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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: The Darn Old Cocked Hat - the sequel 1
From: Hanno Ix
Date: 2013 Mar 11, 15:54 -0700
From: John Karl <jhkarl@att.net>
To: hannoix@att.net
Sent: Monday, March 11, 2013 10:51 AM
Subject: [NavList] Re: The Darn Old Cocked Hat - the sequel 1
From: Hanno Ix
Date: 2013 Mar 11, 15:54 -0700
John:
Thank you for your detailed answer
- I beg to differ, quite a bit, actually:
1. In the posting 122774 you state that the 3D Normal Distribution *is* the prob. density of the ship's location.
Yes, you can make good a-priory assumption
about throwing coins, but I wonder how you can know any
prob. density of a ship's location a-priory ?
2. If you
please will forgive me: below there is a remark of yours that makes no sense to me:
What counts most at sea is not your best estimate of where you are,
but the confident knowledge of where you're not.
Assume you can only be at one of two not-overlapping locations called "A" and "NOT-A", and there is no other
possible location for you to be, then both "alternatives" you cite are not contradictions but synomyms,
and the "but" has no meaning. You can conclude one half of the statement from the other.
Mayby you are trying to say something else that I could not understand?
3. Please, what is the difference of a "probability problem" and the "determination" of the "Maximum Probability Point"?
4. I proceed like this: Using the known coordinates of stars and the center of the Earth as my reference frame, I have CelNav as means,
with known errors and error distributions, to measure those coordinates. I can then deduce with the knowledge accessible to me
where the ship most likely is. Only when demonstrated verifyable errors or ommissions I will yield.
I have seen none yet.
It is however possible to draw different conclusions from different observances and different methods.
You know, my eyes report the sky as blue, space photographs, however as black? So what is the
fact?
.
Best
regards
h
From: John Karl <jhkarl@att.net>
To: hannoix@att.net
Sent: Monday, March 11, 2013 10:51 AM
Subject: [NavList] Re: The Darn Old Cocked Hat - the sequel 1
The "Cocked Hat" 9th March post
of Hanni Xi was an interesting addition to our many, many, posts on the subject. It's an observation of the type made by A. S. Goudsmit (physicist at M.I.T. in 1946), that the chances are 25% that the true fix lies inside a
3-LOP cocked hat. Both are probability problems that calculate the frequency of occurrence of a large number events, given all the data necessary for the calculation. They both start with a known position of a ship and then calculate fix locations relative to that known location, using many repeated observations.
In celnav at sea we are interested in exactly the opposite situation: Our ship is in an unknown position with just a single observation yielding the three-LOP fix. These St. Hilaire LOPs have known random standard deviation (in intercept distance) from many previous sights using a known position, the same sextant, and the same observer. (There's no azimuth error, it's calculated exactly.)
These two cases are very different -- apparently subtly different. In the first case, the ship's position is known and we make many observations. In the second case the ship's location
is unknown,
and we make one observation (the three LOPs). The first is a probability problem, like calculating that a coin flip very many times produces heads and tails, each 50% of the time. It has an exact answer. The second case has no exact answer, it's a problem in estimation theory. We estimate the ship's position using simple reasonable rules: use all the best available information, make no assumptions and no contradictions. The best we can do is determine the location of the Maximum Probability Point (called the MPP).
Attached is an expanded version of my 12/8/2010 post (14687). It discusses the results of the estimation-theory approach to the cocked hat question. The results are all very different from those of Hanno Xi and Goudsmit, after all, they're very different problems. But it's the estimation approach that's relevant for celnav at sea.
And finally,
this is all academic (while still interesting to some of us). What counts most at sea is not your best estimate of where you are, but the confident knowledge of where you're not.
John K
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