# Regression model with heteroskedasticity in both variables

I've been learning (lurking) from this site for a while and I finally have a question I haven't seen answered yet.

I'm doing a flight test and trying to fit the resulting data to linear line. From a combination of ~14 different sensors, I can calculate the 2 values of interest. From uncertainty propagation, I can find the uncertainty in these 2 values (call them x and y), based on the uncertainty in any of the 14 sensors. So, for any point, there will be error in both x and y, and these errors will not be the same in x and y (I think about it as an ellipse). Also, as x and y change, the error in x and y will change, and the ratio will not remain constant. My understanding is these two conditions are called error-in-variables and heteroscedasticity, respectively.

So for any given point, I'll have an expected value and error in both x and y. I'm interested in fitting a curve to this data, and I'm having trouble finding regression models that can handle both error-in-variables and heteroscedasticity. I could use some advice on models or good books for engineers that might help.

I'm also interested in figuring out how good each of the linear coefficients are. I don't think I will be able to get very good accuracy for any given point, but I will have 10,000 to 100,000 points. Is there a way to leverage this fact to get an accurate fit, even though any single point isn't accurate?

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I think I follow, but "I'll have an expected value and error in both x and y" is tripping me up: are you saying that each datum $(x_i,y_i)$ has a distribution with known expectation $(\xi_i, \eta_i)$ and known covariance matrix $\Sigma_i$? It seems so; but if that's the case, you hardly need to do any curve fitting! Perhaps the $(\xi_i, \eta_i)$ are unknown and need to be estimated? – whuber Feb 20 '13 at 22:52
i think you need to solve heterscedasticity first through transformation maybe. – user21085 Feb 21 '13 at 0:17
Your biggest problem seems to be - if you have errors in $x$ - no matter if heteroskedastic or homo - then ordinary regression assumptions are violated. You need something like "errors in variables"/"Model II" regression models. At that point, you might want to look at something like transformation to obtain homoskedasticity. – Glen_b Feb 21 '13 at 0:21
@whuber: Doing an uncertainty propagation I have an estimate of (ξi,ηi) for any (xi,yi). Why do you say I don't need curve fitting? I'm an engineer and statistics is black magic to me so forgive me if that is a dumb question. – achase90 Feb 22 '13 at 1:10
When the expectation is known, you have already specified the curve, so there is nothing left to be done. I suspect you have mis-stated what you know. An expectation is not an estimate; it's a true value (but usually unknown). If you really mean "estimate" instead of "expectation," then it's crucial that you explain how you make that estimate and what data it is based on. Similarly, you need to explain how you know the errors in $x$ and $y$ and what data that knowledge is based on. – whuber Feb 22 '13 at 15:11