Combinatorial Optimization: techniques to reduce variance of a series of angle values
$begingroup$
I am working with a series of angle value $Theta = [theta_{0} ~ theta_{1} cdots theta_{N-1}]$. Each $theta_{n}$ has a value belonging in $[-pi;pi]$.
My problem is that I would like to reduce the variance of the set $Theta$, knowing that the angle value can have an ambiguity of $pmpi$. Or in other words, each angle in the series can be substituted by its 'conjugate' $theta + pi$ or $theta - pi$, as long as it stays within the range.
I know this is a combinatorial optimization process, and I have tried a rather crude approach using this procedure:
(1) Start by selecting $theta_{0}$ as it is (i.e. no conjugations),
(2) For each subsequent $theta_{n}, n in [1;N-1]$, select the best conjugate of the two possible that minimizes the variance $mathrm{Var}[~[theta_{0} cdots theta_{n}]~]$
This can be thought of as adding a new sample to a series $tilde{Theta}$ which was seeded with $theta_{0}$, and this new sample should produce the smallest variance $mathrm{Var}[tilde{Theta}]$ of the two possible sample values.
I noticed that this approach does not totally minimize the variance. So, are there any (proper) techniques to actually solve this problem? A hint of where to start is also appreciated.
Thanks,
optimization combinations
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
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add a comment |
$begingroup$
I am working with a series of angle value $Theta = [theta_{0} ~ theta_{1} cdots theta_{N-1}]$. Each $theta_{n}$ has a value belonging in $[-pi;pi]$.
My problem is that I would like to reduce the variance of the set $Theta$, knowing that the angle value can have an ambiguity of $pmpi$. Or in other words, each angle in the series can be substituted by its 'conjugate' $theta + pi$ or $theta - pi$, as long as it stays within the range.
I know this is a combinatorial optimization process, and I have tried a rather crude approach using this procedure:
(1) Start by selecting $theta_{0}$ as it is (i.e. no conjugations),
(2) For each subsequent $theta_{n}, n in [1;N-1]$, select the best conjugate of the two possible that minimizes the variance $mathrm{Var}[~[theta_{0} cdots theta_{n}]~]$
This can be thought of as adding a new sample to a series $tilde{Theta}$ which was seeded with $theta_{0}$, and this new sample should produce the smallest variance $mathrm{Var}[tilde{Theta}]$ of the two possible sample values.
I noticed that this approach does not totally minimize the variance. So, are there any (proper) techniques to actually solve this problem? A hint of where to start is also appreciated.
Thanks,
optimization combinations
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
$endgroup$
add a comment |
$begingroup$
I am working with a series of angle value $Theta = [theta_{0} ~ theta_{1} cdots theta_{N-1}]$. Each $theta_{n}$ has a value belonging in $[-pi;pi]$.
My problem is that I would like to reduce the variance of the set $Theta$, knowing that the angle value can have an ambiguity of $pmpi$. Or in other words, each angle in the series can be substituted by its 'conjugate' $theta + pi$ or $theta - pi$, as long as it stays within the range.
I know this is a combinatorial optimization process, and I have tried a rather crude approach using this procedure:
(1) Start by selecting $theta_{0}$ as it is (i.e. no conjugations),
(2) For each subsequent $theta_{n}, n in [1;N-1]$, select the best conjugate of the two possible that minimizes the variance $mathrm{Var}[~[theta_{0} cdots theta_{n}]~]$
This can be thought of as adding a new sample to a series $tilde{Theta}$ which was seeded with $theta_{0}$, and this new sample should produce the smallest variance $mathrm{Var}[tilde{Theta}]$ of the two possible sample values.
I noticed that this approach does not totally minimize the variance. So, are there any (proper) techniques to actually solve this problem? A hint of where to start is also appreciated.
Thanks,
optimization combinations
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
$endgroup$
I am working with a series of angle value $Theta = [theta_{0} ~ theta_{1} cdots theta_{N-1}]$. Each $theta_{n}$ has a value belonging in $[-pi;pi]$.
My problem is that I would like to reduce the variance of the set $Theta$, knowing that the angle value can have an ambiguity of $pmpi$. Or in other words, each angle in the series can be substituted by its 'conjugate' $theta + pi$ or $theta - pi$, as long as it stays within the range.
I know this is a combinatorial optimization process, and I have tried a rather crude approach using this procedure:
(1) Start by selecting $theta_{0}$ as it is (i.e. no conjugations),
(2) For each subsequent $theta_{n}, n in [1;N-1]$, select the best conjugate of the two possible that minimizes the variance $mathrm{Var}[~[theta_{0} cdots theta_{n}]~]$
This can be thought of as adding a new sample to a series $tilde{Theta}$ which was seeded with $theta_{0}$, and this new sample should produce the smallest variance $mathrm{Var}[tilde{Theta}]$ of the two possible sample values.
I noticed that this approach does not totally minimize the variance. So, are there any (proper) techniques to actually solve this problem? A hint of where to start is also appreciated.
Thanks,
optimization combinations
optimization combinations
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
asked Dec 10 '18 at 22:17
KaiserHazKaiserHaz
367
367
add a comment |
add a comment |
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