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    Re: Least squares / was: Jupiter's BIG.
    From: Fred Hebard
    Date: 2003 Dec 24, 11:39 -0500

    Thank you for your kind response to my query.  Now if I can work
    through it and understand it, I'll be doing OK!
    I might add that part of my interest in this is delving into how
    physicists and astronomers handle these very accurate and precise data,
    which leads, in part, to how all these navigation procedures were
    developed.  It's outside my area of experience in biology, where often
    we're lucky if the standard deviation of a set of measurements is less
    than mean!
    In addition to the thanks, this post is to answer your simple
    questions, which I do in the edited text below.
    On Dec 24, 2003, at 9:44 AM, Herbert Prinz wrote:
    > Fred,
    > In your original post on this subject, you spoke of using linear
    > regression
    > for choosing the "best" single observation from a set. I assume the
    > "best"
    > would be the one with the smallest distance from the regression line.
    > Yes?
    > You also said, "The data look very nice; not as good as the lunar of
    > that
    > Australian explorer Kieran Kelly discussed a few months ago, but very
    > good."
    > You did not elaborate on what your criteria are to make data look "very
    > good". But you are probably referring to the correlation between
    > distance
    > and time. Yes?
    Correct.  I failed to mention that the correlation coefficient or its
    square also drop out of the regression and give one a number for
    comparing various lunars.  But, again, it's just that the software
    package I use to process data makes it very easy to plot of distance
    versus time and draw a line through the data.  I'm using it as the
    equivalent of plotting the data by hand and drawing a line through the
    points using a ruler.  If all the points are on the line, the data are
    good (with the caveat you discuss in the next paragraph).  If they're
    scattered all around the line, they're not so good.  These quick visual
    checks can be quantified by the correlation coefficient, but it's the
    speed of the quick visual check that is nice.  You can instantaneously
    assess the quality of the data.  I suppose when one becomes proficient,
    such checks are not necessary, but...
    > Of course, you can use linear regression in this way only if there is a
    > linear dependence between distance and time. Given that Frank's seven,
    > or
    > so, observations were taken within twenty minutes and Jupiter was well
    > positioned, that's reasonable. We trust Frank that he checked this
    > before he
    > shot, but you can't really take it always for granted. For small
    > distances
    > or long time intervals, you might consider using a second order
    > polynom for
    > your regression.
    >  However, when I picked up on the term
    > "least squares fit" that you used, which is of course at the base of
    > the
    > linear regression method, I was not speaking of linear regression at
    > all. I
    > was thinking of using a least square fit for the _reduction_ of the
    > sight,
    > not for data preparation. I would not know how to reduce a single set
    > of
    > distance observations by linear (or any other type of)  regression.
    I agree.

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