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    Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT
    From: Brad Morris
    Date: 2009 Apr 16, 13:37 -0400

    Hi Marcel
    
    There is no doubt that the human eye can place a line pretty reasonably.
    I was merely pointing out the difference between a mathematically rigorous
    least squares fit (which is what the LINEST function provides in Excel), and the proposed methodology.
    That isn't subjective or an opinion, rather it is a factual observation.
    
    I eagerly await Jim and Frank's independent results.  At this point, I think
    we can all begin to see that the proposed methods will provide an estimation
    of longitude (latitude is nearly free!).  What remains is a debate regards
    how well it performs. There may be exclusionary limitations, as alluded to
    by Henry and others.
    
    Best Regards
    Brad
    
    
    
    
    
    -----Original Message-----
    From: NavList@fer3.com [mailto:NavList@fer3.com] On Behalf Of Marcel Tschudin
    Sent: Thursday, April 16, 2009 1:04 PM
    To: NavList@fer3.com
    Subject: [NavList 7974] Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT
    
    
    Brad, regarding your comment on the least square fit:
    "The navigator is urged to, in one case, fair a line thru a series of
    data points or in another, to visualize a parabola as we slide the
    paper.  That isn't a least squares fit by any stretch of the
    imagination."
    It's actually surprising how well we can estimate a linear regression
    line through data points with a reasonable correlation only by using a
    ruler. In this case it would only require to adjust the scale of the
    axis accordingly.
    
    Marcel
    
    On Thu, Apr 16, 2009 at 6:55 PM, Brad Morris  wrote:
    >
    > I agree with your result for a least squares fit of the parabola to the 
    measured data. With the variation
    > limited, there is no doubt that the least squares fit will provide a 
    reasonable parabola and therefore a
    > fairly good estimation of Lat and Lon.  If this is how the method is used in 
    practice, you will find little
    > argument from me.
    >
    > However neither procedure, as I understand them to date, uses a least squares fit.  The
    > navigator is urged to, in one case, fair a line thru a series of data points 
    or in another, to visualize
    > a parabola as we slide the paper.  That isn't a least squares fit by any stretch of the imagination.
    > As such, we may find that the deviation in longitude is somewhat greater than your value of 5.68nm.
    >
    > Best Regards
    > Brad
    >
    >
    >
    >
    > -----Original Message-----
    > From: NavList@fer3.com [mailto:NavList@fer3.com] On Behalf Of waldendand{at}YAHOO.COM
    > Sent: Thursday, April 16, 2009 11:23 AM
    > To: NavList@fer3.com
    > Subject: [NavList 7970] Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT
    >
    >
    > The .doc file works great. Thanks.
    >
    > I took George's spreadsheet and automated it with a macro.
    >
    > First, I used it to generate 1000 sets of 13 observations.  For each set, I 
    let excel fit the parabola, and find the time and altitude of its peak.  I 
    then find the difference between the resulting latitude and longitude thus 
    calculated and the true values.  Thus for each set of 1000, I get a mean and 
    standard deviation of the error.
    >
    > Using George's convenient "scatter switches", I tried various combinations.  
    He has four switches. In order, the first turns on/off the scatter in ship 
    speed (if one always uses the nominal 10 knots in the calculation), the 
    second changes the equation of time (as I understand it, changing the value 
    to be calculated as opposed to introducing scatter into the result), the 
    third changes the longitude (same comment), the fourth adds scatter to the 
    observed altitude. In the table below, the switches are listed in order, 
    1=on, 0=off.
    >
    > As can be seen for everything on, the standard deviation of longitude error 
    in nm is 5.68.  It can be seen the ship speed scatter seems to have a 
    slightly greater effect than the altitude reading scatter.
    >
    > The 1 mile lat and 5 mile long estimates look pretty good to me. YMMV.
    >
    >
    >
    >
    > 1,1,1,1 min lat min long        nm long
    > mean    0.00    0.19    0.11
    > stdev   0.40    10.16   5.68
    > max     1.29    48.02   26.85
    > 0 1 1 1
    > mean    0.01    0.19    0.10
    > stdev   0.44    6.57    3.67
    > max     1.55    22.64   12.66
    > 0 0 1 1
    > mean    0.00    -1.00   -0.56
    > stdev   0.42    6.73    3.77
    > max     1.41    20.77   11.61
    > 0 0 0 1
    > mean    -0.01   -0.38   -0.21
    > stdev   0.44    6.72    3.76
    > max     1.14    20.50   11.46
    > 1 1 1 0
    > mean    0.00    -0.22   -0.12
    > stdev   0.05    7.48    4.18
    > max     0.25    24.18   13.52
    > 0 1 1 0
    > mean    -0.01   -0.38   -0.21
    > stdev   0.00    0.01    0.01
    > max     0.00    -0.34   -0.19
    > 1 0 0 1
    > mean    0.01    0.17    0.09
    > stdev   0.42    9.97    5.58
    > max     1.33    32.45   18.14
    > 
    >
    >
    >
    >
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    >
    
    
    
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