Strategies to detect and delete cluttering aggregations of GPS points?

Question

my problem is that I have a large set of GPS tracks from different GPS loggers used in cars. When not turned off these cheap devices log phantom movements even if standing still:

As you can see in the image above, about a thousand points get visualized in a kind of congestion. Now I want to remove all of these points so that the red track coming from the left ends before the jitter starts. My approach is to "draw" two or three circles around each point in the track, check how many other points are located within these circles and check the ratio:

(#points / covered area) > threshold?

If the threshold exceeds a certain ratio (purple circles), I could delete all points within. So: easy method, but has huge disadvantages, e.g. computation time, deleting "innocent" tracks only passing through the circle, doesn't detect outliers like the single points at the bottom of the picture).

I am looking for a better way to detect large heaps of points like in the picture. It should not remove false positives (of perhaps 5 or 10 points, these aggregations don't matter to me). Also, it should not simplify the rest of the track!

Edit: The result in given example should look like this:

Answer 1

My first step would be to investigate the speeds implied by the 'movements' of your stationary car and the changes in altitude. If either of these changes too quickly or too slowly (you'll have to decide the thresholds here) then you can probably conclude that they are due to the GPS jitter.

What information, other than position at time, does your GPS device report ?

EDIT (after OP's comment)

The problem is to characterise part of the log as 'car moving' and part of the log as 'car not moving but GPS location jittering'. I suggested one approach, Benjamin suggested another. If speed doesn't discriminate accurately enough, try acceleration. Try rate of change of heading. If none of these simple approaches work, I think it's time for you to break out your stats textbooks and start figuring out autocorrelation of random processes and the like. At this point I quietly slink away ...