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I have $N$ data points that I am trying to fit using a function of the form

$y_i = \prod_j {X_{i,j}}^{b_j}, \quad j=1..N$

where $\mathbf X$ and $\mathbf y$ are measured values. The form of this function is motivated from physics. I also have a variance-covariance matrix containing the estimated uncertainties of $\mathbf X$ and $\mathbf y$, which I assume to be distributed according to a multivariate normal distribution. Finally, $\mathbf b$ are the exponents I would like to determine.

I am not sure how to handle the fact there are uncertainties in the $\mathbf X$ variables and furthermore that they are correlated with the errors in $\mathbf y$. If the errors in $\mathbf X$ did not exist then I would simply find the $\mathbf b$ that minimize

$\chi^2 = \sum_i \frac{(y_i - \prod_j X_{i,j}^{b_j})^2}{\sigma_y^2}$

What should I do in this case? Is there a similarly simple functional form that I can optimize?

It is not lost on me that I can take the log of both sides and it looks like a standard regression problem, except without an intercept. Is this a useful line to pursue?

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