KernelScale optimization for ARD kernels

Hello,

How are KernelScales determined for ARD kernels since we cannot choose to optimize these kernels?

Constant initial KernelParameters specified by the user are not the same once the metamodel is created using fitrgp.

So there is something happening leading to a change in kernel scales (but not specified in the documentation)..

Answer 1

[akshatsood]

Dear @Rebecca Mazloum,

I understand that you are curious to know about "KernelScales" optimization for ARD kernels.

In Gaussian Process Regression (GPR) models, especially when using Automatic Relevance Determination (ARD) kernels, the "KernelScale" parameters are crucial as they determine the length scales for each dimension of the input data. When you use the "fitrgp" function in MATLAB, the initial "KernelParameters" you specify can indeed change during the fitting process. This is because "fitrgp" typically optimizes these parameters to best fit the training data.

You are indeed corect in you understanding that, "KernelScale" cannot be optimized for any of the ARD kernels. This has been specified in the following part of the documentation KernelScale Optimization

The documentation does not explicitly mention how the value is determined. However, it does provide information about the search range over which the parameter is varied for achieving an optimal fit. Please find below an extract from Release Notes of R2023b concerning search range of "KernelScale" hyperparameter.

Starting from MATLAB R2023b, "KernelScale" hyperparameter search range does not depend upon predictor data during optimization of GPR models i.e. when you specify to optimize the GPR hyperparameter "KernelScale" by leveraging the "OptimizeHyperparameters" name-value argument, it searches among positive values log-scaled in the range instead of where .

I hope ths helps.