# Problem with function handle for Basis function in fitrgp

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Stefan Hövel on 27 Jan 2020
Commented: Adam Danz on 21 Nov 2020
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:
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 on 28 Jan 2020
"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 on 21 Nov 2020
>gprMdl1.Beta is not 3x1, there is still only one beta. Could you help me check this?
Check this out.....
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

Ridwan Alam on 28 Jan 2020
Edited: Ridwan Alam on 28 Jan 2020
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.
fitrgp() is only using your beta0(1) as Adam mentioned in the comment.

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Ridwan Alam on 28 Jan 2020
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 on 28 Jan 2020
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 on 28 Jan 2020