What are the details fitrkernel model and how the predictions are produced?

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Mariam Ahmed 2020-11-9

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https://ww2.mathworks.cn/matlabcentral/answers/641390-what-are-the-details-fitrkernel-model-and-how-the-predictions-are-produced

编辑： Mariam Ahmed 2020-11-9

I am trying to implment my own predict function for a model produced using fitrkernel in which a Gaussian kernel regression model is fit using random feature expansion.

I Have two questions regarding fitrkernel

1) Question 1:

What I understand is that fitrkernel solves for β and b in the linear regression equaition, f(x)=T(x)β+b.

However, the model otuputs the following with no mention of β and b:

  RegressionKernel
              ResponseName: 'Y'
                   Learner: 'svm'
    NumExpansionDimensions: 128
               KernelScale: 1
                    Lambda: 0.000328947368421053
             BoxConstraint: 1
                   Epsilon: 0.0192374351371386

So, how can I know what are the values of β and b?

2) Question 2:

I looked up the documentation of the model and I found in https://www.mathworks.com/help/stats/fitrkernel.html that Gaussian kernel approximation is done using Random Feature Expansion. That is,

T(x)=m^(−1/2)*exp(iZx′)′ --- Eq(1)

where Z∈ℝ^(m×p) is a sample drawn from N(0,σ^−2) and σ^2 is a kernel scale.

The values for m and σ^2 coresponds to the NumExpansionDimensions and KernelScale name-value pair arguments of fitrkernel, respectively.

However, In Eq(1) the sample Z ~ N(0,σ^−2) is drawn at random, so how the model can predict the same output for the same input all the time? and what is i?