Hi y'all, wondering if you could help me with this. I have a data set with a linear relationship between the independent and dependent variables. Both the depended and independent variables have error due to measurement and this error is not constant. For example, {x1, x2, x3, x4, x5} {y1, y2, y3, y4, y5} {dx1, dx2, dx3, dx4, dx5} {dy1, dy2, dy3, dy4, dy5} where one data point would be (x1±dx1, y±dy1), and so on. Assuming the relationship is of the form, y = ax + b, I need both the best value for a, and its uncertainty, (a ± da). I've been scouring the internet for more information on total least squares methods, and generalized method of moments, etc. but I can't find something that works for the case where the error in x and y is just some arbitrary value, like in my case. helpful hints?
I think what you are trying to say is that the variance of the distribution of the errors is not constant with respect to X and Y. You must define what you mean by "best". I'll try to put some words in your mouth. We want the line y = ax + b that minimizes the expected error between data points and the line, when we average these errors over the whole line between X = (some minimum value of interest) and X = (some maximum value of interest), giving all those parts of the line equal weight in this averaging. The error between a data point (x_i,y_i) and the line will be measured by the perpendicular distance (x_i, y_i) and the the line. Let's try to define what you mean by "something that works". Do you mean a computer program that could (by trial and error if necessary) estimate the line? Or do you require some symbolic formula that you can use in a math paper? I assume you are talking about the variances of the errors at various values of (x,y). What exactly do you know about this? For example, if we have a data point (10.0, 50.2), do you have a lot data with similar values, so that we can estimate the variance in X and Y around the value (10.0,50.2)? Or do you only have data with widely separated X and Y values and are basing your assertion that the variances in the errors change with X and Y because of the overall scattered appearance of the data?
Sounds like you have a combination of 'weighted' and 'total / orthogonal' least squares. Following link to MIT lecture on weighted least squares might be helpful: http://academicearth.org/lectures/weighted-least-squares
Look at the Wikipedia article on Total Least Squares http://en.wikipedia.org/wiki/Total_least_squares. I've only scanned the article myself, but it looks like what you want. It has an example written in Octave, which is a free Matlab work-alike.