I have an overdetemined system of linear equation and want to minimize overall error. Up...

Caleb Proctor

Caleb Proctor

Answered

2022-07-10

I have an overdetemined system of linear equation and want to minimize overall error. Up to now, not a problem, I could use least squares. The problem is that I know that some equations in my system are more uncertain, while others are exact. Actually, I have a number of equations with different confidence levels ("low confidence","medium confidence", "high confidence" and so on). In a AX=B system, the solution should take this into account and keep unchanged the B coefficients of the "high confidence" equations, while the B coefficients of "low confidence" equations could be changed more drastically than the B coefficients of "mid confidence" equations.

Answer & Explanation

Elianna Wilkinson

Elianna Wilkinson

Expert

2022-07-11Added 11 answers

an efficient way to give more or less weight to equations in an overdetermined system is to rescale them; that is, multiply both A and b by a diagonal matrix W on the left. Here is an example:
{ x + y = 10 x + 2 y = 14 x + 3 y = 40
Solve this system using least squares,
A = ( 1 1 1 2 1 3 ) , b = ( 10 14 30 )
and got x = 2, y = 10. The right hand side for this solution is ( 8 , 18 , 28 ) T .
But suppose the first equation is very important / certain, while the last one is the least important. If I let W be the diagonal matrix with entries ( 4 , 1 , 1 / 2 ), the result is x = 2.84, y = 7.07. The right hand side for this solution is ( 9.91 , 16.98 , 24.05 ) T . So, the first equation is satisfied almost exactly, while the last equation is pretty far from target.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?