Functional Continuous Decomposition

TL;DR

The paper introduces Functional Continuous Decomposition (FCD), a novel framework for non-stationary time-series analysis that offers continuous, parametric optimization and improves interpretability and performance in various applications.

Contribution

FCD is a new, JAX-accelerated method enabling continuous, parametric decomposition of time-series data with guaranteed smoothness, enhancing analysis and feature extraction capabilities.

Findings

FCD achieves up to $C^1$ continuity in fitting.

FCD provides accurate signal decomposition with an average SRMSE of 0.735.

FCD-enhanced CNNs converge faster and are more accurate.

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.