The dynamics of complex systems, from financial markets to the brain, can be
monitored in terms of multiple time series of activity of the constituent
units, such as stocks or neurons respectively. While the main focus of time
series analysis is on the magnitude of temporal increments, a significant piece
of information is encoded into the binary projection (i.e. the sign) of such
increments. In this paper we provide further evidence of this by showing strong
nonlinear relations between binary and non-binary properties of financial time
series. These relations are a novel quantification of the fact that extreme
price increments occur more often when most stocks move in the same direction.
We then introduce an information-theoretic approach to the analysis of the
binary signature of single and multiple time series. Through the definition of
maximum-entropy ensembles of binary matrices and their mapping to spin models
in statistical physics, we quantify the information encoded into the simplest
binary properties of real time series and identify the most informative
property given a set of measurements. Our formalism is able to accurately
replicate, and mathematically characterize, the observed binary/non-binary
relations. We also obtain a phase diagram allowing us to identify, based only
on the instantaneous aggregate return of a set of multiple time series, a
regime where the so-called `market mode' has an optimal interpretation in terms
of collective (endogenous) effects, a regime where it is parsimoniously
explained by pure noise, and a regime where it can be regarded as a combination
of endogenous and exogenous factors. Our approach allows us to connect spin
models, simple stochastic processes, and ensembles of time series inferred from
partial information