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    Re: Cocked hats, again.
    From: Gary LaPook
    Date: 2007 Mar 13, 17:03 -0700

    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 george---.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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