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    Re: Still on LOPs
    From: Trevor Kenchington
    Date: 2002 Apr 29, 20:37 -0300

    Peter,
    
    
    As a reluctant convert to George's conclusion, I'll try to answer a
    couple of your points:
    
    > What I currently find intriguing is the enlarged hat which retains the same
    > proportions as the original. I suspect that it does not need to be enlarged greatly
    > to increase enormously the likelihood of its containing the actual position, as
    > soon as we accept the notion that the further we move from the LOPs the less the
    > chance of encountering the actual position.
    
    
    This is one aspect that still troubles me, even though I cannot find
    fault with the proof of George's conclusion. The answer may be that the
    true position lies within 25% of the complete set of all possible cocked
    hats that might have resulted from a particular attempt to fix your
    position. For any one cocked hat, the chance isn't 25%, it is either
    zero or 100%. For most of the small cocked hats, the chance is zero
    (i.e. the true position is not within them), while for a fair slice of
    the large 'hats, the chance is one (i.e. the true position is within them).
    
    In practice, of course, we only plot a single cocked hat and we don't
    know where our true position is. Nor do we know whether our 'hat is
    large or small relative to the universe of other ones that we might have
      plotted. So we have to invert the logic leading to 25% of cocked hats
    containing our true position into a 25% chance of that true position
    being within our particular cocked hat.
    
    
    > The next step which is, inevitably, finding the fix position in exactly the same
    > place as it was in the smaller hat of the same proportions, doesn't just beg but
    > fairly shrieks the question, now thrice repeated: is this not all an exercise in
    > futility?  25%, 14.whatever%, what difference does it make if the fix position, the
    > only point we can calculate and use, is exactly the same?
    
    
    I'd not say it is futility. We need to know not only our best estimate
    of where we are but also how imprecise that estimate may be. If our true
    position was always inside any cocked hat that we might draw, we could
    disregard any possibility of our being outside its borders. We would
    know, with absolute certainty, that we were not outside the triangle.
    What George has been trying to tell us (those of us who did not already
    know -- as many evidently did) is that there is actually a high chance
    that we are outside our 'hat and so should take appropriate precautions
    if the borders of that 'hat fall anywhere near any dangers.
    
    > Still suspect that the base for the 25% idea has a slim and tottery foundation: the
    > 50% chance of a RANDOM point falling on either side of a RANDOM line, which becomes
    > 25% when two lines intersect.
    > As somebody else pointed out, they are not random at all, each LOP is an
    > approximation of the actual position, this is why they come together in a
    > (hopefully) small hat.
    
    
    It is not  matter of the lines being random. Of course they are not. It
    is a matter of the _errors_ around the true LOPs (the ones that pass
    through our true position) being random. Actually, even that is not
    required. All that the proof of the 25% requires is that there is an
    equal chance of the error being to one side or the other of the true LOP.
    
    I have previously suggested, and I hold to it, that the errors in real
    LOPs are not symmetrical in that sense. It is likely that real, flesh
    and blood navigators tend to err on one side or the other, on average.
    Still, since the orientation of the navigator and the aids to navigation
    vary from one fix to another, I suspect that those inadvertent biases
    equal out over all of the fixes on a voyage.
    
    > Let's look at this (yet!) another way. Visually. Draw (with your mind's eye if you
    > are lazy) a hat with generous proportions within and without on an A4 sheet (as I
    > have been). Now according to the 25% theory put 25 points at random within the hat.
    > What are we to do with the 75 we have left to distribute? I suppose we have to
    > place them at the same random distances outside the hat. Some of them, since there
    > are so many compared to inside the hat, are going to land an awfully long way from
    > the hat itself. Nevertheless each is presumably as valid as any other. But no other
    > distribution, either, seems to make much sense. According to the 'LOP approximation
    > of AP' fact, and also the 'careful navigator' assumption which accompanied the
    > beginning of this discussion, since we still have only 25 for inside the hat the
    > remaining 75 will have to be thickly clustered all along, and close to, the outside
    > of the LOPs, since the further away from the LOPs we go the fewer points are to be
    > found. This picture is truly absurd, it looks as though the 75 outside are beating
    > at the walls, trying to get inside!
    
    
    Pretty much. If there is only a 25% chance of the true position lying
    inside the cocked hat, there is an additional 50% or so chance of it
    lying outside but not far away and only a small chance that your
    position is wildly wrong.
    
    What we need, but this discussion has not really yet progressed to, is
    some well-established but easily-applied rules of thumb that would tell
    us how far away from the centre of the 'hat we may be -- or rather the
    probability that we are more than some distance from it. A practical way
    to draw a 90% confidence ellipse around the MPP set within a particular
    cocked hat would be really useful!
    
    
    
    Trevor Kenchington
    
    
    
    --
    Trevor J. Kenchington PhD                         Gadus{at}iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
    
                         Science Serving the Fisheries
                          http://home.istar.ca/~gadus
    
    
    

       
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