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    Re: Ellipse of Confidence in position finding
    From: Bill Lionheart
    Date: 2019 Mar 15, 08:58 +0000

    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
    >> View and reply to this message
    > fair winds!
    > --
    > Andrés Ruiz
    > Navigational Algorithms
    > http://sites.google.com/site/navigationalalgorithms/
    > View and reply to this message

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