  Real-time density estimation is ubiquitous in many applications, including
computer vision and signal processing. Kernel density estimation is arguably
one of the most commonly used density estimation techniques, and the use of
"sliding window" mechanism adapts kernel density estimators to dynamic
processes. In this paper, we derive the asymptotic mean integrated squared
error (AMISE) upper bound for the "sliding window" kernel density estimator.
This upper bound provides a principled guide to devise a novel estimator, which
we name the temporal adaptive kernel density estimator (TAKDE). Compared to
heuristic approaches for "sliding window" kernel density estimator, TAKDE is
theoretically optimal in terms of the worst-case AMISE. We provide numerical
experiments using synthetic and real-world datasets, showing that TAKDE
outperforms other state-of-the-art dynamic density estimators (including those
outside of kernel family). In particular, TAKDE achieves a superior test
log-likelihood with a smaller runtime.
