3 ms·
That was a painfully bad explanation of why random errors cause an over-estimate of distances. There's a much better visual explanation: Suppose you have the t
by ThrustVectoring 11y ago
That was a painfully bad explanation of why random errors cause an over-estimate of distances. There's a much better visual explanation:
Suppose you have the true location for both points. Draw a circle with a center at the first point that goes through the second. Replace the second point with a circular distribution of errors around the original location. Any error outside the original circle increases the measured distance, while any error inside the original circle decreases it. More points are going to be outside the circle, since that's the direction it curves.
This explains a qualitative feature about the formula given - that the error tends towards zero as the distance between two points increases. It also maps well with the case where d approaches zero, where any measurement is simply the result of the random effect.
- Someone 11y agoI think both this explanation and the one in the article are correct, but ignore the fact that GPS distance measurements typically are made by summing the distances between many correlated noisy position measurements. [GPS devices must use frequent position measurements because approximating a continuous real path by line segments will underestimate the length of curved paths] Because of that, the largest contribution to the summed error of the distance between points A and B does not occur when the position of B is erroneously measured at a position B' that is farthest from A. Reason is that, likely, the next measurement will happen when the GPS device is at a position C such that B is roughly halfway between A and C. That means B' lies in the direction of C, and the distance between C' and B' will be smaller than that between C and B, correcting for the measurement error between B and A. Because of that I think it is more insightful to describe the measured path as randomly zigzagging around the actual path. For any reasonably smooth real path such a zig-zag path will be longer than the real path. Even if the real path zigzags around a lot, it is very, very unlikely that the measurement errors happen to undo the zigzagging. [paper at http://www.tandfonline.com/doi/full/10.1080/13658816.2015.1086924 http://www.tandfonline.com/doi/full/10.1080/13658816.2015.10..., by the way]
- ThrustVectoring 11y agoI think my explanation, your zig-zag one, and the article's are describing the same thing with slightly different emphasis. The zig-zag is what you get when you compose a bunch of what I'm visualizing together, and it's the sort of thing that gets worse when you add more measurements of shorter distances. Balancing it is the corner-cutting effect which gets better with more measurements (which the paper you linked describes as "interpolation" which it intentionally did not analyze).
- nickff 11y agoThank you for this post; I wish there were more like it. You used incisive logic to critique and explained another, better way of understanding the idea.
- Retric 11y agoRandom errors need not be unbiased errors. Otherwise I think you have am intuitive answer,
- paulddraper 11y agoNice. Great job.