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I want to apply the weighted least squares method in order to identify parameters of a dynamic process. The process is described by a second order differential equation of the form:

$$ \ddot{y}+a_1\dot{y}+a_2y=b_0u $$

where $\ y,u $ are the output and input of the process respectively and are measured as well. The unknown parameters $\ a_1,a_2,b_0 $ are those that have to be identified by processing the input-output data of the system. The process can also be written in the form:

$$ y = [a_1 \ a_2 \ b_0]Ψ \rightarrow y=\theta^{*T}Ψ $$

where $\ Ψ $ is computed through the input-output data of the system.

I have already implemented the Ordinary Least Squares (OLS) method given by the formula:

$$ \hat{\theta}= (Ψ^ΤΨ)^{-1}Ψ^Τy $$

so the matrix $\ Ψ $ is computed and $\ \hat{\theta}=[\hat{a_1} \ \hat{a_2} \ \hat{b_0}] $ are the estimates of the parameters. In order to implemented the weighted method I need to apply the formula:

$$ \hat{\theta}= (Ψ^ΤWΨ)^{-1}Ψ^ΤWy $$

where $\ W $ is a diagonal matrix. I have searched some literature and found out that the "best" choice of $\ W $ is $\ W = (E[ee^T])^{-1} $ where $\ e = y - \hat{y} $ ($\ \hat{y} $ is the estimated output of the process). However, the covariance matrix of the error equation is not known and this form of $\ W $ can't be obtained and used. I also haven't found any good guidance (or an algorithmic one) in order to properly compute the weights' matrix. Is there an algorithmic way by using the vectors $\ y,u $ and the matrix $\ Ψ $ in order to properly compute $\ W $ and apply the WLS method ?

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Weighted least squares is a method to induce a-priori knowledge into the ordinary least-squares procedure. The basic example is a simple scaling of the dimensions, such that each predictor variable has about the same variance (which is up to a factor also what you found in your reference).

In your case, you would use that if the parameters $y$, $\dot y$, and $ \ddot y$ deviated by several orders of magnitude. However, you would do that then before the least squares (which the induces the diagonal matrix $W$).

In my opinion, however, it is not useful to use the weighting parameters itself as a means to fit the model (as you seem to try here). When it's just about fitting, there are other models which are fitted more easily and give a better result. For example, have a look at ridge regression, the Lasso and all those other non-linear methods such as neural networks and so on.

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