[Paper Review] Spectral methods for small sample time series: A complete periodogram approach

Sourav Das, Suhasini Subba Rao+1 more|arXiv (Cornell University)|Jul 1, 2020

Complex Systems and Time Series Analysis32 references4 citations

TL;DR

This paper proposes a novel 'complete periodogram' that reduces finite-sample bias in spectral analysis of small time series by incorporating best linear predictors beyond the observed data boundaries. By replacing the standard discrete Fourier transform with a complete DFT that includes predictions for unobserved lags, the method yields an unbiased estimator of the spectral density, and its estimated version shows significantly lower bias than the classical periodogram, especially in small samples and near spectral peaks.

ABSTRACT

The periodogram is a widely used tool to analyze second order stationary time series. An attractive feature of the periodogram is that the expectation of the periodogram is approximately equal to the underlying spectral density of the time series. However, this is only an approximation, and it is well known that the periodogram has a finite sample bias, which can be severe in small samples. In this paper, we show that the bias arises because of the finite boundary of observation in one of the discrete Fourier transforms which is used in the construction of the periodogram. Moreover, we show that by using the best linear predictors of the time series over the boundary of observation we can obtain a "complete periodogram" that is an unbiased estimator of the spectral density. In practice, the "complete periodogram" cannot be evaluated as the best linear predictors are unknown. We propose a method for estimating the best linear predictors and prove that the resulting "estimated complete periodogram" has a smaller bias than the regular periodogram. The estimated complete periodogram and a tapered version of it are used to estimate parameters, which can be represented in terms of the integrated spectral density. We prove that the resulting estimators have a smaller bias than their regular periodogram counterparts. The proposed method is illustrated with simulations and real data.

Motivation & Objective

• To address the severe finite-sample bias in the classical periodogram, especially when sample sizes are small and spectral densities have sharp peaks.
• To develop a theoretically grounded alternative to the standard periodogram that achieves bias reduction below O(n⁻¹) by incorporating predictions of unobserved time series values.
• To demonstrate that the complete periodogram, based on best linear predictors, is an unbiased estimator of the spectral density under second-order stationarity.
• To propose a feasible estimation procedure for the complete periodogram using estimated predictors, ensuring practical applicability.
• To validate the method through simulations and real-world analysis of sunspot data, showing improved peak amplitude estimation.

Proposed method

• Define the complete discrete Fourier transform (DFT) by extending the regular DFT to include best linear predictors of Xτ for τ < 1 and τ > n, based on the observed series {Xt}t=1n.
• Construct the complete periodogram as the squared magnitude of the complete DFT: Ĩn(ω) = |J̃n(ω)|², which is shown to be an unbiased estimator of the true spectral density f(ω).
• Use the property that cov[J̃n(ω), Jn(ω)] = f(ω) to establish the theoretical foundation for unbiasedness of the complete periodogram.
• Estimate the unknown best linear predictors using sample autocovariances and solve the Yule-Walker equations to form the 'estimated complete periodogram'.
• Apply data tapering to the estimated complete periodogram to further reduce leakage and bias, producing a 'tapered estimated complete periodogram'.
• Use the resulting periodograms to estimate parameters expressible as integrals of the spectral density, proving reduced bias compared to classical periodogram estimators.

Experimental results

Research questions

• RQ1 Can the finite-sample bias of the classical periodogram be reduced below O(n⁻¹) by incorporating predictions of unobserved time series values?
• RQ2 Does the complete periodogram, based on the complete DFT, provide an unbiased estimator of the spectral density under second-order stationarity?
• RQ3 How does the performance of the estimated complete periodogram compare to the classical and tapered periodograms in terms of bias and mean squared error in small samples?
• RQ4 Can the complete periodogram better capture the amplitude of spectral peaks, especially in the presence of sharp or dominant frequencies?
• RQ5 Does the method retain its advantages in large samples, and how does it compare to classical approaches in real-world data such as sunspot activity?

Key findings

• The complete periodogram is theoretically unbiased for the spectral density f(ω), with bias of order less than O(n⁻¹), unlike the classical periodogram whose bias is O(n⁻¹).
• The estimated complete periodogram achieves significantly lower bias than the classical periodogram in small samples, with bias reduced from O(n⁻¹) to O(n⁻²) in some cases.
• In simulations, the estimated complete periodogram showed a 50–70% reduction in bias for AR(1) and ARMA(1,1) models with small sample sizes (n = 20, 50), especially near spectral peaks.
• For the sunspot data (n = 3168), the complete periodogram detected a higher peak amplitude (7.1×10⁵) at the 11-year cycle compared to the regular (6.98×10⁵) and tapered (6.25×10⁵) periodograms, indicating better amplitude estimation.
• After smoothing with a Bartlett window, the complete and tapered complete periodograms revealed a more dominant and clearly defined 11-year peak, while also better capturing the quasi-periodic band between 9.1–12.6 years.
• The method demonstrated robustness in real data, with the complete periodogram outperforming classical methods in preserving the true spectral shape and amplitude, even in large samples.