Combinatorial Optimization: techniques to reduce variance of a series of angle values












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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,










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    $begingroup$


    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,










    share|cite|improve this question









    $endgroup$















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      0








      0





      $begingroup$


      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,










      share|cite|improve this question









      $endgroup$




      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






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      asked Dec 10 '18 at 22:17









      KaiserHazKaiserHaz

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