Least squares fitting where we pre-determine the slope

Hello all,

I have a set of data, say on a scatter plot, and I have a line that has a pre-determined slope (i.e. not determined by the data on the scatter plot but determined by ideal calculations). I would like to move the line through the data until it has the least amount of offset (like least-squares fitting) but without changing the slope of the line. Is there a least-squares fitting function that lets you pre-determine the slope?

Cheers!

Answer 1

[Ameer Hamza]

Yes, you can do that following the pattern of the linear regression fitting with little modification. In linear regression, we fit the equation

a*x+b = y

and in MATLAB we write it as

[x_vector ones(size(x_vector))]\y_vector

to get a and b. But since you already know slope a, your equation become

b = y-a*x

so in MATLAB use

ones(size(x_vector))\[y_vector-a*x_vector]

it will give you the value of b which minimize the least square error.

[Jeff Miller]

Or perhaps slightly more intuitively, in MATLAB use

b_estimated = mean(y-a*x);

[John D'Errico]

As Jeff points out, if the slope is known, then the constant term is simply the mean of y-a*x. That is indeed the least squares estimate. There is no need to use backslash here.

[Barbara Wortham]

Thanks everyone!

[Curtis Baden]

Thank you for the thorough, concise explanation! I'm hoping to perform a similar calculation, except I'd like to employ a weighted least squares regression (using inverse variances associated with my dependent variable as weights). How will this calculation change in this case?

[Ameer Hamza]

For weighted least square you can use lscov(). In term of the backslash operator, you can use the following

(X.'*W*X)\(X.'*W*y)

where W is a diagonal matrix containing the weights.