What resources are available for applying inference/computational statistics to infer the underlying error/noise model, given X measurements from some apparatus? (Below, I am mostly referring to spatially-correlated noise, but spatiotemporal noise works too. This discussion is far simpler in terms of times series alone.)
My idea is this: let's say we use some apparatus (e.g. a telescope, or a medical test) and we make X number of measurements simultaneously. The X measurements were taken at the same time t, and therefore should be biased in some manner (due to the test environment and so on.)
Assume there is some (unknown) stochastic process that governs instrument noise for each sample, and some (unknown) underlying model of the noise between each of the X simultaneous measurements. That is, if we repeat this experiment, the new measurement will continue to be biased by this spatial/temporal noise due to the instrument.
Given thousands of repeated measurements, what methods exist in the literature to infer the noise/error model in this data?
I suspect the most optimal approach is latent variable models, especially characterizing spatially-dependent noise.
Is my question clear?