 5 dB is shown as the gray curve, with the clean signal superimposed as the red curve. The clean signal has one intrinsic mode type (IMT) function, whose fundamental component is weak. See \ref{['Model:equation2']} for the precise definition. (b) The spectrogram of $Y(t)$. (c) The second order SST. (d) The extracted lowest-frequency ridge from the TFR shown in (c) by the existing curve extraction algorithm, Single-RD\ref{['singleCurveExt:theory']}, is superimposed as the red curve. The extracted ridge is biased by the ridges of the harmonics after the 19th second. The true instantaneous frequency (IF) is superimposed as the blue-dashed line. (e) Same as (d), but by another existing RD algorithm, RRP-RD. (f) Same as (d), but by our proposed curve extraction algorithm SAMD-MHRD. (g) The decomposed harmonics of the IMT function are shown in red, and the true harmonics are shown in gray. Note that from the 29th to the 31st second, the fundamental component is less accurately recovered due to the low signal-to-noise ratio. It is the same for the fourth harmonic. (h) The summation of the decomposed harmonics is shown in red, and the true IMT function is shown in black.
  • Figure 2: Top row: an accelerometer signal. Bottom left: the spectrogram of the accelerometer signal. Bottom right: the SST of the accelerometer signal. The fundamental component is indicated by the red arrow, and the harmonics are indicated by the blue arrows.
  • Figure 3: Illustration of Single-RD, FM-RD and RRP-RD on a two-oscillatory components signal. (a)-Left: Spectrogram. We can see two curves that are associated with the fundamental components of the two IMT functions. (a)-Middle: The second-order SST. (a)-Right: The detection result using SAMD-MHRD (resp. FM-RD and RRP-RD) is superimposed on the 2nd-order SST as the red (resp. purple and blue) curves. (b) A portion of the first and second IMT functions is shown for visual inspection.
  • Figure 4: Log-scale distributions of $\delta^{(\square)}$ for various RD algorithms. Top to bottom rows correspond to $D_1=0.1$, $D_1=0.2$, and $D_1=0.5$, respectively. From left to right columns: noise-free, SNR = 5 dB, and SNR = 0 dB.
  • Figure 5: (a) A signal fulfilling the ANHM with $L=2$ contaminated by noise $\Phi$ with an SNR of 5dB is shown as the grey curve. The clean signal is superimposed as the red curve. (b)-Left: the 2nd-order SST of the noisy signal. (b)-Middle: the true IFs of the first three harmonics of the first IMT function are superimposed as blue-dashed curves. (b)-Right: the 2nd-order SST of the extracted first IMT function by SAMD-MHRD with 3 iterations. (c) and (d): The reconstructed first and second IMT functions are shown as the red curves and the true IMT functions are superimposed as the black curves. (e): The superposition of the reconstructed IMT functions is shown as the red curve and the clean signal is superimposed as the black curve.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Remark
  • Proposition 3.1
  • proof