Initial values in multivariate nonlinear regression

I need to do a multivariate nonlinear regression.The regression would be quadratic with three variables (x1, x2, x3) and looks like the following:

y = a0 + a1x1 + a2x1^2 + a3x2 + a4x2^2 + a5x3 + a6x3^2 + a7x1x2 + a8x1x3 + a9x2x3 + a10x1x2x3

As shown above, the model has 11 coefficients to be estimated.If I am correct, the code should look as following:

load mydata

tbl = table(x1,x2,x3,y);

modelfun = @(b,x)b(1) + b(2)*x(:,1) + b(3)*x(:,1)^2 + b(4)*x(:,2) + b(5)*x(:,2)^2 + b(6)*x(:,3) + b(7)*x(:,3)^2 +...

b(8)*x(:,1)*x(:,2) + b(9)*x(:,1)*x(:,3) + b(10)*x(:,2)*x(:,3) + b(11)*x(:,1)*x(:,2)*x(:,3)

beta0 = [b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11];

mdl = fitnlm(tbl,modelfun,beta0)

My question is how to find good starting guess initial values for beta0. I do not want to use random initial values for the coefficients since that might not lead to a solution. I searched Matlab questions/answers and found that I could use functions such as patternsearch (link) or ga (link), but this would require having the Global Optimization Toolbox. Unfortunatelly, I do not have this toolbox.

Answer 1

[Torsten]

Your model equation

y = a0 + a1x1 + a2x1^2 + a3x2 + a4x2^2 + a5x3 + a6x3^2 + a7x1x2 + a8x1x3 + a9x2x3 + a10x1x2x3

is linear in the model parameters a0, a1,...,a10.

Thus no good starting values will be necessary. Each optimizer will reach the optimal parameters in one step (independent of the starting values).

[Moha Singa]

Thank you a lot.