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When you're fitting a model $y=f(x)$ to data (${x_i, y_i}$) with errorbars on both the independent ($x$) and response ($y$) variables, it's standard that you can define an 'effective variance' when calculating the chi-squared

$$(\delta y_i)^2 + \left(\delta x_i \left(\frac{df}{dx}\right)_{x=x_i} \right)^2$$

Here the differential is evaluated at the value of $x_i$ of interest.

My question is if this can be extended to the case where there are two independent variables $(x, z)$, so that you're fitting a function $y=g(x, z)$. Something like:

$$(\delta y_i)^2 + \left(\delta x_i \left(\frac{dg}{dx}\right)_{x=x_i} \right)^2 + \left(\delta z_i \left(\frac{dg}{dz}\right)_{z=z_i} \right)^2$$

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  • $\begingroup$ Is f a function? If so, a function of what. Can you explain what dy, dx are. I assume df/dx is a derivative of the function f. The formula here ned.ipac.caltech.edu/level5/Leo/Stats7_3.html is similar but has differences. Please be more precise $\endgroup$
    – martino
    Commented May 9, 2014 at 15:27
  • $\begingroup$ I've edited some things to hopefully make it clearer. The link you've supplied is the one variable case. $\endgroup$
    – Gremlin
    Commented May 12, 2014 at 12:50
  • $\begingroup$ Where is effective variance standard? In which field of study? $\endgroup$
    – Aksakal
    Commented May 12, 2014 at 12:57
  • $\begingroup$ I'm not sure how to answer that. It's a common tool used in model fitting. Here's a paper - www.jrossmacdonald.com/jrmpubs/192LSFITTING.pdf $\endgroup$
    – Gremlin
    Commented May 12, 2014 at 13:10

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