# Principal Comoponent Analysis for a nonlinear regression / parameter estimation model

#### Entropie_13

##### New Member
Hello,

the matlab code of estimated parameters of an non-linear differential equation model in the added file 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? Alternatively you might have other suggestions for dimension reducing techniques or methods of determining the extent of collinearity between parameters.

The code refers to a set of 240 measurements of y at z=0 and y at z=end of a tube, in which the chemicals at the beginning of the added file are flowing along and reacting according to the reaction laws r. The parameters k,K, EA and dH are supposed to be estimated, while the others are externally given. They are estimated by simulating the equation dy/dz with the ode45 solver in matlab. This is written in the file f which is used by the lsqnonlin command mentioned in the code above.

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#### Dason

What exactly do you hope to achieve by doing a PCA on your estimated parameters?

#### Entropie_13

##### New Member
I would like to reveal whether there is collinearity between parameters, due to the large number of coefficients and their multiplicative and additive relationsships. The large K_s in the denominator of r and E_A have shown low t-values when estimating the model with my experimental values of y at z=0 and z= end of tube. I therefore think I can reduce the number of parameters or make the equations more simple.

#### Entropie_13

##### New Member
I dont know if my formulation "PCA on the estimated parameters" is correct. In the linear model we do PCA on the Covariance Matrix of the observabels because we want eigenvectors of the Variance of the derivative of y with respect to a parameter. This derivative is the covariance matrix of the observables, but I guess in my nonlinear model the terms in this matrix include observables and other parameters from the derivative of y after a parameter.