When solving an overdetermined linear system, is it possible to weight the influence of each equation?
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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.
linear-algebra
asked Mar 12 '14 at 15:55
viypsviyps
889
$endgroup$
add a comment |
$begingroup$
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.
linear-algebra
asked Mar 12 '14 at 15:55
viypsviyps
889
$endgroup$
$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36
add a comment |
$begingroup$
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.
linear-algebra
asked Mar 12 '14 at 15:55
viypsviyps
889
$endgroup$
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.
linear-algebra
linear-algebra
asked Mar 12 '14 at 15:55
viypsviyps
889
asked Mar 12 '14 at 15:55
viypsviyps
889
edited Mar 12 '14 at 16:35
viyps
asked Mar 12 '14 at 15:55
viypsviyps
889
asked Mar 12 '14 at 15:55
viypsviyps
889
asked Mar 12 '14 at 15:55
viypsviyps
889
889
$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36
add a comment |
$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36
$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
$endgroup$
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
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active
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votes
$begingroup$
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.
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
$endgroup$
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
$endgroup$
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
$endgroup$
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.
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
edited Dec 8 '18 at 12:35
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
answered Mar 12 '14 at 16:59
Matt L.Matt L.
8,899822
8,899822
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
add a comment |
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Perfect answer. Do you know any specific bibliographic reference for this?
$endgroup$
– viyps
Mar 12 '14 at 17:18
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
$begingroup$
Check out "weighted least squares": en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
$endgroup$
– Matt L.
Mar 12 '14 at 19:31
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
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.
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
answered Mar 12 '14 at 16:52
Ross MillikanRoss Millikan
294k23198371
294k23198371
add a comment |
add a comment |
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$begingroup$
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)...
$endgroup$
– AlexR
Mar 12 '14 at 16:04
$begingroup$
@AlexR question edited...
$endgroup$
– viyps
Mar 12 '14 at 16:36