I may be wrong, but I understand that you'd like to optimize the kernel parameters such that the GPR (Gaussian Process Regression) evaluated on your X_train data is not lower than 90% of the lowest value in the y_train data. Please let me know if this is not the case or if I misinterpreted your code.

More broadly, the intended function to optimize in GPR is the likelihood. I don't think you'll be able to achieve your goal by replacing the function to optimize as you are doing.

If you want to work with positive data and GPR, I suggest two approaches:

• Implementing a GPR with positive coefficients: I can provide code for this, but it might be slow.

• Transforming and inverse-transforming your data:

  • You can apply a transformation like x: max(0, x) to your predicted data.
  • Alternatively, apply some function f to your data, for instance np.where(x < t, 2*x - t, x), which expands the positive space to include some negative values. Then, apply your GPR and use the inverse function f⁻¹, such as np.where(x < t, 0.5*(x + t), x), to map the results back to the positive space. This gives the GPR some flexibility, and you end up with positive values without hard thresholding as in the previous approach.