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From his message
to Peter Fogg Fri Nov 05 2004 - 05:14:14 EST
I think I can understand now what is Herbert's 3-d
objection against the simple averaging.
(Which is the "most important objection" now, after the previous
ones were dealt with).

He apparently has in mind the following situation.
We observe the altitudes of Sun, for example, at
9 a.m. taking 5 altitudes over 5 minutes.
Then at 11 a.m. we do the same, but because of the clouds
take only 4 observations in 5 minutes.

Then at 2 p.m. we manage to see the Sun again, and take 3 observations.
Now we want to feed all these 12 altitudes and times to
a computer, and the computer is supposed to return
"a fix" that is latitude and longitudes, two numbers.

The question is what algorithm should we program to this computer.
My answer is "Least squares", there is no doubt in it.

Based on such examples, Herbert claims that
"averaging is the thing of the past", that "averaging altitudes
is always wrong" and so on.

OK, maybe indeed it is the "thing of the past". In the same sense
that manual reduction with tables, and plotting position lines
are the "things of the past". Actually I suspect that the whole
subject of CelNav is "the thing of the past" in the same sense.

Few comments.

1. Even in this situation, there is nothing "wrong" with averaging
each of the three series BEFORE you feed them into computer.
Because computer power is cheap, it is just the question of what
you prefer: to do some preliminary computation (averaging) and feed
to your computer 3 pairs of numbers, or to feed all 12.

2. A single position line has independent value.
I would plot my first position line on the map at 9:30 a.m,
without waiting for the other 2 series.
In obtaining this position line IT DOES NOT MATTER whether you
feed to the computer 5 pairs of numbers or one pair, their average.
The result will be the same.
But if you do reduction with tables, reducing 5 altitudes will
take 5 times the time needed to reduce the average. With the same
result if you make no blunder during these 5 reductions.

3. Even after the reduction of all 3 series, the result is NOT
just the pair of numbers computer can give you.
The result is the whole triangle formed by the three position
lines. You DON't really know exactly where you are.
Somewhere inside this triangle or maybe near it.
The size of the triangle tells you approximately the dergee of
indeterminacy of your fix.

3. Herbert says:
"It would be only fair that those who propose averaging should also
provide this analysis"

I am sorry. Those who "proposed averaging" are dead for many
hundreds of years
already. The procedure is so evident and universal in every
situation when we MEASURE something and want to increase
precision, that those people who first "proposed" it are even not
remembered. The procedure is used everywhere in science for
many hundreds years. For rigorous
mathematical justification I recommend to
read Gauss. Or a GOOD modern book on "the theory of errors".
Or to take an
undergraduate statistic course:-)

Alex.

   

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