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    Re: Cocked hats, again.
    From: Fred Hebard
    Date: 2007 Mar 14, 22:58 -0400

    The size of the Gaussian bell curve, or whatever curve, around each  
    position line wouldn't affect the probability of being on one side or  
    another; as long as the curve is symmetrical the probability would  
    still be one half.  The triangle is merely at the intersection of the  
    lines; another of our assumptions are that the errors for each line  
    are independent (and, for some calculations, equal).  A small  
    triangle suggests that sigma is small for each line, and a large  
    triangle that sigma is large.  You appear to be assuming sigma is the  
    same size for different sized triangles.
    I suspect that George's rule may only apply when the lines are 120  
    degrees apart.  Obviously, the degenerate case of two times on bodies  
    at the same azimuth or at azimuths 180 degrees apart would boil down  
    to a two-line fix.
    I can't present the rule exactly, but three lines only partition the  
    plane into seven spaces, not eight.  That eighth possibility may be  
    hidden in the middle of the triangle.
    On Mar 14, 2007, at 4:57 AM, glapook@PACBELL.NET wrote:
    > Gary LaPook wrote:
    > Well at first blush is would seem then that from 3 LOPs with equal
    > chances of being on one side or the other of each LOP that you would
    > have 8 combinations (2^3) so one specific case would occur only 1 out
    > of 8 times for a ratio of 7 to 1 not three to 1. That is my first
    > thought and I haven't made any drawings yet.
    > But take the case of just two LOPs crossing at a 90� angle. You would
    > have a 1 in 4 chance of being in any one of the specified quadrants so
    > the 3 to 1 ratio makes sense in the two LOP case. I will have to give
    > the three LOP situation some more thought.
    > One other problem I have with that idea is that it doesn't take into
    > account the size of the triangle. With a very small triangle it
    > approaches certainty that the observer is outside the triangle. At the
    > other extreme, with a very large triangle it becomes less and less
    > likely that the observer is outside the triangle. When the
    > displacement of the LOPs equals 3.3 sigma the only place the observer
    > can be is in the exact center of the triangle since this is the only
    > location that satisfies the requirement that the observer be within
    > 3.3 sigmas of each LOP. ( I know that 3.3 sigma only equals 99.9% of
    > the cases but this leaves only 1 in one thousand observation with a
    > larger error. You could use 4 sigma,. 1 in 10,000 or 5 sigma, 1 in a
    > million if you like.)  But is is a virtual certainty that with a
    > maximum sized triangle the only place the observer could be is in the
    > center of the triangle.
    > On Mar 13, 6:10 pm, Fred Hebard  wrote:
    >> Gary,
    >> I don't know why George got bogged down in that statistical
    >> argument.  He has previously, several times, presented a much more
    >> elegant one based on probability.  For each position line, with
    >> random error, you have an equal chance of being on one side or
    >> another of the line.  So the probability of being on one side is
    >> 1/2.  Now count up all the different combinations of one side or
    >> another for three position lines, and you end up with the 3 out of 4
    >> proportion.
    >> Fred
    >> On Mar 13, 2007, at 8:03 PM, Gary LaPook wrote:
    >>> Gary LaPook wrote:
    >>> Well good, we seem to agree on most things. And I agree with you  
    >>> that
    >>> many times the position of the observer will be outside of the
    >>> triangle,
    >>> the only reservation I have is with the strict 3 out of four
    >>> proportion,
    >>> although I haven't done any rigorous testing on this point. I had
    >>> reviewed the series of posts in December (number 1908) and saw the
    >>> diagram with eight triangles, six triangles showing the position of
    >>> the
    >>> observer outside and only two showing it inside. I have a question
    >>> about  what these actually show. I also noted that the two
    >>> triangles in
    >>> which the position of the observer is inside are much larger than  
    >>> the
    >>> six triangles showing the position is outside. In fact these two
    >>> triangles appear to have a total area of 18 times larger than  
    >>> each of
    >>> the six other triangles for a ratio of 18 to 6 or 3 to one but I  
    >>> don't
    >>> know if the relative areas signify anything.
    >>> But, I am curious and will investigate further.
    >>> George Huxtable wrote:
    >>>> I hardly ever find anything to disagree with in Gary LaPook's
    >>>> contributions. They are usually full of sense.
    >>>> That applies, too to most of his 6th March posting, Navlist 2236,
    >>>> labelled "resolution of systematic error". Though actually, what he
    >>>> discusses there is the situation of entirely random error, where  
    >>>> all
    >>>> systematic error has been corrected out.
    >>>> But Gary concludes with this-
    >>>> "Again, no one
    >>>> is suggesting that the position of the observer is at the center of
    >>>> the triangle but this represents the center of possible  
    >>>> positions of
    >>>> the observer.  In fact, the position of the observer will be  
    >>>> outside
    >>>> of the triangle often but I don't agree with the three out of four
    >>>> allegation. Counter intuitively, the smaller the triangle the more
    >>>> likely that the position of the observer is outside the  
    >>>> triangle! If
    >>>> you think about it, this should be obvious. Using   reducio ad
    >>>> absurdum, think about a triangle only one inch in size, it would be
    >>>> impossible for the observer to be within the triangle. At the other
    >>>> extreme, a very large triangle with all of the displacements of the
    >>>> LOPs from the center of the triangle equal to 3.3 NM (3.3 sigma's,
    >>>> linear sigma's are slightly different than circular sigma's, see
    >>>> Bowditch), the only place that the position of the observer  
    >>>> could be
    >>>> is at the fix in the center of the triangle!"
    >>>> Here, Gary is wrong. The statement, that "with entirely random
    >>>> errors,
    >>>> three times out of four the triangle will not embrace the  
    >>>> position of
    >>>> the observer", is precisely correct, and based on irrefutable
    >>>> statistical arguments. If Gary doesn't accept this, unlikely
    >>>> though it
    >>>> may seem to him at first sight, he should scan through previous
    >>>> discussion of cocked hats on this list, as Geoffrey Kolbe  
    >>>> suggested.
    >>>> If he remains unconvinced, we can go through the arguments once
    >>>> again,
    >>>> until one way or another, either he accepts it, or else he  
    >>>> convinces
    >>>> us otherwise.
    >>>> The best way to consider the matter is not to take a particular
    >>>> triangle, and then consider "where can the true position be?",  
    >>>> but to
    >>>> take a true, known position of an observer, and three celestial
    >>>> bodies. Now plot in his vicinity a set of three position lines that
    >>>> have been displaced by Gaussian amounts, toward or away from the  
    >>>> GPs
    >>>> of the bodies in question. (Indeed, the argument doesn't rely on a
    >>>> Gaussian distribution, just equal numbers toward and away). And  
    >>>> then,
    >>>> with the same observer position, and the same three bodies, plot
    >>>> another triangle, with different displacements, and another, until
    >>>> you
    >>>> are tired of it. The resulting triangles will be different each  
    >>>> time;
    >>>> some will be large, some will be tiny, some long and thin, others
    >>>> nearly equilateral. And in the end, if you check enough triangles,
    >>>> you
    >>>> find that 25% of those triangles will embrace the true position  
    >>>> that
    >>>> you started with.
    >>>> It doesn't matter how skilled or unskilled the observer is; it
    >>>> remains
    >>>> true. For a skilled navigator, of course, the triangles will
    >>>> indeed be
    >>>> smaller on average, which is where his skill shows itself, but  
    >>>> still,
    >>>> only 25% of those smaller triangles will include the true position.
    >>>> George.
    >>>> contact George Huxtable at geo...@huxtable.u-net.com
    >>>> or at +44 1865 820222 (from UK, 01865 820222)
    >>>> or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    > >
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