Combining observed Gaussian error with common fractional model error

Question

I'm currently trying to incorporate model errors into a likelihood function to fit the model to some results. Let's just call them $x_{m,i}$ for the model and $x_{o,i}$ for the $N$ observations, indexed by $i$.

Each observation has a given Guassian/normally-distributed error $\sigma_{o,i}$. For the model, I know that, in any given model, all the $x_{m,i}$ have the same fractional error. That is, a single fractional error $\sigma_f$ is (for now...) Gaussian/normally-distributed, and each observation has error $\sigma_f\times x_{i,m}$. How do I incorporate this into a likelihood function for fitting the model?

I'm not sure how much detail is needed for this to make sense, but basically I'm trying to do a $\chi^2$-minimization with

$$\chi^2=\sum_{i=1}^N\frac{(x_{o,i}-x_{m,i})^2}{\sigma_{o,i}^2+\sigma_{m,i}^2}$$

but I'm not sure what to do for $\sigma_{m,i}$. My instinct is that just putting in $\sigma_{m,i}=\sigma_fx_{m,i}$ is wrong, because that still allows the model errors to be, in some sense, drawn independently, even though there's only really one random process for the fractional model error, rather than an independent process for each observation.

I tried to formulate something for $\sigma_{m,i}$ with propagation of error by writing

$$\sigma_{\text{total},i}^2=\left(\frac{\partial(x_{o,i}-x_{m,i})}{\partial x_{o,i}}\right)^2\sigma_{o,i}+\left(\frac{\partial(x_{o,i}-x_{o,i}\frac{x_{m,i}}{x_{o,i}})}{\partial \frac{x_{m,i}}{x_{o,i}}}\right)^2\sigma_{f}^2 \\=\sigma_{o,i}^2+x_{o,i}^2\sigma_f^2$$ but, as you can see, that just got me back to this answer I don't believe.

I suspect a better way of thinking about it is in terms of a perfectly correlated error component. Well, this is what I'll think about next, anyway!

Thanks in advance for any advice. I'm not a statistician, nor particularly well-versed in statistics, so I apologize if my description is incomprehensible to an expert. I'll try to clarify if necessary.

Answer 1

My solution is to formulate the common fractional error as a perfectly correlated error between the parameters. To calculate the corresponding covariance matrix, it's easiest to start from the correlation matrix. Because everything is perfectly correlated, this should be everywhere one. So, by definition,

$$\rho_{i,j}=\frac{\sigma_{i,j}}{\sqrt{\sigma_{i,i}\sigma_{j,j}}}=1$$

Now, we also know that the fractional error is constant, so $\sigma_{i,i}=\sigma_fx_i$. Thus,

$$\sigma_{i,j}=\sqrt{\sigma_{i,i}\sigma_{j,j}}=\sigma_f\sqrt{x_ix_j}$$

This defines the new covariance matrix, which can be added to the (diagonal) covariance matrix of the uncorrelated observed errors.

I think that if one writes out the propagation of error formula properly, with all the observations, then one should recover this formula...