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    Re: Resume of "Averaging"
    From: Alexandre Eremenko
    Date: 2004 Nov 5, 23:25 -0500

    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:-)

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