  Multiscale entropy (MSE) has been widely used to examine nonlinear systems
involving multiple time scales, such as biological and economic systems.
Conversely, Allan variance has been used to evaluate the stability of
oscillators, such as clocks and lasers, ranging from short to long time scales.
Although these two statistical measures were developed independently for
different purposes in different fields in the literature, their interest is to
examine multiscale temporal structures of physical phenomena under study. We
show that, from an information-theoretical perspective, they share some
foundations and exhibit similar tendencies. We experimentally confirmed that
similar properties of the MSE and Allan variance can be observed in
low-frequency fluctuations (LFF) in chaotic lasers and physiological heartbeat
data. Furthermore, we calculated the condition under which this consistency
between the MSE and Allan variance exists, which is related to certain
conditional probabilities. Heuristically, physical systems in nature including
the aforementioned LFF and heartbeat data mostly satisfy this condition, and
hence the MSE and Allan variance demonstrate similar properties. As a
counterexample, an artificially constructed random sequence is demonstrated,
for which the MSE and Allan variance exhibit different trends.
