Detecting Periodicity of a General Stationary Time Series via AR(2)-Model Fitting

TL;DR

The authors propose using AR(2) fitting to detect periodicity in general stationary time series by exploiting a peaked spectral density around the target frequency. They introduce a near-pole spectral class $f_\delta$ and show that when the peak is sharp, the AR(2) best approximation recovers the true frequency $\lambda_0$, with the AR(2) parameters converging to $(2\cos\lambda_0,-1)$. Under a close-to-pole regime where $\delta_n\to0$ with sample size, the AR(2)-based estimator of the peak frequency is consistent and achieves a rate faster than $n^{-1/2}$, approaching $n^{-2/3}$ as peak sharpness increases. The Sunspot data example demonstrates the method's practical performance, yielding a periodicity around 11 years that aligns with the well-known solar cycle. Overall, the paper provides a theoretically grounded, simple AR(2)-based approach with provable consistency and favorable rates for identifying spectral peaks in broad stationary settings.

Abstract

Estimating the periodicity of a stationary time series via fitting a second order stationary autoregressive (AR(2)) model has been initiated by the seminal paper of Yule(1927).. We investigate properties of this procedure when applied to a general stationary processes possessing a spectral density with a dominant peak at some frequency $λ_0\in(0,π)$. We show that if the peak of the spectral density is sharp enough (in a way to be specified) then the AR(2) model, which best (in mean square sense) approximates the underlying process, correctly identifies the frequency $λ_0$. To investigate consistency properties of the AR(2) based estimator of $λ_0$, a near to pole framework is adopted. Triangular arrays of stationary stochastic processes are considered that possess a spectral density the peak of which at $λ_0$ becomes more pronounced as the sample size $n$ of the observed time series increases to infinity. It is shown in this set up, that the AR(2) based estimator achieves a rate of convergence which is larger than the parametric $n^{-1/2}$ rate and which can be arbitrarily close to $ n^{-2/3}$, the best rate that can be achieved by this estimator.

Figures (2)

• Figure 1: Plot of the times series of Sunspot Numbers from 1700 to 2020.
• Figure 2: Periodogram of Sunspot Series (black solid line) and the Spectral Density of the fitted AR(2) model (blue line).

Theorems & Definitions (8)

• Lemma 1
• Lemma 2
• Proposition 1
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 1
• Lemma 6