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I just started studying the theory of Gaussian processes. I'm mainly interested in studying functional data and I haven't found the answer to my doubt. Let's say I have some curves that I consider as realizations of a Gaussian process with mean m and covariance matrix C. If I suppose that my data have been generated using a certain covariance kernel k, is it possible to estimate the parameters of k? Is there any R package to do it? Thanks a lot in advance!

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It is indeed possible to estimate the parameters of the Covariance kernel of a Gaussian process.

You can for instance use maximum likelihood methods or Bayesian methods to do so. Maximum Likelihood methods (including the urgently needed gradients for numerical optimisation) are discussed in this in this PhD thesis for instance. The Bayesian approach for essentially the same problem has for instance been discussed in this paper. In this particular case, the Bayesian approach might be even easier, since there are no gradients required.

Depending on how many data points you have, the estimation procedure can be computationally very demanding. This is mostly the case, since many large and dense covariance matrices have to be constructed over and over again, essentially, whenever you change the parameter. Some colleagues and me have been working on this problem - therefore I hope you do not mind me advertising this work.

I do not know of any R-Package providing this. When you have large datasets, R may also be not the right choice, from an efficiency point of view.

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