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Thanks for that detailed list to the history. Its not so easy to catch
up with previous NavList discussions! One of the article says "In
celestial navigation, the bisector of the azimuths of two Marcq Saint
Hilaire LoPs ia a true line of position that cancels out systematic
errors."  I see  that is interesting - if the only error was for
example a fixed index error, or a habit of misestimating the horizon
one way, and direction towards the GP was opposite for the two lines
this makes sense. But then the incentre (Kimberling centre X(1) )
gives the intersection of the angle bisectors as the true position
assuming the direction of the GPs is either towards the inside of the
hat or away from it.  The Index error can be measured though, and
perhaps once can assume that eventually one would estimate other
systematic errors either from either taking sights on more bodies or
from taking sights when the position is known be an independent fix of
some kind.  One is then left with zero man random errors, and the
(possibly weighted) symmedian.

If we want to model the errors with a systematic and random error, eg
Gaussian with the the same non zero mean and variance in each LOP,
variance known but not the mean, we need more than three LOPS. I have
not finished scouring the archive, has someone explained how to get
the ML estimate in this case yet?

Bill L




On Thu, 14 Mar 2019 at 21:32, Andrés Ruiz  wrote:
>
> Bill, see only:
>
> An outside fix example
> http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39403 see "CelNav.bisectors.pdf"
> http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39410
> http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39411
> Fix NOT inside EoC
> Fix = intersection of bisectors
>
> An example of an inside fix
> http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39404
> Fix IS inside EoC
> Fix = intersection of bisectors = CoG( cocked hat) = symmedian =...
>
>
> El mié., 13 mar. 2019 a las 15:33, Bill Lionheart () escribió:
>>
>>
>>>
>>> Another important thing about ellipses. Remember my post with an example 
where the EoC, the symmedian point, and others give us the wrong position.
>>> Fix by bisectors
>>> http://fer3.com/arc/sort2.aspx?subj=3+sights+SR+example&author=&y=201401&y2=201912
>>> http://fer3.com/arc/sort2.aspx?subj=That+darned+old+cocked+hat&author=&y=201001&y2=201912
>>>
>>
>> I had a bit of a look at that thread and saw a lot of confusion.  You get a 
probability density function and that is correct given the assumptions. As 
navigators we have sometimes to choose a point estimate from the PDF, for 
example as departure point for our next dead reckoning. One should always 
understand a "fix" as just a point estimate of a probability distribution and 
so consider the risks to navigation of being at the "most dangerous" location 
within a probability contour.
>>
>> I think practically the common reason the symmedian is not a good point 
estimate is that the variances of the LoPs are different, eg they were 
obtained from linear regression fit for a different number of altitude 
observation of the same body. Then we need the weighted symmedian (weighted 
least squares solution) as our maximum likelihood estimate. It is still 
inside the cocked hat for three LoPs but it could be anywhere in the interior 
depending on the weights.
>>
>> Bill Lionheart
>>
>> 
>
>
> fair winds!
> --
> Andrés Ruiz
> Navigational Algorithms
> http://sites.google.com/site/navigationalalgorithms/
>
> 

   

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