# Methods for smearing multidimensional data

I have a dataset $X \in \mathbb{R}^{2p}$. Half of the parameters $X_{pre} \in \mathbb{R}^p$ represent true values without measurement biases or errors. The other half $X_{post} \in \mathbb{R}^p$ are the measured values of the same variables. I have paired data because this dataset is the result of simulation. The problem I have is that it is computationally much easier to generate $X_{pre}$ than $X_{post}$. I would like to use my complete dataset $X$ to smear future simulated datasets $X'_{pre}$ by applying the measurement bias and error.

Unfortunately the relationship between these variables is intricate. Generally the error dominates over the bias in these measurements; the median/mode are fairly close to the true value, though not always the mean because the errors can be highly skewed. The errors in the measurements of the $p$ variables are also correlated. I have tried using the gamlss to model these relationships one variable at a time, but this ignores the correlated errors -- and also the errors are difficult to model with gamlss in some cases because of how highly skewed they are, the topic of another question. Because of the difficulty, it doesn't seem feasible to me to scale this up to measuring all $p$ variables simultaneously.

One thing I have thought of is a nearest neighbors technique. Given a value $x \in \mathbb{R}^p$, I can lookup the nearest $N$ points in $X_{pre}$ and find the associated values in $X_{post}$. Rather than simply taking the mean or median of these values, however, I will sample from them, to model the smearing of the error. A few questions:

• Is this a valid procedure?
• Is there a cross-validation metric I could use to optimize $N$? I would need to compare two $p$-dimensional datasets (smeared and fully simulated), and since $p>1$ the KS test would not necessarily work.
• Are there other approaches that I could try?