From time series to stochastics: A theoretical framework with applications on time scales spanning from microseconds to megayears

“Time series” has been an ambiguous term, sometimes referring to a series of measurements and other times used as synonymous to a stochastic process in discrete time. This ambiguity has been harmful to several scientific disciplines, theoretical and applied including hydrology, as it has hampered the understanding of the difference between a number and the abstract object called a random variable. Furthermore, what has been known as “time series models”, such as ARMA models have been equally misleading, as they are often non-parsimonious or overfitted, unnatural or artificial, theoretically unjustified and, eventually, unnecessary.

We present a general methodology for more theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, thereby enabling parsimonious modelling. The discrete-time properties of the processes are theoretically derived from the continuous-time ones and a general simulation methodology in discrete time is built, which explicitly handles the effects of discretization and truncation. Some additional modelling issues are discussed with focus on model identification and fitting, which are often made using inappropriate methods.

We apply the theoretical framework for several processes, including turbulent velocities measured every several microseconds and hydroclimatic processes, whose proxy reconstructions can provide information for time scales up to millions of years.