Noise reduction for functional time series

ArXiv ID: 2307.02154 “View on arXiv”

Authors: Unknown

Abstract

A novel method for noise reduction in the setting of curve time series with error contamination is proposed, based on extending the framework of functional principal component analysis (FPCA). We employ the underlying, finite-dimensional dynamics of the functional time series to separate the serially dependent dynamical part of the observed curves from the noise. Upon identifying the subspaces of the signal and idiosyncratic components, we construct a projection of the observed curve time series along the noise subspace, resulting in an estimate of the underlying denoised curves. This projection is optimal in the sense that it minimizes the mean integrated squared error. By applying our method to similated and real data, we show the denoising estimator is consistent and outperforms existing denoising techniques. Furthermore, we show it can be used as a pre-processing step to improve forecasting.

Keywords: Functional Principal Component Analysis (FPCA), Curve Time Series, Noise Reduction, Forecasting, Time Series Dynamics

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper relies heavily on advanced mathematical concepts like functional Hilbert spaces, FPCA, and MISE-optimal projections, earning a high math score. While it includes simulations and real data applications, the focus is on theoretical consistency and methodology rather than providing executable code, detailed backtesting frameworks, or specific financial implementation details, resulting in a lower empirical rigor score.

  flowchart TD
    A["Research Goal<br>Develop optimal noise reduction<br>for functional time series"] --> B
    subgraph B["Methodology: Extended FPCA"]
        B1["Model Curve Time Series<br>with Signal + Noise"] --> B2["Identify Subspaces<br>Signal vs. Noise"] --> B3["Project Data onto<br>Signal Subspace"]
    end
    B --> C["Data Processing<br>Simulated & Real Data"]
    C --> D["Computational Process<br>Projection & Denoising"]
    D --> E["Findings: Optimal Denoising"]
    D --> F["Findings: Improved Forecasting"]
    
    style A fill:#e1f5e1,stroke:#2e7d32
    style B fill:#e3f2fd,stroke:#1565c0
    style C fill:#fff3e0,stroke:#ef6c00
    style D fill:#f3e5f5,stroke:#6a1b9a
    style E fill:#fce4ec,stroke:#c2185b
    style F fill:#fce4ec,stroke:#c2185b