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    Re: Still on LOP's
    From: Michael Wescott
    Date: 2002 Apr 25, 16:41 -0400

    Edited to re-introduce some brevity.
    Trevor: >>> and >
    Me: >> and none.
    > We art not dealing with quantum mechanics and the positions and
    > velocities of electrons. (Fortunately!)
    Might be easier if we were. Intuition doesn't try to "help" out
    as much.
    >> What we can talk about is the pattern or distributions of 'hats
    >> that get generated near a true position. As shown before, 25% of
    >> the generated 'hats will enclose the true position and 75% will
    >> not. There are two assumptions that have to be met for this to be
    >> true. First, the distribution of the error must be symmetric
    >> around 0, meaning there is no inherent bias in the errors in
    >> either direction (toward or away).  Second, the errors must be
    >> independent in each of the observations.
    >> If we add the addition assumption that smaller errors are more
    >> likely than larger errors, and we generate a large number of hats
    >> then we'll notice several things. First, 25% of the hats will
    >> contain the true position. Second, the hats vary quite a bit in
    >> size but larger hats enclose the true position more often than
    >> smaller ones. And third, there are more smaller hats than larger
    >> ones.
    >> More smaller hats which tend to be near but not enclosing the
    >> true position. Larger ones are rarer but tend to surround the
    >> true position. That should help the intuition.
    > I'm not so sure that that argument addresses the point that I was
    > trying to judge intuitively but it does raise a very important point
    > that should have been evident from inspection of Geoffrey Kolbe's
    > diagrams but which I, for one, did not notice: Given the same
    > magnitude (but not direction) of error in the three LOPs, those
    > cocked hats which enclose the true position are LARGER than those
    > which do not. (I have not checked that for non-equilateral 'hats,
    > nor for those in which the errors differ among the three LOPs. Is it
    > true with those too?)
    Always true. Take any triangle with a dot in it (not necessarily
    centered).  Now move one of the lines to the other side of the dot at
    the same distance.  The line will now be closer to the opposing vertex
    and the triangle smaller.
    > Hence, the long-standing assumption that a smaller cocked hat
    > indicates a "better" fix needs to be reconsidered. If it is smaller
    > because the three LOPs were more precise than usual, that is correct
    > -- whether the precision is due to improved measurement or just to
    > the LOPs chancing to lie closer to their correct positions. But a
    > small 'hat could also be a warning that your true position is
    > unlikely to be inside it. A disturbing revelation!
    It indicates a better fix; it doesn't guarentee a better fix.
    See below.
    > Turning back to my earlier point that Michael thought he had helped
    > with: My concern was not with the size of the 'hat changing, despite
    > constant absolute values of errors, depending on where the 'hat fell
    > relative to the true position. I was concerned with changing sizes
    > of the 'hat caused by different absolute values of the errors drawn
    > from constant error distributions. I'm not about to consider the
    > limiting cases of zero error (which I had avoided before Rodney
    > pointed out that it should not be used) or an infinitely-small
    > 'hat. However, there are real cases of very, very small 'hats
    > arising by chance (NOT by geometry). The logic that most
    > contributors to this debate seem ready to accept tells us that 25%
    > of the complete universe of those contain the true positions that
    > they were estimates of. I'm still having trouble accepting that
    > while, if it has to be rejected, then it seems to me that the whole
    > proof of the 25% will go down in flames.
    With a single 'hat there is no big or small; you've got a sample of 1
    and it is the normal size.  You can still do things like calculating
    the confidence contours.  But if you double the size of the hat, the
    size of the contour changes to match. Why? Because our estimation of
    the sigma of the distribution of the errors is based on only that
    sample, and everything scales the same.
    On the other hand, bigger and smaller do have meaning when you have a
    history. This is where experience comes in. With history (experience)
    you now have a handle on sigma that's independent of this particular
    set of observations. Now the relationship of the contour and the hat
    will change as the the size of the hat changes because sigma won't
    change. Relatively smaller (than expected) hats will have smaller,
    more sharply peaked contours, while larger ones will have broader,
    flatter contours.
    I think you're putting too much emphasis on inside versus outside. The
    real question is how close is the MPP to the real position. There's no
    direct evidence of how close, or in which direction the real position
    is, given just the one hat. But if you plot the universe of hats
    around a true position, smaller hats will be more common close to the
    true point than farther away. That's no guarantee that a small hat is
    a from good sights, but you do get more small hats from good sights
    than from bad ones. (The assumption that has to be made to support
    that assertion is that small errors are more likely that large ones.)
    We practice to get better and to get to know how good we are. But it's
    also good to know that what looks like a very good fix might still,
    just by random chance, still be way off.  Somebody has to win the
    >>>> Similarly, if you only have one 'hat, then placing the MPP in the
    >>>> centre of it is the best you can do.
    >>> That is indeed the MPP, if the 'hat is equilateral.
    >> No. Given 3 LOPs and a 'hat, the MPP is equidistant from each side
    >> of the hat; in the center of the inscribed circle. The mutual
    >> orientation of the LOPs won't alter that.
    > Steven Tripp's calculations suggest that it does. [...]
    > If there are mathematical reasons to reject that conclusion, I fear
    > that we will need to take this discussion to a higher level of
    > complexity.
    Sorry, but Steven Tripp's calculations don't find the MPP. They find
    the least squares point, a computationally more tractable exercise and
    in this case, they aren't the same.
    >> There remains one question. Is the contour around the MPP
    >> elliptical?  For that to be true, we have to be more specific about
    >> the distribution.  If it's Gaussian then the contour is elliptical.
    > That is a point which has come up before on this thread. If there
    > are but two LOPs, it is clear that the contour is elliptical with
    > Gaussian distributions. Are we to understand that that is also true
    > if there are three LOPs? More than three?
    > I'm not asking for the mathematical proof, which I likely wouldn't
    > understand anyway. But I would welcome somebody laying out his or
    > her credentials as an authoritative voice and then telling me one
    > way or another.
    The answer is yes and another has staked credentials on it. I do have
    a degree in Math; but I preferred analysis and abstract algebra to
    statistics so the credentials I will stake on it might be somewhat
            Mike Wescott

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