In this paper, we introduce a periodogram-like function, called expectile periodograms, for detecting and estimating hidden periodicity from observations with asymmetrically distributed noise. The expectile periodograms are constructed from trigonometric expectile regression where a specially designed objective function is used to substitute the squared $l_2$ norm that leads to the ordinary periodograms. The expectile periodograms have properties which are analogous to quantile periodograms, which provide a broader view of the time series by examining different expectile levels, but are much faster to calculate. The asymptotic properties are discussed and simulations show its efficiency and robustness in the presence of hidden periodicities with asymmetric or heavy-tailed noise. Finally, we leverage the inherent two-dimensional characteristics of the expectile periodograms and train a deep-learning (DL) model to classify the earthquake waveform data. Remarkably, our approach achieves heightened classification testing accuracy when juxtaposed with alternative periodogram-based methodologies.

翻译：本文提出了一种名为“期望周期图”的类周期图函数，用于从具有非对称分布噪声的观测中检测和估计隐藏周期性。期望周期图基于三角期望回归构建，其中采用专门设计的目标函数替代了导致普通周期图的平方$l_2$范数。该函数具有与分位数周期图相似的特性，可通过分析不同期望水平提供更广泛的时间序列视角，但计算速度显著提升。本文讨论了其渐近性质，并通过模拟实验展示了其在处理含非对称或重尾噪声的隐藏周期性时的效率与稳健性。最后，我们利用期望周期图固有的二维特征，训练了一个深度学习模型对地震波形数据进行分类。值得注意的是，与基于周期图的其他方法相比，我们的方法在分类测试中实现了更高的准确率。