Problem with function handle for Basis function in fitrgp

I like to use a custom Basis function for fitrgp.

beta0 = [1.5,2.0];
hfcn = @(X) (beta0(1).*(X).^beta0(2)); 

This should be a powerMean-function.

If I call:

gprMdl1 = fitrgp(x,y,'BasisFunction',hfcn,'Beta',beta0) 

with dummy-data from:

load(fullfile(matlabroot,'examples','stats','gprdata2.mat'))

I get

gprMdl1.Beta = 0.0019

Why is gprMdl1.Beta not 1x2 like beta0? Is it posible to optimize custom parameters in Basis function (like beta0)?

[Adam Danz]

"Why is gprMdl1.Beta not 1x2 like beta0?"

That basis function will return as many values as provided in the input X. X is multiplied by 1 value (beta0(1)) and added to one value (beta0(2)). So if X is only 1 value, the output should only be one value. If x is 1x5, the output will be 1x5; if x is 3x4, so is the output.

[Adam Danz]

>gprMdl1.Beta is not 3x1, there is still only one beta. Could you help me check this?

Check this out.....

% Your function
beta0 = [1;2;3];
hfcn = @(X) ([ones(size(X,1),1),X,X.^2]*beta0);

% Test some inputs-outputs
hfcn(1)

ans = 6

hfcn([1;2;3])

ans = 3×1

6 17 34

●

Answer 1

[Ridwan Alam]

load(fullfile(matlabroot,'examples','stats','gprdata2.mat'))

In this case, your x is n x 1;

hfcn returns H, which has size n x 1; i.e. p = 1;

So if you want to provide 'Beta' to fitrgp(), it should be of size p x 1, i.e 1 x 1.

https://www.mathworks.com/help/stats/fitrgp.html#d117e376477

fitrgp() is only using your beta0(1) as Adam mentioned in the comment.

[Stefan Hövel]

So, the number of coefficents/parameter within the Basic function is restricted to the number of dimensions of the input-vector? In other words powerMean is not available as a custom Basic function for a single dimension vector or univariate regression. Is that correct?

[Ridwan Alam]

Well ... you are still achieving power mean from the hfcn, except the fact that for a 1-d vector, you are taking the mean of only one element per instance, right? So your H is still returning 1.5*x^2. The 'Beta' you provide to fitrgp() has no relation to the way your hfcn is defined, except working as an initial value for finding the optimal beta of H*beta. Hope it makes sense.

[Stefan Hövel]

Okay, now I get it. Beta can be interpreted as a weight on the Basis function H. I would like fitrgp to find the optimal values for beta0 within hfcn. For the KernelFunction it is posible see Example: Fit GPR Model Using Custom Kernel Function:

kfcn = @(XN,XM,theta) (exp(theta(2))^2)*exp(-(pdist2(XN,XM).^2)/(2*exp(theta(1))^2));

Here fitrgp optimizes the theta-values (see KernelInformation). I suppose it is not posible for Basis function.

Thanks a lot!

[Ridwan Alam]

Sure. Glad to help.