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Linear regression, error in both variables

by fhqwgads2005
Tags: error, linear, regression, variables
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fhqwgads2005
#1
Apr10-11, 11:41 PM
P: 23
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 (x1dx1, ydy1), 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?
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Stephen Tashi
#2
Apr11-11, 11:06 AM
Sci Advisor
P: 3,252
Quote Quote by fhqwgads2005 View Post
this error is not constant.
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.

y = ax + b, I need both the best value for a, and its uncertainty, (a da).
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.

I but I can't find something that works
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?

for the case where the error in x and y is just some arbitrary value, like in my case.
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?
hotvette
#3
Apr11-11, 12:22 PM
HW Helper
P: 925
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/we...-least-squares

elhef
#4
Jun2-11, 06:50 PM
P: 1
Linear regression, error in both variables

I was hoping to find help on the same topic! Any ideas?

Quote Quote by Stephen Tashi View Post
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.



Do you mean a computer program that could (by trial and error if necessary) estimate the line?
yes and yes.

thanks in advance
Stephen Tashi
#5
Jun3-11, 12:42 AM
Sci Advisor
P: 3,252
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.


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