-1

I tried to improve trilateration accuracy by doing least square method. For initial estimation, I get the average value of the cluster points. This value is then increased until the distance to the next estimation is small enough. The increment value is calculated using this formula,

formula

My question is, why most of the times the final answer diverts quite significant from what it should be? The initial estimation is even better, although not that accurate. Do I miss something here?

Edit

The formula is as explained here. I hope this picture explains better,

least square

See the final point is even outside the intersection area.

springrolls
  • 1,331
  • 1
  • 14
  • 40
  • 1
    Please explain this in detail or post a picture. What is *A* and *b*? Also, you might have a more receptive audience at math.stackexchange.com – Jacob Aug 06 '11 at 14:09
  • 2
    Least squares approximations are sensitive to bad measurements (outliers), so the choice of approximation has to be weighed against the quality of your data. – hardmath Aug 06 '11 at 14:49
  • @hardmath: Sorry, I'm not good at math. What do you mean by bad measurements? Like inaccurate circle radii? How can I calculate the weight? – springrolls Aug 06 '11 at 15:03
  • The revised problem/picture make clear the connection to your [earlier question](http://stackoverflow.com/q/6927889/487781), which mentions the inaccurate circle radii. But unfortunately I'm in the dark about what "trilateration" problem you are trying to solve and what effect the inaccuracies of data have on the answer. Nonlinear regression is a technique, but it doesn't tell us what problem it's being applied to. – hardmath Aug 06 '11 at 15:14
  • @hardmath: Ah, you still remember my previous question. Okay, let me explain the problem. So, I'm trying to locate people based on Bluetooth signal strength. Since I use trilateration, I need the distance between the senders (the circle centers) and the receiver, which could be inaccurate due to the signal interference. So I thought using least square will improve the result of receiver's location, instead of simply calculating the average values. – springrolls Aug 06 '11 at 15:27

2 Answers2

2

I think I have a clear enough idea of the problem to offer an answer.

Basically the three-cornered region's interior consists entirely of points that undershoot the three estimated distances (inaccurate circle radii). So it is not surprising that an iterative improvement to the solution, seeking a minimum squared error approximation, will move the point outside that three-cornered region.

More about why the points inside the region give distances that are below the given estimates: These points are exactly those that are inside all three circles (if such an arrangement holds). Therefore the three distances from such a point to the circles' centers are all below their respective radii.

Using the average of the three corner points (is this what is meant by cluster points in the question?) is probably a pretty good way to start. If there is a simple place to improve the calculation, it might lie in using a weighted least squares criterion rather than an absolute least squares criterion.

What I mean by this is that if one radius is 10 yards, and the other two radii are much larger (say 200 and 300 yards for the sake of discussion), it probably doesn't make sense to assume the estimated distances all have errors of about equal size (which is what an absolute least squares fit looks for). Instead assuming the error in the estimated distances is roughly proportion to each distance (a relative error criterion) is more likely to produce a better solution, e.g. giving a greater weight to the shorter distance (because a proportional error in that would be less in absolute magnitude than the proportional error in the longer distances).

This is just a sketch of one idea you might want to incorporate in your solution. I take you have only the three data to work with (locations known to fairly good accuracy as centers of circles, greater uncertainty in the three radii). So it doesn't make sense to try and apply methods that are sophisticated in respect of accuracy, but rather to prefer methods that give a robust solution. I think the relative error criterion would move you in that direction.

hardmath
  • 8,753
  • 2
  • 37
  • 65
  • Yes, the cluster points I meant is the three corner points. With this weighted method, then in the case where the radii estimation are quite accurate and thus, the initial estimation itself is sufficient, doing least square (the iterative approximation) will not divert the result. Is it correct? – springrolls Aug 06 '11 at 16:47
  • 1
    You seem to have some notion of what the correct result is apart from just fitting the least squares solution. I think a weighted least squares solution would be more robust (to errors in the radii), but if you had exact centers and exact radii, you would get the same solution using either weighted or unweighted least squares fitting (namely a zero sum of squares error). Here's what I think might happen in practice. The main source of error is interference, which weakens the Bluetooth signals and gives radii estimates that are too large. So there's an asymmetry in the error. – hardmath Aug 06 '11 at 23:16
0

Least squares minimizes the overall square of the errors, but it says nothing about how close an individual point will be to the true value. The coefficients are affected by all the points, not just a few.

duffymo
  • 305,152
  • 44
  • 369
  • 561