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Generalized dynamic functional principal component analysis

Tzung Hsuen Khoo, Issa-Mbenard Dabo, Dharini Pathmanathan, Sophie Dabo-Niang

TL;DR

Simulations show that GDFPCA performs comparably to dynamic functional principal component analysis (DFPCA) for stationary data, while providing improved reconstruction accuracy in nonstationary settings, where both DFPCA and functional principal component analysis (FPCA) deteriorate.

Abstract

In this paper, we explore dimension reduction for functional time series. We propose a generalized dynamic functional principal component analysis (GDFPCA) which does not rely on spectral density estimation and demonstrates strong empirical performance for both stationary and nonstationary functional time series. We define the generalized dynamic functional principal components (GDFPCs) as static factor time series in a functional dynamic factor model and obtain their multivariate representation from a truncation of the functional dynamic factor model. Estimation is based on a least-squares reconstruction criterion and implemented via a two-step procedure for the coefficient vectors of the loading curves under a basis expansion. We establish mean-square consistency of the reconstructed functional time series under weak stationarity. Simulation studies show that GDFPCA performs comparably to dynamic functional principal component analysis (DFPCA) for stationary data, while providing improved reconstruction accuracy in nonstationary settings, where both DFPCA and functional principal component analysis (FPCA) deteriorate. Applications to real datasets support the empirical advantages observed in the simulations.

Generalized dynamic functional principal component analysis

TL;DR

Simulations show that GDFPCA performs comparably to dynamic functional principal component analysis (DFPCA) for stationary data, while providing improved reconstruction accuracy in nonstationary settings, where both DFPCA and functional principal component analysis (FPCA) deteriorate.

Abstract

In this paper, we explore dimension reduction for functional time series. We propose a generalized dynamic functional principal component analysis (GDFPCA) which does not rely on spectral density estimation and demonstrates strong empirical performance for both stationary and nonstationary functional time series. We define the generalized dynamic functional principal components (GDFPCs) as static factor time series in a functional dynamic factor model and obtain their multivariate representation from a truncation of the functional dynamic factor model. Estimation is based on a least-squares reconstruction criterion and implemented via a two-step procedure for the coefficient vectors of the loading curves under a basis expansion. We establish mean-square consistency of the reconstructed functional time series under weak stationarity. Simulation studies show that GDFPCA performs comparably to dynamic functional principal component analysis (DFPCA) for stationary data, while providing improved reconstruction accuracy in nonstationary settings, where both DFPCA and functional principal component analysis (FPCA) deteriorate. Applications to real datasets support the empirical advantages observed in the simulations.
Paper Structure (14 sections, 2 theorems, 49 equations, 6 figures, 3 tables)

This paper contains 14 sections, 2 theorems, 49 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Truncated GDFPC
  4. Computation of GDFPC
  5. Consistency of GDFPC
  6. Simulation Studies
  7. Simulation results for a stationary first order functional autoregressive FAR(1) process
  8. Simulation results for a non-stationary Wiener process
  9. Simulation results for a dynamic factor model
  10. Applications to real data
  11. Particles with a diameter of 10 micrometers or less (PM10)
  12. Standard & Poor's 100 index (S&P100)
  13. Trading Volume of Bitcoin
  14. Conclusions

Key Result

Theorem 1

Assume that A1-A7 hold. Let the common part of a dynamic factor model be defined by $\psi_t^m=(\boldsymbol{\beta}^m)^{\top}F_{t}$, where $\boldsymbol{\beta}^m := (\beta_{h,j}) \in \mathbb{R}^{(K+1) \times m}$ and $F_{t}=(f_{t},\cdots,f_{t-K})^{\top}$. Secondly, let $\chi^{R,b}_{t}(\hat{\mathbf{f}},\

Figures (6)

  • Figure 1: Boxplots of NMSE of 100 iterations of simulated stationary FAR(1) processes.
  • Figure 2: Boxplots of NMSE over 100 Monte Carlo replications for simulated Wiener processes, comparing FPCA, DFPCA, and GDFPCA.
  • Figure 3: Boxplots of NMSE over 100 Monte Carlo replications for the dynamic factor model, comparing FPCA, DFPCA, and GDFPCA.
  • Figure 4: A plot of the data set PM10 in the form of functional time series.
  • Figure 5: A plot of the centered, scaled and smoothed time series of the 100 stocks in S&P100.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Lemma 1
  • proof : Proof of Theorem \ref{['thm1']}
  • proof : Proof of Theorem \ref{['thm2']}