I came across definition of covariance matrix that is defined from best fit error equation. I would like to clarify correctnes of procedure.

We define equation for error: \begin{equation}\label{eq:fitgood} \chi^2 = \sum _{\text{i=0}}^\text{N} \frac{1}{\sigma_\text{i}^2}( I_\text{i}-I_\text{D}(\theta_\text{i}, \textbf{S}) )^2 \end{equation}

where: $I_\text{i}$ is measured signal and $I_\text{D}$ predicted signal. $\sigma_\text{i}^2$ is variance of measured signal. It is calculated from repeated measurements.

Model is defined as: \begin{equation}\label{eq:enacbafurr} I_\text{D}(\theta_\text{i}, \textbf{S}) = \frac{1}{2}( S_0 - S_1 \,\text{cos}^22\theta_i - S_2 \, \text{sin}2\theta_i \text{cos}2\theta_i + S_3 \, \text{sin}2\theta_i ) \end{equation}

Covariance $\textbf{C}$ of parameters $\textbf{S}$ is than calculated as: \begin{equation}\label{eq:curvaturmtx} \alpha_\text{k,l} = \frac{1}{2} \bigg( \frac{\partial \chi^2}{\partial S_\text{k} \partial S_\text{l}} \bigg)_\textbf{S} \end{equation} \begin{equation}\label{eq:covariance} \textbf{C} = \alpha^{-1} \end{equation}

I would like to know if there are limitations of using this procedure. I would be really pleased if someone can redirect me to some literature where this procedure is described. I would like to know why this is true and not just blindly follow the recipe.

Another thing. I know that from LSE method perspective covariance matrix could be calculated as:

\begin{equation} \text{cov}{\textbf{(S)}} = \sigma^2(\textbf{X}^T\textbf{X})^{-1} \end{equation}

where: \begin{equation} \sigma^2 = \frac{e^Te}{N-n} \end{equation}

but I was unable to calculate it this way. Can someone tell me the approach I should take to modify equations properly in the form of matrices. Thank you in advance!


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