# How to recover the underlying observations from the noisy ones using gaussian processes

I have some simulated experiments where I generate some samples with an exponential correlation function. I am assuming a spatial grid whose variables form a multivariate gaussian distribution with an exponential correlation function and range r. I am assuming the mean of the gaussian is u.

Now if I add some noise to each of the observations like lets say noise with difference variances to each variable. Also suppose I also shift the mean by adding some noise in the mean as well.

How can I estimated the actual underlying values from these noisy observations using gaussian process.

In the case where I add the gaussian white noise, I could just plot the semi variogram fit a model and get the nugget parameter to know the amount of noise. But what in the case where I add noise with different variances to each variable.

Adding different noise levels will definitely make the observations non homogeneous.

Suggestions?

A gaussian process will put a gaussian distribution over all possible outputs including points where you have observations. The MLE of the outputs at the observations will be the value of the mean function at that point. Regardless if you add varying levels of noise to observations or not. Simple way of adding varying noise: $K^+ = K + diag(\sigma_{1:n})$.