Process vs. observational error

[mjzahn]

Several of us took Tim's modeling course last quarter and we talked about "process error" and "observational error" in model design, the first being error in nature (natural stochasticity) and the latter being error in measurement. In class so far, am I right that we're considering our models to include the error term for process error (i.e. our measurements are assumed to be perfectly accurate)?

[mdscheuerell]

Now we're getting into the really interesting stuff...

Yes, as you learned in Tim's class, we typically think of observation errors as arising during sampling or identification, and process errors as arising due to the stochastic nature of the system/environment, whether or not we observe the process of interest. In this class, we are assuming that the predictors in X are measured without error, but the response (y) is measured with error. Both of these are forms of observation errors.

If we were interested in a simple regression model, but we had measurement errors in the predictors, we could model that as something like

y_i = alpha + beta * x_i + e_i with e_i ~ N(0, sigma^2) x_i = z_i + w_t with w_t ~ N(0, tau^2)

where z_t is the true value of our predictor. This is called an "error in variables" problem, which we deal with a lot in time series analysis (eg, "state-space" models).

[mjzahn]

Gotcha! It makes sense that the error term we are dealing with is only pertaining to the response and for error to be included for X, we'd need another equation equal to x_i that includes an error term. I'm curious about the difference in sigma^2 vs. tau^2 for the distributions but that may be more than I need to think about right now.

Just to confirm: we're not including any stochastic processes within nature for our models? Only observation error in the response (y)?

[mdscheuerell]

Just to confirm: we're not including any stochastic processes within nature for our models?

To be fair, the models we are (will be) using can represent stochastic natural processes, but most often we think of ecological processes operating over time and/or space. In those cases, we think of the process errors as something that disrupts the deterministic way of things as they progress/change over time/space.

Only observation error in the response (y)?

Yes, we are only considering situations where the imprecision in our information only applies to the response (y).