Circular Error Probability in polar error expression

In summary, the circular error probability (CEP) is often expressed in terms of \sigma_x and \sigma_y, assuming a normal distribution. This can also be written in terms of polar coordinates, but does not account for any correlation between \sigma_x and \sigma_y. To express the CEP in terms of r and \theta only, we can use the equations x=r\cos(\theta) and y=r\sin(\theta). However, this may result in a more complex integral. Additionally, when considering the spherical error probability (SEP), the complexity increases significantly as the 3-dimensional case must be accounted for. One possible solution is to calculate the r and \theta variances in terms of the x and y var
  • #1
dbeeo
2
0
After doing various searching through the google, most of the circular error probability I found are expressed interm of [tex]\sigma_x[/tex] and [tex]\sigma_y[/tex], which the CEP(circular error probability) usually looks like (assume normal distribution):

[tex]P(r) = \int\int \exp^{-.5(\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2})} dxdy[/tex]

or if we integrate the equation in term of polar coordinate

[tex]P(r) = \int\int \exp^{-.5(\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2})} rd\theta dr[/tex]

However, these equations are based on no correlation between [tex]\sigma_x[/tex] and [tex]\sigma_y[/tex]. And also, both [tex]\sigma_x[/tex] and [tex]\sigma_y[/tex] remains constant disregarding the change of [tex]r[/tex] and [tex]\theta[/tex]. Although we usually can express both [tex]x[/tex] and [tex]y[/tex] in term of [tex]r[/tex] and [tex]\theta[/tex],

[tex]x=r\cos(\theta)[/tex] and [tex]y=r\sin(t\theta)[/tex]

But right now, I would like to have the CEP to express in [tex]r[/tex] and [tex]\theta[/tex] only, since the error I will have are [tex]\sigma_\theta[/tex] and [tex]\sigma_r[/tex], and I would like to avoid the correlation issue. So, I'm just not know that if this equation will make sense or not,

[tex]P(r) = \int\int \exp^{-.5(\frac{r^2}{\sigma_r^2}+\frac{\theta^2}{\sigma_\theta^2})} rd\theta dr[/tex]

Anyone has any input/idea about this? One of the other problem is that I need to expand the CEP into spherical error probability (SEP), which is in the 3dimensional. Although I have a paper to show somewhat a close form solution for this problem, however they still consider [tex]\sigma_x[/tex] and [tex]\sigma_y[/tex] with correlation as their error instead of [tex]\sigma_\theta[/tex] and [tex]\sigma_r[/tex]. But the complexity just increases way too high if I'm following this method.
 
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  • #2
Your y expression has a t in it - seems to have come from nowhere. Your final P(r) is just wrong. Plug in the expressions for x and y as functions of r and θ to get the correct integral in polar coordinates.
 
  • #3
it was a typo for y expression, it should be,

[tex]y=r\sin(\theta)[/tex]

For the final P(r), I'm trying to find the other way that involve both [tex]\sigma_r[/tex] and [tex]\sigma_\theta[/tex]. So it seems like it is impossible unless i substitute them into the following expression?

[tex]P(r) = \int\int \exp^{-.5(\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2})} rd\theta dr[/tex]
 
  • #4
Have you tried calculating the r and θ variances in terms of the x and y variances? Off hand it looks messy.
 
  • #5
If you consider the case where the x and y variances are the same, the resultant polar coordinates have a distribution where the angle is uniform over a circle and r2 has an exponential distribution.
 

What is Circular Error Probability (CEP)?

Circular Error Probability (CEP) is a measure of the accuracy of a weapon system or measurement device. It represents the radius of a circle within which 50% of all rounds or measurements will fall. In other words, it is a measure of the precision of a system.

How is CEP calculated?

CEP is typically calculated by taking the square root of the sum of the squared errors from the mean (also known as the Root Mean Square Error or RMSE). The errors can be in any direction, but are most commonly measured in a two-dimensional plane.

What is the polar error expression for CEP?

The polar error expression for CEP is a representation of the probability of a weapon system or measurement device being accurate within a certain distance and angle from the intended target or measurement point. It takes into account the direction of the errors, rather than just the distance.

How is CEP used in military applications?

CEP is used in military applications to determine the accuracy of weapons, such as missiles and bombs. It is also used in targeting systems, to determine the likelihood of hitting a specific target. CEP is also used to assess the performance of military equipment and to evaluate the effectiveness of training and tactics.

How does CEP differ from other accuracy measures?

CEP is different from other accuracy measures, such as Mean Absolute Error (MAE) and Mean Squared Error (MSE), because it takes into account both the magnitude and direction of errors. This makes it a more comprehensive measure of accuracy. Additionally, CEP is a circular measure, meaning it takes into account errors in all directions, rather than just along a single axis.

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