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In reply to my comment about the presentation of multi-LOP results by the
AstronavPC procedure-
"The problem is that [...] the confidence-ellipse, is based on only a single
sample. It's relying on the statistics of a single event. "

Frank commented-
"Right. But I wouldn't describe this as a fundamental flaw. It's just a
particular choice in the way they're calculating the confidence ellipse."

===============

If you calculate the confidence ellipse from the way that three LOPs
intersect, AND NOTHING ELSE, then there is no other choice. And as a result,
there is the significant chance that they will intersect in such a way as to
give a seriously misleadingly-small answer for the confidence ellipse.

The answer, of course, is to introduce additional information, and a dash of
common sense. To treat each position line as being surrounded by an
error-band, just as Frank suggests. And if a computer-analysis program for
multiple sights could do just that, all would be well.

But what I was pointing out was that the error-ellipse was indeed
calculated, by that program, without reference to any such commonsense
notions. As a result, significantly often (from a three-sight cocked-hat),
it will so happen that the user will be presented with a confidence-ellipse
that bears no relation to the expected scatter in the deduced position. If
the user takes that ellipse, drawn enticingly on each plot, at face value,
and takes it to really mean what it looks as if it means, he will be badly
misled. And could be led into danger. And there is no warning provided about
the matter, in that document, or (as far as I can tell) on the computer
screen. That is the fundamental flaw.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Frank Reed" 
To: 
Sent: Tuesday, March 02, 2010 2:30 AM
Subject: [NavList] Re: How good is St. Hilaire?


George, you wrote:
"The problem is that [...] the confidence-ellipse, is based on only a single
sample. It's relying on the statistics of a single event. "

Right. But I wouldn't describe this as a fundamental flaw. It's just a
particular choice in the way they're calculating the confidence ellipse. The
way the analysis is done, as presented, is by treating the standard
deviation of the observations as an *output* of the statistical analysis.
But you can also treat it as an input. And in fact, when I've coded this up
for myself, that's how I do it.

Consider the case of a single LOP. We can put a confidence "band" along that
line with a width based on previous experience with that instrument and that
observer. For example, we might have some data from the previous week's
worth of fixes. We use that to "smudge" the LOP and give it some thickness.
Everybody does this, even if it's just a mental adjustment. And if a
confidence "band" doesn't sound very confident, then, of course, if we also
have a DR position with some confidence limits around it, then we can
combine those two and get a long and thin confidence ellipse where the
"smudged" LOP overlaps the DR confidence ellipse (I assume most navigators
do this part in their heads).

With a two-body fix, we can draw a confidence ellipse again based on the
standard deviation of observations as an input. It's an ellipse about the
same size as the "overlap box" for the error bands around each LOP. And with
a three-body fix, this answers your concern about a confidence ellipse being
drawn too small when the LOPs just happen by chance to coincide in a small
triangle. That small triangle should not imply a small confidence ellipse,
and it doesn't if the s.d. is an input rather than calculated from the
observations themselves.

Eventually we get to cases with larger numbers of observations. For example,
there's the case that I call a "rapid-fire fix" where an observer is
shooting one body (presumably the Sun) every few minutes over an extended
period of time. Or we might be interested in an automated system that is
shooting the stars several times per second. In these cases where the number
of observations is much greater than one, it does make sense to treat the
standard deviation as an output of the data analysis. This observed s.d.
might not match our input s.d. perhaps because there's a hazy horizon or
some other factor making conditions significantly worse than expected or
occasionally better than expected. An automated system with a large number
of observations would have a confidence ellipse that fluctuates in size as
observational conditions change. This would be similar to the fluctuating
size of the confidence ellipse around a GPS ellipse as the number of
available satellite signals changes.

There's probably some ideal N of observations where the calculated standard
deviation is better than an input standard deviation. I will vote for five.
If we have fewer than five observations, use an input s.d. for the error
ellipse calculations. If more than five, use a weighted average of some
sort. After ten or twenty observations, the observed s.d. should dominate.

There's no reason to treat that publication as a bible, right? Re-derive its
math from first principles, and try out the different ways of treating the
s.d.

By the way, back to that bit I mentioned about combining a celestial LOP
with a DR error ellipse. Suppose we have two LOPs with a very low angle
between them. The confidence ellipse around them is rather long and thin. If
we ALSO have a confidence ellipse around a DR position, derived in some
meaningful way, then the best estimate of position lies in the area where
those ellipses overlap, not at the center of the celestial fix. That, it
seems to me, is one area where traditional navigation does not have any
systematic rules in place, despite the fact that this case is open to
relatively simple statistical analysis. Instead it's just assumed that the
navigator will figure it out by intuition or "common sense".

-FER


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