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I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, θ_, d_] := {xCentre + d Sin[θ], yCentre + d Cos[θ]} fTorusRand[r_, σ_] := Module[{θ = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, σ], {1}][[1]]}, fTorusProjector[r Sin[θ], r Cos[θ], θ, d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Edit 2: Since $\theta$ and $d$ are independent, I can derive the joint PDF in these coordinates:

$p(\theta,d) = \frac{1}{2\pi} \times \frac{1}{2\pi} exp(-d^2/2\sigma^2)$

I suppose I can then use Jacobians to transform back into the $(x,y)$ frame, although am not sure how to do this?

Best,

Ben

I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, θ_, d_] := {xCentre + d Sin[θ], yCentre + d Cos[θ]} fTorusRand[r_, σ_] := Module[{θ = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, σ], {1}][[1]]}, fTorusProjector[r Sin[θ], r Cos[θ], θ, d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Edit 2: Since $\theta$ and $d$ are independent, I can derive the joint PDF in these coordinates:

$p(\theta,d) = \frac{1}{2\pi} \times \frac{1}{2\pi} \exp(-d^2/2\sigma^2)$

I suppose I can then use Jacobians to transform back into the $(x,y)$ frame, although am not sure how to do this?

Best,

Ben

I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, \[Theta]_, d_] := {xCentre + d Sin[\[Theta]], yCentre + d Cos[\[Theta]]} fTorusRand[r_, \[Sigma]_] := Module[{\[Theta] = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, \[Sigma]], {1}][[1]]}, fTorusProjector[r Sin[\[Theta]], r Cos[\[Theta]], \[Theta], d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Edit 2: Since $\theta$ and $d$ are independent, I can derive the joint PDF in these coordinates:

$p(\theta,d) = \frac{1}{2\pi} \times \frac{1}{2\pi} exp(-d^2/2\sigma^2)$

I suppose I can then use Jacobians to transform back into the $(x,y)$ frame, although am not sure how to do this?

Best,

Ben

I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, θ_, d_] := {xCentre + d Sin[θ], yCentre + d Cos[θ]} fTorusRand[r_, σ_] := Module[{θ = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, σ], {1}][[1]]}, fTorusProjector[r Sin[θ], r Cos[θ], θ, d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Edit 2: Since $\theta$ and $d$ are independent, I can derive the joint PDF in these coordinates:

$p(\theta,d) = \frac{1}{2\pi} \times \frac{1}{2\pi} exp(-d^2/2\sigma^2)$

I suppose I can then use Jacobians to transform back into the $(x,y)$ frame, although am not sure how to do this?

Best,

Ben

I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, \[Theta]_, d_] := {xCentre + d Sin[\[Theta]], yCentre + d Cos[\[Theta]]} fTorusRand[r_, \[Sigma]_] := Module[{\[Theta] = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, \[Sigma]], {1}][[1]]}, fTorusProjector[r Sin[\[Theta]], r Cos[\[Theta]], \[Theta], d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Best,

Ben

I would like to derive the PDF for an ''annulus'' type distribution, defined by the parameters $\theta\sim U(0,2 \pi)$ and $d\sim N(0,\sigma)$; where $\theta$ is the angle round a circle of radius $r$, and $d$ is the deviation normal to the circle's perimeter.

I can generate sample points easy enough, by the following:

fTorusProjector[xCentre_, yCentre_, \[Theta]_, d_] := {xCentre + d Sin[\[Theta]], yCentre + d Cos[\[Theta]]} fTorusRand[r_, \[Sigma]_] := Module[{\[Theta] = RandomReal[{0, 2 Pi}], d = RandomVariate[NormalDistribution[0, \[Sigma]], {1}][[1]]}, fTorusProjector[r Sin[\[Theta]], r Cos[\[Theta]], \[Theta], d]] ListPlot[Table[fTorusRand[5, 0.5], {i, 1, 10000}]] 

Which results in a plot similar to the one below:

My question is, is there a way to derive an analytic form for the PDF of this distribution? I can't help thinking that there's probably a way I'm missing, but am not sure how to proceed here.

Even an approximate PDF / one that mimics the above behaviour would suffice.

Edit: I have tried using the following to get Mathematica to approximate a PDF of the above sample:

 empD = EmpiricalDistribution[data]; 

However, when I try to draw this using:

ContourPlot[PDF[empD, {x, y}], {x, -7, 7}, {y, -7, 7}] 

I just get a blank plot.

Edit 2: Since $\theta$ and $d$ are independent, I can derive the joint PDF in these coordinates:

$p(\theta,d) = \frac{1}{2\pi} \times \frac{1}{2\pi} exp(-d^2/2\sigma^2)$

I suppose I can then use Jacobians to transform back into the $(x,y)$ frame, although am not sure how to do this?

Best,

Ben