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    Re: The cocked hat, again. Was: "100 Problems in Celestial Navigation"
    From: Joel Jacobs
    Date: 2004 Apr 3, 07:58 -0500

    In an off group exchange with Herbert Prinz, to which I have added a few
    words to for clarity, I said this:
    "It is really a matter of personal judgment. The navigator's selection of an
    MPP will vary as to how reliable or accurate the navigator thinks the sights
    were based in the conditions he encountered when taking the sights. A
    mathematical solution may not be appropriate."
    I submit that this is what George is correctly introducing.
    Let me add this thought. When I took the USCG Celestial Exam, year ago, one
    of the questions had a series of sights that when plotted, had one sight's
    LOP distant from the others. If  you answered the question including that
    sight in your calculations, even by averaging, you were wrong. If you
    rejected it as a bad sight, you were right. As a matter of pride, I got
    100%. Of course I was a lot younger and sharper.
    Joel Jacobs
    ----- Original Message -----
    From: "George Huxtable" 
    Sent: Saturday, April 03, 2004 4:23 AM
    Subject: The cocked hat, again. Was: "100 Problems in Celestial Navigation"
    > Joel Jacobs wrote-
    > >I remember using the least squares method in statistical analysis to
    > >establish a line of central tendency, used it to separate the fixed and
    > >variable elements of costs in financial analysis, and in a nautical
    > >to determine fuel burn at various RPM and speeds. I discovered the bother
    > >was NOT worth the effort, and did it visually using a see through or
    > >rule which seemed to work equally well.  You visually average the
    > >of the observations around a line, that on the average, would split them
    > >equally above and below what be came the line of best fit.
    > I think Joel has the right attitude here. The human brain is good at
    > instinctive analysis of such data.
    > >As far as the cocked hat, did you mention that bisecting the apex of all
    > >angles would produce a point that has some mathematical, though not
    > >necessarily accurate, MPP? I think that it would work for more than three
    > >angles.
    > That has been proposed, but applies only to the unlikely situation that
    > there's a common systematic error, applying exactly the same to the three
    > observations, and no random error, such as would be caused by motion in a
    > seaway. It could apply, in smooth conditions, if a mistake was made in
    > measuring index error, or if anomalous dip affected every measured
    > altitude. In real life at sea, most of the errors will be random ones, and
    > that method will not produce the most probable position.
    > ====================
    > But I wish to argue here against putting undue emphasis on a calculated
    > "most probable position" (MPP) within the cocked hat. There remains much
    > misunderstanding, among navigators, about the interpretation of a
    > cocked-hat error-triangle. The argument applies just the same to triangles
    > formed by three intersecting intercepts, or triangles formed by three
    > intersecting compass bearings. Long-standing listmembers will recall that
    > this question was given an energetic thrashing a couple of years ago, and
    > there's no need to go over those arguments again.
    > Making certain reasonable simplifying assumptions, which are-
    > 1. That all systematic errors have been corrected, so all that remains is
    > random error in the intercept, which is just as likely to be "toward" as
    > is "away".
    > 2. We can neglect the unlikely event of there being no error at all in the
    > intercept.
    > Then the likelihood of the true position lying within the triangle is one
    > in four, and correspondingly, it's three times more likely to be outside
    > the triangle (three times out of four).
    > This surprising (to many) conclusion is based entirely on the statistics
    > the problem, and is quite unrelated to the skill of the navigator. A good
    > navigator will produce smaller error-triangles than a bad navigator will,
    > but still, any such triangle is only 25% likely to embrace the true
    > position.
    > Many navigators still retain the delusion that their true position MUST
    > somewhere within the cocked hat, and the only question is exactly where,
    > within that triangle, it may be. Nothing could be further from the truth.
    > That's why a prudent navigator, having drawn a cocked hat on his chart,
    > then (mentally, at least) applies a sizeable thumb-smudge to his chart,
    > embracing the triangle and a lot more sea besides, and says "I'm somewhere
    > here. I think."
    > I accept that for some purposes, particularly when navigating by computer
    > or electronic chart-plotter, it may be necessary to establish some unique
    > mathematical point, about which an error-ellipse is to be centred. In that
    > case the MPP, calculated in the way that Herbert Prinz suggests, will
    > provide the best answer. What I wish to emphasize is that this MPP should
    > not be invested with any magical significance, and the "navigator's
    > thumbprint" should be kept firmly in mind.
    > In my opinion, for chart-navigation purposes, the intuition of the human
    > mind is sufficiently accurate to estimate some sort of most-probable
    > position from a cocked hat, without any need for statistical analysis or
    > geometry. To that extent, I agree with Joel (if that represents his view).
    > George.
    > ================================================================
    > contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    > 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    > Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    > ================================================================

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