Directional symmetry (time series)

Directional symmetry in a time series refers to a specific type of symmetry where the statistical properties of the series are invariant under a reversal of the time axis only when combined with a specific transformation of the data values. Unlike simple time-reversal symmetry (also known as reversibility or stationarity in the reversed time), directional symmetry requires a compensating adjustment to the series' values to maintain its overall statistical characteristics when the time axis is flipped.

More formally, a time series X(t) exhibits directional symmetry if there exists a transformation T such that the transformed time series T(X(t)) is statistically equivalent to the time-reversed version of the original series, X(-t). Crucially, T is not the identity transformation; if it were, the series would simply be reversible.

The transformation T can take many forms, depending on the specific properties of the time series. It might involve a negation of the values, a shift in the values, or a more complex nonlinear function. The key aspect is that it's a necessary component for achieving statistical equivalence under time reversal.

Directional symmetry is less common than simple time-reversal symmetry and often indicates specific underlying dynamics within the system generating the time series. Identifying directional symmetry can provide insights into the nature of these dynamics and can be useful for modeling and forecasting the series' behavior. It is particularly relevant in contexts where the underlying processes are known to exhibit asymmetry or are influenced by directional forces.