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How to generate a data sample along a annulus (between fixed radii)?


How to understand the drawbacks of K-meansSimulate a uniform distribution on a discGenerate covariance matrix with fixed values in certain cellsAdvice on comparing methods of generating random vectorsPyMC for nonparametric clustering: Dirichlet process to estimate Gaussian mixture's parameters fails to clustersampling for estimation - using random numbers - homework helpGenerate binomial sample with (pretty) exact probabilitySimulate a uniform distribution on a discGenerate synthetic data to match sample dataGenerate sample data from Gaussian mixture modelhow to compare different sets of time series dataGood choices of PRNG for uniform circular (directional) data?






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;








2












$begingroup$


I would like to create a random data sample to test a clustering algorithm using python. One specific data sample I would like to generate consists of two non-overlapping annuluses (annuli?); however, it's okay if the inner-most annuli is a circle - as in the image below (taken from this question).



example data



I realize that I can sample N values of angles ti such that 0 ≤ ti ≤ 2 pi. Based on some reading online, I think one can generate N samples of circular data by specifying a fixed radius r such that xi = r cos(ti) and yi = r sin(ti). Applying this reasoning to the desired annulus, I am thinking one can repeat the process for variable radii rmin and rmax; however, I am not sure how to proceed from this point. My thinking is to generate n samples of radii ri (where n < N, the annulus cluster is a subset of all the data) such that rmin ≤ ri ≤ rmax; doing so necessarily means the first cluster should be the n points within the the outer-most annulus and the second cluster should be the N-n points within the inner-most annulus/circle.



So my questions are as follows:



1) Is my approach so far reasonable? Are there (dis)advantages to this approach, such as clustering more near centers or borders?



2) If my approach is wrong, why is it wrong and what would be a better approach?



EDIT:



The original post that serves as a potential duplicate actually shows how to sample points within a circle, not an annulus. I am concerned with the case of an annulus.






python random-generation circular-statistics







share|cite|improve this question











$endgroup$











  • $begingroup$
    I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
    $endgroup$
    – Stephan Kolassa
    45 mins ago






  • 1




    $begingroup$
    Possible duplicate of Simulate a uniform distribution on a disc
    $endgroup$
    – Stephan Kolassa
    39 mins ago










  • $begingroup$
    That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
    $endgroup$
    – allthemikeysaretaken
    39 mins ago






  • 1




    $begingroup$
    Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
    $endgroup$
    – Stephan Kolassa
    37 mins ago










  • $begingroup$
    Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
    $endgroup$
    – allthemikeysaretaken
    34 mins ago

















2












$begingroup$


I would like to create a random data sample to test a clustering algorithm using python. One specific data sample I would like to generate consists of two non-overlapping annuluses (annuli?); however, it's okay if the inner-most annuli is a circle - as in the image below (taken from this question).



example data



I realize that I can sample N values of angles ti such that 0 ≤ ti ≤ 2 pi. Based on some reading online, I think one can generate N samples of circular data by specifying a fixed radius r such that xi = r cos(ti) and yi = r sin(ti). Applying this reasoning to the desired annulus, I am thinking one can repeat the process for variable radii rmin and rmax; however, I am not sure how to proceed from this point. My thinking is to generate n samples of radii ri (where n < N, the annulus cluster is a subset of all the data) such that rmin ≤ ri ≤ rmax; doing so necessarily means the first cluster should be the n points within the the outer-most annulus and the second cluster should be the N-n points within the inner-most annulus/circle.



So my questions are as follows:



1) Is my approach so far reasonable? Are there (dis)advantages to this approach, such as clustering more near centers or borders?



2) If my approach is wrong, why is it wrong and what would be a better approach?



EDIT:



The original post that serves as a potential duplicate actually shows how to sample points within a circle, not an annulus. I am concerned with the case of an annulus.






python random-generation circular-statistics







share|cite|improve this question











$endgroup$











  • $begingroup$
    I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
    $endgroup$
    – Stephan Kolassa
    45 mins ago






  • 1




    $begingroup$
    Possible duplicate of Simulate a uniform distribution on a disc
    $endgroup$
    – Stephan Kolassa
    39 mins ago










  • $begingroup$
    That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
    $endgroup$
    – allthemikeysaretaken
    39 mins ago






  • 1




    $begingroup$
    Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
    $endgroup$
    – Stephan Kolassa
    37 mins ago










  • $begingroup$
    Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
    $endgroup$
    – allthemikeysaretaken
    34 mins ago













2












2








2





$begingroup$


I would like to create a random data sample to test a clustering algorithm using python. One specific data sample I would like to generate consists of two non-overlapping annuluses (annuli?); however, it's okay if the inner-most annuli is a circle - as in the image below (taken from this question).



example data



I realize that I can sample N values of angles ti such that 0 ≤ ti ≤ 2 pi. Based on some reading online, I think one can generate N samples of circular data by specifying a fixed radius r such that xi = r cos(ti) and yi = r sin(ti). Applying this reasoning to the desired annulus, I am thinking one can repeat the process for variable radii rmin and rmax; however, I am not sure how to proceed from this point. My thinking is to generate n samples of radii ri (where n < N, the annulus cluster is a subset of all the data) such that rmin ≤ ri ≤ rmax; doing so necessarily means the first cluster should be the n points within the the outer-most annulus and the second cluster should be the N-n points within the inner-most annulus/circle.



So my questions are as follows:



1) Is my approach so far reasonable? Are there (dis)advantages to this approach, such as clustering more near centers or borders?



2) If my approach is wrong, why is it wrong and what would be a better approach?



EDIT:



The original post that serves as a potential duplicate actually shows how to sample points within a circle, not an annulus. I am concerned with the case of an annulus.






python random-generation circular-statistics







share|cite|improve this question











$endgroup$




I would like to create a random data sample to test a clustering algorithm using python. One specific data sample I would like to generate consists of two non-overlapping annuluses (annuli?); however, it's okay if the inner-most annuli is a circle - as in the image below (taken from this question).



example data



I realize that I can sample N values of angles ti such that 0 ≤ ti ≤ 2 pi. Based on some reading online, I think one can generate N samples of circular data by specifying a fixed radius r such that xi = r cos(ti) and yi = r sin(ti). Applying this reasoning to the desired annulus, I am thinking one can repeat the process for variable radii rmin and rmax; however, I am not sure how to proceed from this point. My thinking is to generate n samples of radii ri (where n < N, the annulus cluster is a subset of all the data) such that rmin ≤ ri ≤ rmax; doing so necessarily means the first cluster should be the n points within the the outer-most annulus and the second cluster should be the N-n points within the inner-most annulus/circle.



So my questions are as follows:



1) Is my approach so far reasonable? Are there (dis)advantages to this approach, such as clustering more near centers or borders?



2) If my approach is wrong, why is it wrong and what would be a better approach?



EDIT:



The original post that serves as a potential duplicate actually shows how to sample points within a circle, not an annulus. I am concerned with the case of an annulus.






python random-generation circular-statistics




python random-generation circular-statistics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 35 mins ago









Stephan Kolassa

49.4k8103187




49.4k8103187










asked 48 mins ago









allthemikeysaretakenallthemikeysaretaken

134




134











  • $begingroup$
    I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
    $endgroup$
    – Stephan Kolassa
    45 mins ago






  • 1




    $begingroup$
    Possible duplicate of Simulate a uniform distribution on a disc
    $endgroup$
    – Stephan Kolassa
    39 mins ago










  • $begingroup$
    That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
    $endgroup$
    – allthemikeysaretaken
    39 mins ago






  • 1




    $begingroup$
    Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
    $endgroup$
    – Stephan Kolassa
    37 mins ago










  • $begingroup$
    Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
    $endgroup$
    – allthemikeysaretaken
    34 mins ago
















  • $begingroup$
    I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
    $endgroup$
    – Stephan Kolassa
    45 mins ago






  • 1




    $begingroup$
    Possible duplicate of Simulate a uniform distribution on a disc
    $endgroup$
    – Stephan Kolassa
    39 mins ago










  • $begingroup$
    That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
    $endgroup$
    – allthemikeysaretaken
    39 mins ago






  • 1




    $begingroup$
    Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
    $endgroup$
    – Stephan Kolassa
    37 mins ago










  • $begingroup$
    Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
    $endgroup$
    – allthemikeysaretaken
    34 mins ago















$begingroup$
I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
$endgroup$
– Stephan Kolassa
45 mins ago




$begingroup$
I'm a bit confused. You write of "two non-overlapping disks", but the innermost disk can be a "circle". Circles are one-dimensional. Your actual picture shows an annulus which is concentric with a disk. If this is what you actually want, please edit your post to clarify. Thank you!
$endgroup$
– Stephan Kolassa
45 mins ago




1




1




$begingroup$
Possible duplicate of Simulate a uniform distribution on a disc
$endgroup$
– Stephan Kolassa
39 mins ago




$begingroup$
Possible duplicate of Simulate a uniform distribution on a disc
$endgroup$
– Stephan Kolassa
39 mins ago












$begingroup$
That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
$endgroup$
– allthemikeysaretaken
39 mins ago




$begingroup$
That was bad terminology on my part. I edited the post to reflect that I would like to sample points within an annulus (not a disk).
$endgroup$
– allthemikeysaretaken
39 mins ago




1




1




$begingroup$
Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
$endgroup$
– Stephan Kolassa
37 mins ago




$begingroup$
Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk. whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus.
$endgroup$
– Stephan Kolassa
37 mins ago












$begingroup$
Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
$endgroup$
– allthemikeysaretaken
34 mins ago




$begingroup$
Is my approach of varying the radius between rmin and rmax the proper way to "constrain the part where rho is generated"?
$endgroup$
– allthemikeysaretaken
34 mins ago










1 Answer
1






active

oldest

votes


















1












$begingroup$

Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk.



whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus. Note that you need to constrain "on the square root scale".



Here is an adoption of whuber's original R code, which should be easy to translate to Python:



outer_radius <- 1
inner_radius <- 0.7
n <- 1e4
rho <- sqrt(runif(n,inner_radius^2,outer_radius^2))
theta <- runif(n, 0, 2*pi)
x <- rho * cos(theta)
y <- rho * sin(theta)
plot(x, y, pch=19, cex=0.6, col="#00000020")


annulus






share|cite|improve this answer











$endgroup$













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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk.



    whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus. Note that you need to constrain "on the square root scale".



    Here is an adoption of whuber's original R code, which should be easy to translate to Python:



    outer_radius <- 1
    inner_radius <- 0.7
    n <- 1e4
    rho <- sqrt(runif(n,inner_radius^2,outer_radius^2))
    theta <- runif(n, 0, 2*pi)
    x <- rho * cos(theta)
    y <- rho * sin(theta)
    plot(x, y, pch=19, cex=0.6, col="#00000020")


    annulus






    share|cite|improve this answer











    $endgroup$

















      1












      $begingroup$

      Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk.



      whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus. Note that you need to constrain "on the square root scale".



      Here is an adoption of whuber's original R code, which should be easy to translate to Python:



      outer_radius <- 1
      inner_radius <- 0.7
      n <- 1e4
      rho <- sqrt(runif(n,inner_radius^2,outer_radius^2))
      theta <- runif(n, 0, 2*pi)
      x <- rho * cos(theta)
      y <- rho * sin(theta)
      plot(x, y, pch=19, cex=0.6, col="#00000020")


      annulus






      share|cite|improve this answer











      $endgroup$















        1












        1








        1





        $begingroup$

        Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk.



        whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus. Note that you need to constrain "on the square root scale".



        Here is an adoption of whuber's original R code, which should be easy to translate to Python:



        outer_radius <- 1
        inner_radius <- 0.7
        n <- 1e4
        rho <- sqrt(runif(n,inner_radius^2,outer_radius^2))
        theta <- runif(n, 0, 2*pi)
        x <- rho * cos(theta)
        y <- rho * sin(theta)
        plot(x, y, pch=19, cex=0.6, col="#00000020")


        annulus






        share|cite|improve this answer











        $endgroup$



        Your approach will suffer from the problem the poster in the proposed duplicate pointed out, with a higher density near the center than near the outer boundary. This will afflict both the annulus and the disk, but more so the disk.



        whuber's answer in the proposed duplicate directly gives you the disk part, and by constraining the part where rho is generated, you get the annulus. Note that you need to constrain "on the square root scale".



        Here is an adoption of whuber's original R code, which should be easy to translate to Python:



        outer_radius <- 1
        inner_radius <- 0.7
        n <- 1e4
        rho <- sqrt(runif(n,inner_radius^2,outer_radius^2))
        theta <- runif(n, 0, 2*pi)
        x <- rho * cos(theta)
        y <- rho * sin(theta)
        plot(x, y, pch=19, cex=0.6, col="#00000020")


        annulus







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 4 mins ago

























        answered 31 mins ago









        Stephan KolassaStephan Kolassa

        49.4k8103187




        49.4k8103187



























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