I'm writing a Monte Carlo simulation and it involves a number of different variables which can exhibit both inter-person and intra-person variability. These variables are modeled using Gaussian distributions. Generally, population mean and standard deviation are known, and I have some idea of the typical extent of intra-person variability.

I'll use menstrual cycle length as a concrete example.

For a given simulated woman, I need to produce length samples for multiple cycles. I need those samples to respect the known population mean and SD while also expressing inter-cycle variability.

I can easily sample the population-wide distribution using a method such as the Ziggurat algorithm, and this could work for a very rough simulation, but it will be inaccurate because a single woman usually doesn't have as much cycle length variation as the whole population (although some women have dramatic variations).

I initially experimented with taking a sample from the population distribution and storing this as the individual's base value and then mixing this using some arbitrary weighting with per-event samples that were also obtained from the population distribution. This seemed to have acceptable results for a given individual, but it was globally inaccurate because mixing samples tends to pull them towards the mean and so the simulation's population-wide SD fell well below the actual population SD.

I could perhaps use a zero-mean Gaussian distribution to add some per-event offset to the individual's base value, but I intuitively suspect this will violate the population-wide distribution as well because individuals having their base value at the extremities could then produce extremely unlikely samples when the offset randomly pushes them substantially further in the same direction. Perhaps a non-zero mean could solve that, but I'm not nearly well versed enough in stats to have any idea how to pick the mean and SD for the offset distribution so as to keep the population-wide stats intact.

To make matters worse, in reality, the intra-person variability is itself variable, but actual data on that is limited so the best I could do there would be some kind of heuristic.

Is there some relatively simple way to model intra-person variability, while preserving the known population-wide normal distribution?


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