Expectile Periodograms

TL;DR

The paper introduces the Expectile Periodogram (EP), a robust, non-parametric spectral estimator built from trigonometric expectile regression to analyze time series across the full range of expectile levels $\alpha\in(0,1)$. It establishes a rigorous asymptotic framework linking EP to the expectile spectrum $g(\omega,\alpha)$, showing EP ordinates converge in distribution to scaled $\chi^2$ variables and that smoothing yields a consistent spectrum estimator $\hat{g}(\omega,\alpha)$. Through simulations, EP detects hidden periodicities that ordinary PG misses and outperforms the Quantile Periodogram (QP) under Fisher’s test, while providing smoother behavior across $\alpha$. Real-data applications include earthquake waveform classification using a deep learning model on EP-derived features, achieving superior accuracy compared to PG and QP, underscoring the practical value of the two-dimensional EP representation for distribution-aware spectral analysis. Overall, EP offers a distributionally rich, tail-sensitive alternative to traditional spectral tools with demonstrated robustness and predictive utility.

Abstract

This paper introduces a novel periodogram-like function, called the expectile periodogram, for modeling spectral features of time series and detecting hidden periodicities. The expectile periodogram is constructed from trigonometric expectile regression, in which a specially designed check function is used to substitute the squared $l_2$ norm that leads to the ordinary periodogram. The expectile periodogram retains the key properties of the ordinary periodogram as a frequency-domain representation of serial dependence in time series, while offering a more comprehensive understanding by examining the data across the entire range of expectile levels. We establish the asymptotic theory and investigate the relationship between the expectile periodogram and the so called expectile spectrum. Simulations demonstrate the efficiency of the expectile periodogram in the presence of hidden periodicities. Finally, by leveraging the inherent two-dimensional nature of the expectile periodogram, we train a deep learning (DL) model to classify earthquake waveform data. Remarkably, our approach outperforms alternative periodogram-based methods in terms of classification accuracy.

Figures (11)

• Figure 1: (a) Loss functions for the QR at $\theta =0.25,0.5,0.75$. (b) Loss functions for the ER at $\alpha =0.25,0.5,0.75$. (c) The quantile function $\mu^*(\theta)$ and the expectile function $\mu(\alpha)$ for the standard Gaussian distribution.
• Figure 2: (a) and (b): Ensemble means of smoothed periodograms of model (\ref{['ar2']}) with $\omega_{\nu_c} = 2 \pi \times 0.25$ and $\omega_{\nu_c} = 2 \pi \times 0.3$, respectively; (c) and (d): the EPs of model (\ref{['ar2']}) at $\alpha = \{0.05, 0.06,...,0.95\}$ for $\omega_{\nu_c} = 2 \pi \times 0.25$ and $2 \pi \times 0.3$, respectively; (e) and (f): ensemble means of periodograms of model (\ref{['hidden']}) with $\omega_{\nu_c} = 2 \pi \times 0.25$ and $\omega_{\nu_c} = 2 \pi \times 0.3$, respectively.
• Figure 3: The periodograms of the mixture model (\ref{['eq-mix']}). (a) The EP with asymmetric pattern across the expectile levels, and (b) the EPs at $\alpha =0.1$ and $0.9$, along with the PG. The number of realizations is 5,000 and the sample size is $n=200$.
• Figure 4: MSE and KL divergence of smoothed periodogram. (a) AR(2) model (\ref{['ar2']}); (b) mixture model (\ref{['eq-mix']}). The results are based on 5,000 simulation runs.
• Figure 5: Left panel: expectile; right panel: quantile. First row: time series and their corresponding 0.9 expectile and quantile; second row: the ASECP and LCP of the time series; third row: the spectra and the averaged periodograms of 500 realizations; fourth row: the periodograms across the expectile and quantile levels at $\omega = 0.1 \times 2\pi$ of one realization. The spectra are computed by averaging 5,000 smoothed periodograms.