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An information criterion for detecting periodicities in functional time series

Rinka Sagawa, Yan Liu, Valentin Patilea

Abstract

We propose an information criterion for determining an unknown number of periodic components in functional time series. Identifying the number of frequencies in large-scale time series has been a central focus. To achieve this goal, we suggest an iterative procedure, utilizing the residual process obtained through least squares fitting. This iterative approach demonstrates broad applicability. We establish the consistency of the estimated number of periodic components by minimizing the information criterion. The efficacy of the procedure is illustrated through numerical simulations. In real data analysis, we apply this information criterion to temperature data and sunspot data.

An information criterion for detecting periodicities in functional time series

Abstract

We propose an information criterion for determining an unknown number of periodic components in functional time series. Identifying the number of frequencies in large-scale time series has been a central focus. To achieve this goal, we suggest an iterative procedure, utilizing the residual process obtained through least squares fitting. This iterative approach demonstrates broad applicability. We establish the consistency of the estimated number of periodic components by minimizing the information criterion. The efficacy of the procedure is illustrated through numerical simulations. In real data analysis, we apply this information criterion to temperature data and sunspot data.
Paper Structure (19 sections, 8 theorems, 195 equations, 2 figures, 31 tables, 1 algorithm)

This paper contains 19 sections, 8 theorems, 195 equations, 2 figures, 31 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Trigonometric regression models
  3. Sample-based selection of the number of periodicities
  4. Simulation
  5. Data Analysis
  6. Sunspot data
  7. Temperature data
  8. Conclusion
  9. Proof of Theorem \ref{['criterion_thm']}
  10. Proofs of Lemmas in Section \ref{['sec2']}
  11. Proof of Lemma \ref{['para_as_converge']}
  12. Proof of Lemma \ref{['p_y_approx']}
  13. Proof of Lemma \ref{['theta_estimate']}
  14. Technical results for Lemma \ref{['error_evaluate']}
  15. Additional simulation results for choice of $\kappa$
  16. ...and 4 more sections

Key Result

Lemma 2.5

Suppose $\{X_t; \, t \in \mathbb{Z}\}$ is a zero-mean stationary process satisfying Assumption k_cum_ass. Under Assumption asp:iden, if $0 \leq r \leq r_0$, then the least squares estimates $\hat{\bm{\bm{\psi}}}(r)$ converges to the true vector $\bm{\bm{\psi}}(r)$ in probability; if $r > r_0$, then

Figures (2)

  • Figure 1: The daily average temperature data of Kyoto in Japan, from January 1, 2018 to December 25, 2020. The dashed red lines indicate the segmentation of the data into 3 intervals of $365$ days each.
  • Figure 2: The rate when the periodicity is correctly estimated across all simulations for each $\kappa$ within the "stable" range when $N=120$, $480$, and $960$, respectively.

Theorems & Definitions (21)

  • Remark 2.2
  • Remark 2.3
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 3.2
  • Theorem 3.3
  • Remark 4.1
  • proof
  • ...and 11 more