# validating a gaussian process fitted to data

I am relatively new to applying Gaussian processes to data. I come from a math background but the most popular literature on it seems to be from a machine learning perspective and not from a stochastic process/measure theoretic perspective.

Anyway, I am asked to fit a Gaussian process to data, i.e. $$GP(x) \sim N(\mu(x),\sigma^2(x)).$$ The model upon which the data is based assumes that for $x_1 \neq x_2$, $GP(x_1)$ and $GP(x_2)$ are independent. In addition, we specify parametric forms to $\mu(x),\sigma^2(x)$. These parametric forms are provided since we have restrictions on how the mean and variance behave across $x$.

I have training data and validation data -- samples of $(x_i,GP(x_i))$ for $i=1,\dots,N$ for training and samples of $(x_j,GP(x_j))$ for $j=1,\dots,M$ for validation distinct from training.

Using the training data, I formulate the negative log likelihood (just product of pdf's) and use MLE to obtain the parameters needed for the mean and the variance functions above.

Now, I want to assess the validity of my model for $\mu(x)$ and $\sigma^2(x)$ by taking each sample from the validation set, applying the transformation $\frac{GP(x_j) - \mu(x_j)}{\sigma(x_j)}$, and plotting the histogram to see if it's $N(0,1)$.

I am not so successful in achieving a nice looking histogram that matches the N(0,1) pdf so far. My question is: is it realistic to expect that one can actually find parametric forms for $\mu(x),\sigma^2(x)$ in order to "validate" the use of a Gaussian Process to fit the data? How does one even check whether a Gaussian Process of any sort can be applied to the data in the first place? (My data set takes on positive and negative values so that's a first step).

... is it realistic to expect that one can actually find parametric forms for $\mu(x)$, $\sigma^2 (x)$ in order to "validate" the use of a Gaussian Process to fit the data?