[Paper Review] Quantifying non-periodicity of non-stationary time series through wavelets

Vicente J. Bolós, Rafael Benı́tez+1 more|arXiv (Cornell University)|Dec 16, 2019

Complex Systems and Time Series Analysis28 references4 citations

TL;DR

This paper introduces the windowed scale index, a wavelet-based method that quantifies non-periodicity in non-stationary time series by analyzing localized scalogram energy across time and scale. It proves that for the Haar wavelet, vanishing wavelet transform at scale 2T implies periodicity, validating the scale index as a reliable non-periodicity measure, and demonstrates superior sensitivity and interpretability compared to sample entropy in detecting regime shifts in financial time series.

ABSTRACT

In this paper, we introduce a new wavelet tool for studying the degree of non-periodicity of time series that is based on some recently defined tools, such as the extit{windowed scalogram} and the extit{scale index}. It is especially appropriate for non-stationary time series whose characteristics change over time and so, it can be applied to a wide variety of disciplines. In addition, we revise the concept of the scale index and pose a theoretical problem: it is known that if the scale index of a function is not zero then it is non-periodic, but if the scale index of a function is zero, then it is not proved that it has to be periodic. This problem is solved for the particular case of the Haar wavelet, thus reinforcing the interpretation and applicability of the scale index as a useful tool for measuring non-periodicity. Finally, we discuss the relationship between non-periodicity and unpredictability, comparing the new wavelet tool with the sample entropy.

Motivation & Objective

• To develop a time-localized measure of non-periodicity for non-stationary time series, addressing the limitation of global scale indices.
• To resolve the theoretical ambiguity of whether a zero scale index implies periodicity by proving it holds for the Haar wavelet.
• To compare the windowed scale index with sample entropy in measuring unpredictability, emphasizing interpretability and sensitivity.
• To apply the method to financial time series (crude oil and gold futures) to detect changes in predictability during financial crises.

Proposed method

• Proposes the windowed scalogram as a localized time-scale representation of wavelet transform energy.
• Defines the windowed scale index as the normalized $ L^2 $-norm of the wavelet transform within a time window centered at a given time with a specified time radius.
• Uses the Haar wavelet to prove that if the wavelet transform vanishes at scale $ 2T $ for all times, the signal is periodic, thus validating the scale index.
• Applies the windowed scale index with a time radius $ \tau $ to analyze subseries of financial time series, enabling localized non-periodicity assessment.
• Compares the windowed scale index with sample entropy (SampEn) using rolling windows on financial data, evaluating sensitivity and interpretability.
• Employs base-2 logarithmic scaling of the time-scale plane to ensure equidistance between dyadic scales, facilitating visual and analytical interpretation.

Experimental results

Research questions

• RQ1 Can the scale index be rigorously interpreted as a non-periodicity measure, particularly in the case where it vanishes?
• RQ2 Does the vanishing of the wavelet transform at scale $ 2T $ for all times imply periodicity, especially for the Haar wavelet?
• RQ3 How does the windowed scale index compare to sample entropy in detecting changes in unpredictability in non-stationary time series?
• RQ4 Can the windowed scale index detect shifts in predictability regimes during financial crises, such as the 2007–2008 global financial crisis?
• RQ5 Is the windowed scale index more interpretable and sensitive than sample entropy for quantifying non-periodicity in time series?

Key findings

• For the Haar wavelet, if the continuous wavelet transform vanishes at scale $ 2T $ for all times, the signal is periodic, thus validating the scale index as a sufficient condition for periodicity.
• The windowed scale index is bounded between 0 and 1, providing a more interpretable measure of non-periodicity than unbounded entropy measures like sample entropy.
• In the BvP oscillator example, the windowed scale index clearly distinguishes between low and high non-periodicity, while sample entropy yields nearly identical values.
• During the 2007–2008 financial crisis, the windowed scale index of crude oil futures contracts increased significantly, indicating higher non-periodicity, whereas gold futures remained more stable.
• The global scale index of gold futures is slightly higher than that of crude oil, but the windowed scale index reveals a more nuanced and accurate picture of regime shifts during crises.
• The windowed scale index shows higher sensitivity to changes in non-periodicity than sample entropy, especially in detecting short-term shifts in predictability during financial turbulence.