# How to fit simulated with measured data having systematic errors?

Consider a given time series of measured data, i.e. the number of persons tested positive for COVID-19. And consider a simulation - consisting of a model, possibly fed with some parameters estimated a priori (not via the measured data) - which gives another time series.

More often than not these two time series won't fit at once. But the goal of a simulation usually is to have a fit with measured data, and finally one wants to show: "See, these two histograms do match essentially."

That they don't match may have two reasons:

• the measured data are erroneous (possibly in a systematic way) and/or

• the model and/or the estimated parameters are erroneous

Assuming that the model is roughly correct (i.e. an advanced SEIR model) what are acceptable ways to "explain away" the gap between measured and simulated data?

The measured data may have two sources of error that are known in principle but not quantitatively:

• a reporting delay (which includes incubation and latency time, time for testing, and time for proper reporting), giving rise to a shift of data.

• a dark figure (i.e. cases that were not detected, falsely attributed, or false negative or positive cases), giving rise to typically lower values of the measured data (than in reality).

What is the "right way" to reflect these sources of errors, especially when drawing diagrams?

1. First correct and adjust the measured histogram, assuming some estimated reporting delays and dark figures, and then compare the corrected histogram with the simulated histogram?

2. First adjust the simulated data in a reciprocal way (by artificially adding some fictuous estimated reporting delay and dark figures) and then compare adjusted histogram with the non-corrected histogram of measured data?

Is one of these the one and only "good practice" - or are both or none of them scientifically tenable.

• This is an interesting question, though I think you're more likely to get a rigorous answer addressing your problems (time series modeling, error correction) on stats.SE, no? The bits of your question concerning COVID could be removed without changing the meaning. Jul 7 '20 at 16:42
• Am I understanding the question correctly: to remove errors, should changes be made to the stimulated model or to the data? Wouldn't it depend on the phenomena you are trying to model, and what the sources of error are? Like here, both the mentioned errors appear to be due to the data, not the epidemic that you're modelling. Jul 14 '20 at 10:39
• @AP2261: You are understanding correctly. And yes, you are right: both errors are due to the data, but they are only examples. Possible sources of errors of the model are countless. Jul 14 '20 at 14:42