Posted by This Christmas I got a little handheld GPS, which I’ve been using mostly for geocaching. As the device acquires signals from the various satellites dutifully orbiting overhead, it displays your position coordinates and a figure indicating the estimated uncertainty. At the beginning of the acquisition or if the view of the sky is poor, it might be something like 50 feet. If you have a clear view of the sky and are receiving signals from many satellites, it might be as low as 11 or 12 feet.*
This figure is called the circular error probable. In essence, there is a 50% probability that your GPS-reported position is within one CEP distance of the actual position. The terminology originates in military bombing of ground targets, and was used in the context of “50% of bombs will land within one CEP of the target”. The assumption is that the position error obeys a Gaussian distribution, which is not such a bad assumption for GPS, which is subject to a lot of small error sources and thus more or less subject to the central limit theorem.
However, we’re measuring a distance on a 2-d plane. Distances can’t be negative, so the standard bell curve doesn’t work. We have to use the 2-dimensional Gaussian function, which quoting Wikipedia is:
This is more complicated than we need. The uncertainties are the same in both directions, and since we have that symmetry we don’t need to bother with x and y anyway. We’ll just assume the function is centered at the origin and use x^2 + y^2 = r^2 and write the function in terms of r. This gives:
We’ll go ahead and call this our Sunday Function. This is positive everywhere and decays smoothly out from the origin, as it should. I’ll graph it, picking A to normalize the function and σ so that the CEP is 1:
The ordinary 1-d Gaussian distribution has its width given by its standard deviation, but we could give the width in terms of something like a CEP, where 50% of the population is within a distance D of the mean. If we did this, we could calculate that 82% are within 2D of the mean, 95.6% are within 3D of the mean, and so forth. But it turns out that the 2-d Gaussian function that characterizes GPS error doesn’t generate the same values, (Because there’s an extra factor of r inside the integral, due to the Jacobian. Might make a good though somewhat technical post later.)
As it happens, if the CEP is 1, there’s already a 93.7% chance you’re within twice the CEP, and a 99.8% chance you’re within three times the CEP. If we graph both the cumulative probability distributions for being within x of the mean for both the 1-d and 2-d Gaussians, they look like this:
Here I’ve picked the widths for each so that both cumulative distributions are equal to 1/2 at x = 1. The 2-d function approaches 1 considerably faster. This is pretty convenient for GPS users, for instance. Even if there’s only a 50% chance you’re within (say) 20 feet of the reported position, it’s highly likely that you won’t be all that much farther away even in the worst case.
January 17, 2011
January 17, 2011
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January 17, 2011
January 17, 2011
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