Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Functional Continuous Decomposition

Teymur Aghayev

TL;DR

This paper proposes Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions and demonstrates that a Convolutional Neural Network enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

Abstract

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to $C^1$ continuous fitting, FCD transforms raw time-series data into $M$ modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

Functional Continuous Decomposition

TL;DR

This paper proposes Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions and demonstrates that a Convolutional Neural Network enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

Abstract

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to continuous fitting, FCD transforms raw time-series data into modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.
Paper Structure (26 sections, 20 equations, 9 figures, 6 tables)

This paper contains 26 sections, 20 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Background
  4. Mode Decomposition
  5. Local Smoothing Algorithms
  6. Summary of Research Gap
  7. Methodology
  8. Normalization
  9. Uniform Segmentation and Mode Calculation
  10. Local Translation
  11. Levenberg-Marquardt Optimization
  12. Ensuring Continuity
  13. Forward Fit
  14. Unscaling Parameters
  15. Computational Optimization
  16. ...and 11 more sections

Figures (9)

  • Figure 1: FCD example on Bitcoin 1-minute data
  • Figure 2: Decomposition of UAH-DriveSet velocity dataset.
  • Figure 3: Normalized derivative of decomposition for UAH-DriveSet velocity dataset.
  • Figure 4: Decomposition performed on EEG data with sine model.
  • Figure 5: Optimized Frequency for EEG dataset.
  • ...and 4 more figures