Appropriate distribution for minimum distance

[belliott4488]

I have what I think is probably a basic question from probability and statistics (about which I'm pretty ignorant).

If I have a set of projectile trajectories that were generated by a Monte Carlo process, and I'd like to know the probability the projectile will come within distance d of some fixed point in space, is there a distribution that is naturally appropriate for this?

I'm thinking that I'll find the minimum distance to the point in question for each trajectory, and then see if I can fit those distances to the appropriate distribution. I just don't know what that would be. (Rayleigh, maybe??)

Thanks for any suggestions.

EDIT: I probably should have stated that we can assume that the Monte Carlo variations used to generate the sample set were normally distributed.

 

[Stephen Tashi]

You may get some help if you describe how the trajectories were distributed relative to the point in question. "Normally distributed" doesn't explain how a 2-D trajectory is distributed.

 

[belliott4488]

You may get some help if you describe how the trajectories were distributed relative to the point in question. "Normally distributed" doesn't explain how a 2-D trajectory is distributed.

Sorry - didn't mean to imply that they're 2-D. In fact, they're 3-D trajectories, and they're not truly ballistic (there's powered flight involved). For the sake of this question, I think we can make the approximation that they are projectiles that originate from a fixed point in space, and the initial velocity vector has variations that are normally distributed about a mean vector. If it matters, then we can specify that the magnitude, elevation, and azimuth angles are what are drawn from normal distributions. I am aware that my result will be only as good as this approximation.

To further complicate matters, what I described as a fixed point in space in the original post is actually another trajectory for a different object. The actual problem is to estimate the probability of collision between two objects, given Monte Carlo sets of trajectories for each. So what I'm doing is finding the separation vector at the point of closest approach for each pairing of trajectories. Given this set of vectors, I'd like to estimate the probability that the minimum separation will be less than a given distance.

Now that I've said that this is a 3-D problem, does that invalidate using a Rayleigh distribution? I've read that Rayleigh is good for describing the magnitude of a vector whose components are independently normally distributed, but the examples are always 2-D. Is that a limitation of Rayleigh? If so, is there a 3-D analog?

Thanks very much for any guidance you can offer.

 

[Stephen Tashi]

Perhaps I don't understand your goal. The minimum distance between two 3-D trajectories as curves isn't necessarily the minimum distance between two objects that follow those trajectories since time is also involved. For example two 3D curves may intersect, but objects can follow those curves without colldiing.

I suggest you present some simulation results of the data and people will probably have suggestions about what distribution to fit to it.

Actually giving an angle a normal distribution won't be possible due to the "wrap around" at intervals of two pi.

 

[belliott4488]

I apologize again for over-simplifying the problem. These trajectories are actually the output of fairly complex 6-degree-of-freedom simulations, so I can't really even attempt to give the all the details. (Truthfully, I don't know all of them - there are all kinds of mechanical systems, the details of which I don't know, that are being simulated.)

Nonetheless, I think it would be informative to know if there's a reasonably straight-forward approach to the simplified problem I'm trying to describe. So, let me try to be a bit more specific.

Suppose I have two projectiles, one being launched from point A and the other from point B. I generate a set of trajectories for each projectile as follows:

Projectile from A:
velocity magnitude sampled from a normal distribution with mean V_a = 50 m/s, sigma = 5 m/s
velocity elevation angle sampled from a normal distribution with mean theta_a = 60 deg., sigma = 3 deg
velocity azimuth angle sampled from a normal distribution with mean phi_a = 45 deg., sigma = 5 deg

Projectile from B:
V_b = 40 m/s, sigma = 3 m/s
theta_b = 70 deg, sigma = 5 deg.
phi_b = -10 deg, sigma = 5 deg

These are totally made-up numbers, in case that wasn't obvious. I hope there isn't something inherently stupid in my choices.

[EDIT: It just occurred to me why you made the comment about a normal distribution for an angular variable - that's probably ever not well-defined, is it? In practice I think people fudge this by treating the angles as if they're non-cyclic and making the sigma small enough that all the sampled values are within a few degrees of the mean. There's always a non-zero probability of getting an outlier that wraps around, but in practice, I don't think that happens. This is actually done in the software run by people in another organization, and I don't have much insight into it. They probably have better-defined angular distributions, perhaps empirically derived.]

I didn't pick locations for A and B, but let's just say that they're such that the trajectories and launch times of projectiles A and B will be such that it's reasonable to ask what is the probability of a collision between them.

I have sets of such trajectories, all time-tagged in reference to the same t0, so I can easily compute the separation vector between the two projectiles at each time step for every combination of a trajectory for projectile A and one for projectile B. Taking the vector with the smallest magnitude for each such pairing gives me a set of m*n vectors, where m is the number of trajectories from A and n is the number of trajectories from B.

Is it a sensible question to ask what is the probability of collision given these minimum separation vectors? In reality there are possible correlations between the variations in the two trajectories, but if possible I'd like to pretend that they are completely independent, to be a little more general.

If that question does not in fact make sense, let me ask where I am going wrong in my reasoning, which goes as follows:

I have a "nominal" trajectory for each projectile, which is the trajectory generated from the mean velocity vector. I can calculate the minimum separation distance between the projectiles based on the nominal trajectories. In addition, I have a set of off-nominal min. separation vectors that correspond to samples from distributions of trajectories about the nominal trajectories. Can I not examine those samples, maybe make some reasonable assumptions about the underlying distributions (which will likely be gross approximations), and from those estimate the probability that the minimum separation will have a magnitude less than some value?

Thanks again.

 

[Stephen Tashi]

. Can I not examine those samples, maybe make some reasonable assumptions about the underlying distributions (which will likely be gross approximations), and from those estimate the probability that the minimum separation will have a magnitude less than some value?

Of course you can do that. There isn't any pure mathematical theorem that says you can or can't proceed that way. This is a question about how mathematics is applied. I think you must post a graph of the data to get suggestions about what family of distributions fit it. As far as I know, there isn't any theoretical result that says your simplified trajectory problem must have a particular family of distribuitons as its outcome.

 

[belliott4488]

Of course you can do that. There isn't any pure mathematical theorem that says you can or can't proceed that way. This is a question about how mathematics is applied. I think you must post a graph of the data to get suggestions about what family of distributions fit it. As far as I know, there isn't any theoretical result that says your simplified trajectory problem must have a particular family of distribuitons as its outcome.

Ah, okay - that makes sense.

I was kind of hoping someone might say, "Oh, this is what so-and-so developed the so-and-so distribution to describe." There are so many distribution functions out there, and I know very little about most of them.

I wish I could post some data or plots or something, but unfortunately, I'm not free to do that. I guess I'll just have to make some educated guesses at possible distributions and see if any of them fit the data reasonably well.

Thanks again for your help.

 

[Hornbein]

Is it a sensible question to ask what is the probability of collision given these minimum separation vectors? In reality there are possible correlations between the variations in the two trajectories, but if possible I'd like to pretend that they are completely independent, to be a little more general.

I don't see any advantage to being "more general," but let's assume that they are independent to make the problem easier. You seem to have gravity in there, but it should affect both missiles equally so it shouldn't matter. So what I would do is this.

Everything is linear. That's good. Essentially what you have for each missile with a point of certainty, which is the launch point, where the standard deviation is zero. After launch the standard deviation of the location of the missile increases linearly. Since you have three dimensions, you want a three dimensional multivariate Gaussian. If you have more than three variables you should boil them down to get their effect on the three dimensions.

What I would do is first forget about all the randomness and just use the mean values of all the parameters. This is your best estimate of where the two missiles will be at any time. Use that, and use numerical methods to determine the time of minimum distance T. T let's you calculate the means standard deviations at that time. So you have the means and standard deviations of both multivariate random variables. You are in luck! All you have to do is subtract one from the other X(T) - Y(T) = Z(T) and you get a new multivariate Gaussian. The mean of Z(T) is simply the difference of the means. The variances of Z(T() is Var(X(T)) + Var(Y(T)). Then you have the distribution of the distance. A minor wrinkle is that you don't really want Z(T), you want the absolute value |Z(T)|, but you can deal with that to get the probability that |Z(T)| < whatever, which I assume is what you want to know.

I'm neglecting the fact that T is really an interval, not a constant. This should be OK as long as the standard deviation doesn't change significantly during the interval. Even if it does, this is the sort of error that tends to cancel out, since it is smaller before T and larger afterward.

The correlations complicate things, but I'll leave that up to you.

 

[FactChecker]

This sounds like a typical Circular Error Probable (CEP) problem. The CEP is the radius that contains half the sample. It is the parameter needed for a circular bivariate normal distribution (see http://en.wikipedia.org/wiki/Circular_error_probable)