# Error propagation in a linear model

I am currently interested in learning more on error propagation. At the moment I am trying to find out how to calculate the uncertainty of a value that is obtained from a linear model. For the linear model, the uncertainty of the slope and of the intercept with the x-axis are given. I have not found any useful explanation or I did not recognize one, because they were too complex for me. The Model would be:

y = s * (x-X0)


where:

s=slope (=0.9)
x=value on the x-axis (=25)
X0=intercept with the x-axis (= 5)


I want to know dy, the uncertainty of y. I know that:

ds = uncertainty of s (= 0.01)
dx = uncertainty of the actual x-value (= 2)
dX0 = uncertainty of the intercept with the x-axis (= 1.5)


This appears to me as a simple problem, so it should not necessary to solve the model with all possible values s, x and X0 could have. Or am I wrong here? I also appreciate recommendations on text-books, if you have any. Kind regards, S.

## 2 Answers

Yes you are correct.

For other readers, there are a number of clear summaries of how to propagate errors through a linear system.

Here are three of the best ones I found:

Okay, I eventually figured it out by myself: The possible errors of the components have to be propagated separately and then summed up:

model error=slope error+value error+intercept error


This looks like this:

dy = (ds*(x-X0)) + (s*dx) + (s*dX0)


So this would yield for my example:

dy=(0.01*(25-5)) + (0.9*2) + (0.9*1.5)


thus:

dy=3.35


This is consistent with the value I would obtain when solving the linear model with all possible combinations of values (results range between 16.333 and 19.684).