# Time Series Shocks with Exponential Decay

Imagine a piano key played in an auditorium: The amplitude of the sound wave is perhaps highest in the first milliseconds, then slowly decays to zero if no other notes are played. If other notes are played, there is a complex signal where notes are decaying as incoming notes re-peak the signal. This is not necessarily the application I'm interested in, but it describes the kind of process I'm hoping to estimate.

I have background in time series, but none in statistical signal processing. My goal is not to be able to forecast this signal, rather, I'm hoping for some references or examples that will explain how to estimate the data-generation equation of this process given a set of features. In other words, how can I parametrize this signal and how can I estimate the decay and other parameters from the data.

SOME IDEAS I HAVE SO FAR: From time series, I know that I can index each covariate by time. In other words, I can posit that the signal is a function of its last $p$ values as well as $m$ lags of some other exogenous (also time-varying) information $$signal_t = \alpha + \sum_{i=1}^p \theta_i signal_{t-i} + \sum_{j=1}^m \lambda_j exogenous_{t-j}$$ My issue here, however, is that (given the data I have), capturing the signal properly would require many lags (high $p$, high $m$) and I do not have enough data to do this. Furthermore, my exogenous covariates are sparse (usually an indicator representing the occurrence of some event in period $t-j$). Finally, even if I did have enough data for this, I wouldn't necessarily see the signal decay smoothly over time.

I've considered moving average type modeling, but the data set is too large for such a thing (MA estimation takes a long time and is a non-convex procedure). I've also thought about how duration modeling can be applied, but haven't really figured it out yet. I think the way to deal with this is to somehow parametrize the decay and learn it from the data (I'm just not sure how to do this).