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

Question

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?

Answer 1

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:

The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.

Somewhat ironically, an important and magisterial text on time series has been written by James D. Hamilton (1994).