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Subject: Re: Still on LOP's
From: George Huxtable (george@XXX.XXX)
Date: Thu May 02 2002 - 06:22:45 EDT
I am beginning to wonder whether we are reaching the point of diminishing
returns in this discussion about cocked hats. However, it seems to be still
attracting some interest, even if many of the comments are, or seem to me,
quite misplaced, and so I will respond.
First, to deal with Dov Kruger's contribution. I would prefer to deal with
Dov's own views directly, views he can defend himself, rather than those of
a third-party guru who is "thinking about it". However...
On his guru's behalf, Dov said-
>You can't assume the errors are normally distributed.
There's no need to assume that. Let me quote my statement about the cocked
hat,made back on 13th April in reply to Peter Fogg.
>It's a surprising fact that no matter how good the navigator, only one
>time >in four will his cocked hat embrace his actual position, which is
>three >times more likely to lie outside it. This is a universal truth,
>relying in >no more than this proposition: that each position line, being
>the best >estimate that can be made, is just as likely to lie to the left
>of the true >position as to the right.
That is the only assumption that is needed, equal numbers right and left.
He added-
>You further have to assume equal errors on all readings for the moment.
No, you don't have to. Why should you?
and he said-
>...and observations of phenomena where your only possible error is
>>one-sided (I don't think that second one is too relevant for navigation).
Well, it might well be relevant for navigation, if you think of the way
that sideways tilt error of a sextant always gives rise to larger measured
altitudes, never smaller. But error distributions of that type might well
be contrary to the 'equal right and left' assumption stated above.
Dov continues-
> Making a statement like p(inside the triangle) = 0.25 assumes that
>each reading is independent. They are not.
Yes they are. The actual position is the actual position. You can't do
anything to change it by taking a bearing. It stays fixed, though the
observer may or may not know where it is. So taking one bearing on a
landmark has not the slightest effect on what the observer will measure
when he takes another bearing, on that same landmark or another one, from
that same position.
On his own behaf, Dov goes on to say-
>You can't talk about the probability of being outside the triangle without
>>knowing the variance of your measurements.
Yes you can. That's what the logic of the cocked hat problem predicts.
I am in agreement with Dov's 4-step proposal for a statistical test. It
needs someone who is better-equipped for computer graphics than I am.
When he says-
>I will also guess that the probability of being within the triangle goes
>up at least quadratically if the triangle is increased in size by
>projecting lines parallel to the sides outside it. So if p(outside) =
What does this mean? What is a "quadratic" increase on a probability of
less than 1?
Nevertheless, it's pleasing to note that both Dov and Jared Sherman accept
that the probability of the cocked hat embracing the actual position is
going to be less than 1, even if they disagree with the value of 25%..
Jared Sherman said-
>George, all this talk of simulations and errors makes me wonder how anyone
>>can discuss simluations at all before one has even enumerated the
>possible >errors and error conditions.
Well, if he wishes to challenge the validity of the 25% probability of
being within the cocked hat, under the conditions as defined above, then to
test that there is no need to consider "possible errors and error
conditions". All that is needed is to reproduce those specified conditions,
equal numbers left and right, and nothing more. And if he thinks those
specified conditions are unreasonable, let him say so, but then he would be
attacking something different from the proposition that I set out.
Jared added-
>Which brings me back to the real point: None of this matters, and it is
>>arguably of no real use at all to argue whether the exact position lies
>>here or there.
Well, I think it does matter, though most of the juice has been squeezed
out of this argument by now. None of it tells you where the exact position
lies, just the probability of finding it within a particular zone of a
plot. If this prolonged discussion has disabused a few navigators about the
common misconception that a cocked hat must contain their true position, it
will have done some good. Whether they accept that the true figure for that
probability is 25% or some other figure doesn't concern me greatly.
On that note, I would go along with Jared's comment-
>If you've plotted a cocked hat, you also plot a circle of error around it,
>>and you say "I am in this circle" not in (or out) of the cocked hat.
George Huxtable.
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george@XXX.XXX
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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