If you make a measurement and it is subject to a random unbiased
error then you generally are safe in assuming that the random component
will make the quantity larger as often as it makes it smaller. This is
how it seems to be with GPS there are errors in positioning that are
inherent in the system but certainly don't show any particular bias.
Given this observation you would expect the distance between two points
located with unbiased random error would also be unbiased, i.e. it would
be on average bigger as often as it was smaller than the true value.
However, you would be wrong.
Researchers at the University of Salzburg (UoS), Salzburg Forschungsgesellchaft (SFG), and the Delft University of Technology have done some fairly simple calculations that prove that this is not the case. Irrespective of the distribution of the errors, the expected measured length squared between two points is bigger than the true length squared unless the errors at both points are identical.
That is, if you have two points p1 and p2 and errors in measuring x and y at each, the squared distance measured between them will come out as bigger than the true distance unless the errors are such that they move both points by the same amount - which is highly unlikely in practice.
How can this be?
Consider the two points and the straight line between them. This straight line is the shortest distance between the two points. Now consider random displacements of the two points. The only displacements that reduce the distance are those that move the two points closer together, for example displacements along the line towards each other. The majority of random displacements end up increasing the distance.
This is the reason that unbiased errors end up biasing the distance measurement.
So given that the GPS path is just a sum of distances computed between pairs of points, the total estimated distance is going to be bigger than the true distance because of random errors. A little more work and the researchers derive a formula for how much of an Over Estimate of Distance OED is produced:
OED= (d2 + var - C)1/2 - d
where var is the variance in the GPS position and C is the autocovariance (correlation) between the errors. Notice that the more correlated the errors, the smaller the over estimate.
- More Here
However, you would be wrong.
Researchers at the University of Salzburg (UoS), Salzburg Forschungsgesellchaft (SFG), and the Delft University of Technology have done some fairly simple calculations that prove that this is not the case. Irrespective of the distribution of the errors, the expected measured length squared between two points is bigger than the true length squared unless the errors at both points are identical.
That is, if you have two points p1 and p2 and errors in measuring x and y at each, the squared distance measured between them will come out as bigger than the true distance unless the errors are such that they move both points by the same amount - which is highly unlikely in practice.
How can this be?
Consider the two points and the straight line between them. This straight line is the shortest distance between the two points. Now consider random displacements of the two points. The only displacements that reduce the distance are those that move the two points closer together, for example displacements along the line towards each other. The majority of random displacements end up increasing the distance.
This is the reason that unbiased errors end up biasing the distance measurement.
So given that the GPS path is just a sum of distances computed between pairs of points, the total estimated distance is going to be bigger than the true distance because of random errors. A little more work and the researchers derive a formula for how much of an Over Estimate of Distance OED is produced:
OED= (d2 + var - C)1/2 - d
where var is the variance in the GPS position and C is the autocovariance (correlation) between the errors. Notice that the more correlated the errors, the smaller the over estimate.
- More Here
No comments:
Post a Comment