Cross-posted from physics stackexchange
Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). Can we interpret this to obtain meaningful statistical thresholds?
Recently I've been doing some volunteer work on an open-source system identification tool for mechanisms driven by permanent-magnet DC motors. These are fairly close to an ideal LTI system, and accordingly we've had success approaching the problem with OLS and derived techniques.
Our analysis involves time-series measurements of a mechanical system's voltage, velocity, and acceleration as a driving signal is applied. We perform an OLS fit of the motor's voltage balance equation, with the highest-variance measured variable (typically acceleration) as the dependent variable, and the others as predictors. The parameters from this OLS are then used to compute control gains for the motor.
This methodology works quite well in most cases, but it can be hard for inexperienced users to diagnose when the fit is invalid due to undersampling and a calculated control gain cannot be trusted. One promising approach seems to be to perform a principal component analysis to determine which signals are well-sampled enough to warrant inclusion in the OLS - for example, we can look for eigenvalues of the non-normalized covariance matrix (the data matrix multiplied by its transpose) that are below a significance threshold, and remove from the fit the parameter that is "closest" to the matching eigenvector.
But, while we've found this to work well in practice, I'm stumped as to how to interpret the (experimentally-determined) threshold for the above procedure. A naive dimensional analysis suggests that, after appropriate physical conversions, the "threshold" has units of joule * second - physical units of Action. What does this mean? Can we estimate an appropriate threshold from first principles?
More generally, is there a conceptual reason that Action seems to be the natural unit of "sample size" in a statistical physics experiment like this? I know system identification is not "traditionally" viewed as statistical physics, but there seem to be parallels here and I wonder if interpretation can be helped by the analogy.