  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.
