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As part of the Improved Machine Learning in Mathematica 11, Wolfram advertises the Method "GaussianProcess" which can be used with Predict[] to obtain smooth Gaussian random fields. The screenshot below from the official documentation shows how this is applied.

However, I was wondering if I can somehow specify the mean function $\boldsymbol{\mu}$ and covariance kernel function $k(x_i, x_j)$? I cannot find it in the documentation of Predict[], or anywhere else in the Wolfram documentation. Perhaps there is an undocumented option to provide the mean and kernel function?

I am referring $\mu$ and $k$ as used in this paper this paper.

enter image description here

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As far as I know, it's not yet possible to specify arbitrary covariance kernels. However, there are a few ways to control the kernel type. You can specify the type of covariance function using the "CovarianceType" suboption like so:

pred = Predict[data, 
  Method -> {
      "GaussianProcess", 
      "CovarianceType" -> "Linear"
  }
]

The possible kernels for "CovarianceType" are:

{
   "Periodic", "SquaredExponential", "RationalQuadratic", "Linear", 
   "Mattern5/2", "Mattern3/2", "NN", "WN"
}

You can also specify algebraic combinations of these kernels, such as:

"CovarianceType" -> "Linear" + "Periodic" * "RationalQuadratic"

Finally, there is the "Composite" option value, which will try and find a sensible combination of kernels that works well. Furthermore, there is the "SearchMethod" suboption for composite kernels, which can be set to "Greedy" or "SimulatedAnnealing". E.g.:

Method -> {
      "GaussianProcess", 
      "CovarianceType" -> "Composite",
      "SearchMethod" -> "SimulatedAnnealing"
  }

After the predictor pred has been computed, you can investigate what it's using by evaluating:

pred[[1]]

edit As far as I know, there is currently no implementation of mean functions, but I might be wrong about that.

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