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    Re: Position lines, crossing
    From: Frank Reed CT
    Date: 2006 Dec 10, 22:37 EST
    Geoffrey, you wrote:
    "The justification for assuming that the separation of the
    Moon-Rasalhague pair of sightings (which were almost exactly in
    opposition) was due to a systematic error of some sort is the fact
    that the final "cocked hat" ended up so small after applying the same
    systematic error to all the sightings (which is effectively what I did)."
     
    I agree completely. As soon as I saw those two parallel LOPs so far apart, my immediate thought was that someone forgot to apply the correction for dip. And sure enough, when you adjust the other LOPs, the net crossing point for all of the LOPs fits inside a much smaller box. This is good evidence of a systematic error (probably a dip error, maybe a bad index correction...). With fewer sights I would be less likely to consider this possibility.
     
    So this raises an interesting question: how many sights do you need before you can make statistically significant judgements about a systematic error? Clearly if I only have a pair of LOPs, I have no evidence. If I throw in a third LOP, and it misses the crossing point of the first two by 5 miles, I may be suspicious, but there's no way of knowing whether I've got one bad sight or a systematic error in all three. But if we keep taking sights, and let's say for an example I accumulate eight LOPs symmetrically placed in azimuth, then I can surely make some kind of statistically reasonable judgement about systematic error. If the LOPs from eight sights give me an almost perfectly symmetrical octagon ten miles across, then I would feel justified in adjusting each sight by five miles (or whatever number yields the smallest box around the area of intersection --better yet, whatever distance leaves me with the most "random-looking" set of intersection points). So what's the magic minimum number of sights that makes this procedure a good idea at the 50% confidence level? What's the minimum for 95% confidence?
     
    Peter Fogg, you wrote:
    "George is well and truly launched on one of his favourite hobby-horses. One that seems to me to be very much a case of creating an unnecessary mountain out of a mole-hill."
     
    Mole-hill mountainized, it may be, yet we must climb it! :-)
     
    And you wrote:
    "What else has changed? Nothing. The only position that can be calculated from this shape, or the original, is exactly the same, and lies at the centre (least squares) of either the small or the large shape. One is virtually certain to contain the actual position, the other less so, but nothing else has changed or can be changed without additional data. "
     
    There are two things we should get from a set of sights: a best estimate of position and an expectation of the error in that position. Your point is certainly correct that the center of the triangle is the center no matter what. But part of George's point is that the size of the triangle is no measure of error --a small triangle does not imply an accurate fix, below a certain minimum size. And that's true. When we DO draw a reasonable error circle around the fix at the center of the triangle, it turns out that it encompasses considerably more area than the triangle itself.
     
    The "popular delusion", I think, arises from the idea that a three-star fix yields an automatic measure of error. The best way to cure yourself of that delusion is to do more five and six star fixes. At least do some simulations with reasonable values for the error distribution of each sight, and take note of the frequency with which you get an accidentally small triangle. This is something that can be done relatively easily with a simple bit of software. Anyone want to volunteer to make something web-based? I will do so later this week if no one else beats me to it. Or does it already exist?
     
    -FER
    42.0N 87.7W, or 41.4N 72.1W.
    www.HistoricalAtlas.com/lunars
    PS: Saw the Space Shuttle just 8-9 minutes after launch last night from here in Mystic. Very bright (magnitude -2), yellow in color, and moving fast low in the southeast sky.

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