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My last contribution to this thread before heading off on another
business trip:

Michael Wescott wrote:

> Trevor wrote:
>
>>Geoffrey wrote:
>>
>>>My "proof", as you were kind enough to call it, has nothing to say
>>>regarding the size of the 'hat.
>>>
>
>>I did not suggest that it had. But your proof leads to the conclusion
>>that there is a 0.25 probability that the true position lies within the
>>cocked hat _regardless_ of the size of that 'hat. That is
>>counter-intuitive. Either there is a reason why what seems intuitive is
>>wrong or else there is some error in your proof that I cannot see. I was
>>wondering whether you could dismiss the latter possibility by pointing
>>to the reason for disregarding my argument that the number cannot always
>>be 0.25.
>>
>
> Once you have a specific 'hat and a specific true position there
> is no matter of probability. The true position either lies in the
> 'hat or it doesn't. It makes no sense to talk of the probability
> of the true position being in the 'hat; it's either inside or
> outside.


Obviously. But we are considering the real situation in which you have a
cocked hat and you know that there is a true position somewhere but you
do not know where. In order to quantify the probabilities of that true
position being in various places, we are postulating various
hypothetical possibilities. But nobody has suggested that a specific
true position has some probability, other than zero or one, of being
inside or outside some specific 'hat.

We art not dealing with quantum mechanics and the positions and
velocities of electrons. (Fortunately!)


> What we can talk about is the pattern or distributions of 'hats
> that get generated near a true position. As shown before, 25% of
> the generated 'hats will enclose the true position and 75% will
> not. There are two assumptions that have to be met for this to be
> true. First, the distribution of the error must be symmetric
> around 0, meaning there is no inherent bias in the errors in
> either direction (toward or away).  Second, the errors must be
> independent in each of the observations.
>
> If we add the addition assumption that smaller errors are more
> likely than larger errors, and we generate a large number of hats
> then we'll notice several things. First, 25% of the hats will
> contain the true position. Second, the hats vary quite a bit in
> size but larger hats enclose the true position more often than
> smaller ones. And third, there are more smaller hats than larger
> ones.
>
> More smaller hats which tend to be near but not enclosing the
> true position. Larger ones are rarer but tend to surround the
> true position. That should help the intuition.


I'm not so sure that that argument addresses the point that I was trying
to judge intuitively but it does raise a very important point that
should have been evident from inspection of Geoffrey Kolbe's diagrams
but which I, for one, did not notice: Given the same magnitude (but not
direction) of error in the three LOPs, those cocked hats which enclose
the true position are LARGER than those which do not. (I have not
checked that for non-equilateral 'hats, nor for those in which the
errors differ among the three LOPs. Is it true with those too?)

Hence, the long-standing assumption that a smaller cocked hat indicates
a "better" fix needs to be reconsidered. If it is smaller because the
three LOPs were more precise than usual, that is correct -- whether the
precision is due to improved measurement or just to the LOPs chancing to
lie closer to their correct positions. But a small 'hat could also be a
warning that your true position is unlikely to be inside it. A
disturbing revelation!

Turning back to my earlier point that Michael thought he had helped
with: My concern was not with the size of the 'hat changing, despite
constant absolute values of errors, depending on where the 'hat fell
relative to the true position. I was concerned with changing sizes of
the 'hat caused by different absolute values of the errors drawn from
constant error distributions. I'm not about to consider the limiting
cases of zero error (which I had avoided before Rodney pointed out that
it should not be used) or an infinitely-small 'hat. However, there are
real cases of very, very small 'hats arising by chance (NOT by
geometry). The logic that most contributors to this debate seem ready to
accept tells us that 25% of the complete universe of those contain the
true positions that they were estimates of. I'm still having trouble
accepting that while, if it has to be rejected, then it seems to me that
the whole proof of the 25% will go down in flames.


Michael's response to mine continued:

>>>Similarly, if you only have one 'hat, then placing the MPP in the centre of
>>>it is the best you can do.
>>>
>
>>That is indeed the MPP, if the 'hat is equilateral.
>>
>
> No. Given 3 LOPs and a 'hat, the MPP is equidistant from each side of
> the hat; in the center of the inscribed circle. The mutual orientation
> of the LOPs won't alter that.

Steven Tripp's calculations suggest that it does. With a malformed cocked hat, 
taking the shape of a narrow isosceles triangle, the two near-parallel sides 
serve to place the MPP on one dimension but their intersection point provides 
little information relevant to the MPP's position on an axis perpendicular to 
that first dimension. Hence, the MPP lies close to the third LOP and to the 
bisector of the first two but not at the geometric centre of the triangle.

If there are mathematical reasons to reject that conclusion, I fear that we 
will need to take this discussion to a higher level of complexity.


> There remains one question. Is the contour around the MPP elliptical?
> For that to be true, we have to be more specific about the distribution.
> If it's Gaussian then the contour is elliptical.

That is a point which has come up before on this thread. If there are but two 
LOPs, it is clear that the contour is elliptical with Gaussian distributions. 
Are we to understand that that is also true if there are three LOPs? More 
than three?

I'm not asking for the mathematical proof, which I likely wouldn't understand 
anyway. But I would welcome somebody laying out his or her credentials as an 
authoritative voice and then telling me one way or another.


Turning to Rodney Myrvaagnes' posting, which although addressed to Geoffrey Kolbe

deals with a more general point:

> Your statement is:
>
> 1) that in a long enough run of observations, 25% of triangles of a
> given size will contain the true location.
>
> OR
>
> 2) that in a long enough run of observations, not more than 25% of
> triangles of a given size will contain the true location.
>
> If 2 is the assertion, there must be some size of triangle that
> maximizes the function P(inside) or approaches most closely to 25%. I
> would be much happier with an analytic answer to this one, since I have
> not seen anything yet that doesn't appear to assume a flat, rather than
> normal, distribution. If I have missed something that does take care of
> this, perhaps someone will send it to me off list.

The claim that Geoffrey, George and others have made is clearly #1. Their 
proofs have led to the conclusion that 25% of cocked hats must enclose the 
true positions being estimated, not that a maximum (nor a minimum) of 25% of 
them do.

The proofs that they have offered are not, however, dependent on the 
distribution of errors being rectangular, triangular, Gaussian, Poisson, 
Gouchy or anything else. All those proofs require is that there be an equal 
chance that the observed LOP lies on either side of the true LOP and that the 
side on which a particular observed LOP lies is independent of the side on 
which the other ones do. (We have been talking symmetrical errors but even 
that assumption is not strictly necessary to the proofs. The 25% would still 
arise if errors to the left tended to be smaller than those to the right, 
provided that there are an equal number to each side when summed over a very 
large number of observations.)


Those proofs, at least as presented by Geoffrey, are fairly
straightforward. I'm not comfortable with the conclusion that they lead
to but I still can't see any fault in the logic.


Trevor Kenchington


--
Trevor J. Kenchington PhD                         Gadus@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
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