Fraction-of-Time Probability: Advancing Beyond the Need for Stationarity and Ergodicity Assumptions

Time series arising from measurements in many fields of physics, engineering, chemistry, biology, and econometrics, are commonly modeled as sample paths from an ensemble which, together with a probability measure, is called a stochastic process. Stationarity and ergodicity assumptions about this model are generally made for analytical convenience and mathematical tractability of the model. In this article, it is shown that a dichotomy, which can be very misleading in practice, exists between the properties of a stochastic process and those of its individual sample paths. This dichotomy can be eliminated by adopting the fraction-of-time (FOT) probability approach reviewed in this article for which a probabilistic model is constructed from a single time series without introducing the abstraction of the stochastic process. Two FOT-probability models are reviewed. The first considers probabilistic functions that do not depend on time and employs the relative measure on the real line as a probability measure and the time average as an expectation operator. Such time series are called stationary signals. The second considers periodic, poly-periodic, and almost periodic probabilistic functions and employs the operator that extracts the finite-strength additive sine-wave components of its argument as an expectation operator. This latter model is appropriate for describing time series originating from phenomena involving a combination of periodic and random phenomena. Such time series are called cyclostationary, poly-cyclostationary, and almost cyclostationary signals. The FOT-probability alternative provides a means for circumventing two standard but undesirable practices: (1) Adopting the Kolmogorov stochastic process model by using its Axiom VI without being able to verify its validity for the specific application and (2) Assuming Birkhoff’s ergodicity condition holds without being able to verify its validity for the specific application.