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    Re: Averaging
    From: Fred Hebard
    Date: 2004 Oct 20, 15:08 -0400

    On Oct 20, 2004, at 12:37 PM, Chuck Taylor wrote:
    
    > Certainly a straight-line fit, whether by eyeball or
    > by linear regression, is not rigorously correct in
    > this situation.  Still, within appropriate limits, it
    > is useful for highlighting possible outliers.  As
    > Kempthorne pointed out, a good statistician does not
    > necessarily *believe* his or her model.
    >
    > Standard procedure for plotting a line of position
    > using the St. Hilaire method calls for plotting a
    > straight line, when we know that what we "should" be
    > plotting is the arc of a great circle.  Still, the
    > straight line is useful.
    >
    
    I think one can make a stronger statement than that a straight line is
    useful; it _is_ the appropriate model to be fitting in these
    circumstances.  In that sense, it is rigorously correct.  Within the
    limits of accuracy of graphical techniques or measurement by any
    instrument, a plot of altitude versus time is a straight line with the
    following two caveats: over a short interval of time and away from the
    meridian.
    
    As Herbert Prinz quite rightfully pointed out, there are exceptions
    where one indeed could introduce systematic error in excess of the
    random error that is removed, so one has to be cautious.  These
    exceptions are not at meridian passage and over a short time interval,
    such as 5-10 minutes.  Even at high altitudes and about 10 minutes
    after meridian passage, the straight line fits all the variation, as
    shown in the second graph below.
    
    Here in the first graph are some data illustrating this, among the best
    shots I've taken.  Note that the R-squared is almost 1: there is no
    residual variation that could be usefully fit to another term.  The
    date was 10/1/04, the object the sun.  Time is UT1.  The residuals
    (Hc-Ho) were -.08, .10, .01, -.06, .18 and .02 arc minutes.  When
    fitting Hc by time for the same data, the R-squared was 0.999999 and
    the F value 2911325.
    
    
    
    
    
    Even 10 minutes after meridian passage close to the zenith, Hc versus
    time fits a straight line.  The following data are for the sun on
    10/1/04 at 4*48.6'S, 129*50.7'W.  Time is UT1.
    
    
    
    
    
    

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