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Currently, I am calculating the pseudo-inverse to solve the system, which is simple and fast for my purpose (with a computer, obviously). However, each equation represents an individual from a population. For reasons outside the scope of this site, I need to give more importance to some individuals than others.

E.g., consider a total of $N$ individuals. $0.1N$ would be special and have weight $2$. The only way I can think of to accomplish that is to duplicate the $0.1N$ corresponding equations. Since I am no math expert, I don't know if it is effective or with no effect at all. If equation duplication is effective, ok, that's a solution, but with a minor problem: it does not allow to use non-integer weights.

obs.: I don't know for sure, but the answer for non-overdetermined linear systems might also apply.

dawid
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  • What is your goal and what do you mean by "weight"? You can, of course minimise another quantifier for the error other than sum of errors squared (the usual approach for an underdetermined system)... – AlexR Mar 12 '14 at 16:04
  • @AlexR question edited... – dawid Mar 12 '14 at 16:36

2 Answers2

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If you have an overdetermined system of linear equations

$$Ax=b$$

you can weigh the error with a diagonal weight matrix $W$

$$e=W(Ax-b)$$

This matrix assigns different weights to each equation. Now you can solve the problem by minimizing

$e^Te=(Ax-b)^TW^TW(Ax-b)$

which gives you a system of linear equations for $x$:

$$A^TW^TWAx=A^TW^TWb$$

This system can be solved for $x$ assuming matrix $A$ has full rank.

Matt L.
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You can use a linear or non-linear fitting package. Normally these have a way to specify the measurement error of each point. Points with lower measurement error are weighted more heavily. A discussion is in chapter 15 of Numerical Recipes as well as most numerical analysis texts.

Ross Millikan
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