# Are similarities in notation between Hamiltonian mechanics and Box-Jenkins time series forecasting accidental?

The 1833 formalism of Hamiltonian mechanics prescribes phase space coordinates (p,q) as momenta. https://en.wikipedia.org/wiki/Hamiltonian_mechanics

The Box-Jenkins method (1970) of time series forecasting uses similar notation (p,q) defined as "once stationarity and seasonality have been addressed, the next step is to identify the order (i.e. the p and q) of the autoregressive and moving average terms." https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method

There is nothing in Box-Jenkins 1970 book suggesting awareness of Hamiltonian mechanics as a precedent.

Regardless, and as noted, the similarities in notation are striking. But are they accidental?

The notational similarity is purely accidental. The phase space coordinates $$(p,q)$$ describe a manifold of arbitrary dimensions, typically with the same number of $$p$$'s and $$q$$'s, i.e. $$p = (p_1, ..., p_n)$$, $$q = (q_1, ..., q_n)$$. In the statistical example, the single integers $$p$$ and $$q$$ denote the order of the auto-regressive and moving-average polynomials defining the underlying ARMA model for a time series. Another example for the use of $$p$$ and $$q$$ in statistics can be inferred from PP and QQ plots (abbreviating probabilities and quantiles, respectively). That said, one can of course use a Hamiltonian $$H$$ to create a time series, say for coordinates $$q$$, via $$q_{t+1} = q_t + \{q_t, H\}$$, the curly bracket representing a Poisson bracket. To add historical flavour, the Wikipedia page linked above states: