Hello,
the matlab code of estimated parameters of an non-linear differential equation model is
options=optimset('MaxFunEvals', 10000, 'MaxIter', 10000, 'TolFun', 0.0001, 'TolX',0.0001,'Display','on'); [k_optim, resnorm, residual, exitflag, output, lambda, jacobian] = lsqnonlin(f,k0,lb,ub,options);
From a theoretical point of view: Can I use the Jacobian matrix from my output to perform what is called Principal Component Analysis for the world of linear regression models? For those models you use the covariance Matrix of the observables, because this appears in the equation for the Variance of the parameters. In case of non-linear Models the parameter-variance is defined as J_T * J (but Im not sure), so I thought I might could use the one from my output. Is anyone sure and knows code in Matlab to do this?
PCA exploits the Eigenspace of the covariance matrix - as you mentioned -, thus gives you a mean to reduce dimensionality to the most important features in the system. Without knowing your PDE problem in detail, yes for certain problems you can utilise the Jacobian and its Eigenspace for such an approach, but you must ensure to properly scale/normalize the matrix. Moreover, I would also recommend looking into using autoencoders for investigating that eigenspace. They are sort of a nonlinear pendant for PCA...
Hope that helps.
Thank you very much for your response. Do you mean by normalization to divide the rows or columns of the matrix by its length in order to get values in between 0/1?
My parameters to estimate are ths ks, Ks, EAs and dHs in r. The other variables in r are externally given. The differential equation is dy/dz in which r appears. I got 240 measurements of y at z=0, which is "part" of my independent variables. To those ys, I got their change at z=end of the reactor tube. These are my dependents. (This model is discussed in my question: https://www.researchgate.net/post/How_to_assess_improve_my_parameter_estimation_for_dimethyl_ether_synth_reaction_rate_model_by_statistical_interference_better_regression_etc)
Do you think it is somehow possible to replace the observation matrix with the projection of the observations onto the eigenvectors as you can do it in the linear model? As you can see from the added equations my models is far more complex and the observants become a mixture between observed input data multiplied or summed with other coefficients. But I thought of replacing the denominator of r_meOH and r_WGS, since the results show a high correlation between the Ks.
If I use the code mentioned above for the Jacobi matrix I get a 720 x12 sparse double matrix. Since J = dy / dx my number of rows should equal number of measurements (240). Anyone has any idea why?
Beware, on its foundation the Principal Component Analysis is not done for the estimation of nonlinear parameters. It is primarily based on linear algebra (endomorphisms) and assumes that the variables to be analyzed are independent. To treat the case of parameters or variables with non-linear coupling, there are neural networks (deep learning ...)
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