# Best practices for calculating Sobol Indices for time-series data

I'm looking to implement a global sensitivity analysis (GSA) study with what I'll treat as a black-box simulation. My main inspiration is this paper where they use the Sobol indices as their GSA method.

In the paper, they describe a quantity of interest, that is a scalar quantity (I believe they have three) which they study the variance of throughout the domain of the problem. This includes the variance of the elementary effects, as well as the variance of interactions up to the total variance for that parameter.

In some of my simulations, the result is indeed a scalar quantity. Thus, it makes sense to me how to use Sobol indices (or the Morris screening method that they mention is computationally more efficient). But in others, I end up with a time series such as a pressure history for multiple locations, not necessarily with the same time steps across simulations.

I've looked at this course materials document: which seems to qualitatively study the sensitivity of time series, but I'm hoping for a Sobol index-type description.

A previous question asked a similar question (with fewer details): Sensitivity analysis for modeled time series

I'm willing to try the PCA suggestion in the first answer, although based on the references, I'd likely have to use more than one quantity of interest to describe the same pressure history at a particular location, which makes it less desirable.

My ultimate goal is to speed up our model calibration process by calibrating only subsets of parameters to subsets of simulations, so having as simple as possible descriptions of the sensitivities is preferred. Is there some way to combine the variances of the PCA-GSA method?

Is the PCA method of projecting a time series onto an orthogonal basis the simplest way to capture the variance or are there other methodologies?

Edit: One thought I had was to define some physically important features of the time series. For example, we expect a rapid increase early in the pressure history, and we might be interested in the rate of that increase. So we could definitely look at the parameter sensitivity of that increase as a single quantity of interest. But we want to calibrate our model to the entire pressure history so that doesn't match our calibration process.

1: Leon, L., Smith, R. C., Oates, W. S., & Miles, P. (2018). Analysis of a multi-axial quantum-informed ferroelectric continuum model: Part 2—sensitivity analysis. Journal of Intelligent Material Systems and Structures, 29(13), 2840–2860. https://doi.org/10.1177/1045389X18781024