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Regression is the task of modeling the response $r$ to the exploratory set of variables $X$ such that:

$r=f(X)+ \epsilon$

Assume the regression function $f(X)$ is already known, and for any future sample $X^t$:

$f(X^t)=\bar{r}^t$

Where $\bar{r}$ is always measurable in the sample data. What I would like to model is the noise term that represents the residual of any data point.

$r= \bar{r} + \epsilon(X')$

$r-\bar{r}=\epsilon(X')$

If we make the assumption that $\bar{r}$ is an unbiased estimator of $r$, then the error can be summarized as:

$\epsilon(X') \sim \mathfrak{N}(0,\sigma^2)$

In words what I am trying to do is quite simple: Given a model, describe the noise ($\sigma^2$) of the model as it relates to a set of variables $X'$. i.e. some variables produce noise with higher variance from the model than others, but all noise is normally distributed.

I am at a complete loss as to how to describe this type of problem. I suppose you could say the difference between what I am trying to do and typical linear regression would be that I am trying to correlate a set of variables $X'$ with a stochastic model for noise, instead of with a scalar prediction.

If someone could point me in the direction of a proper formulation for what I am attempting to describe, or techniques to look at for solving this type of problem, I would highly appreciate it. I am fairly new to ML and statistics, I apologize for the poor/incorrect formulation.

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