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    Re: That darned old cocked hat
    From: Antoine Couëtte
    Date: 2010 Dec 13, 21:23 -0800

    To he attention of John Karl, and George Brandeburg


    Re 1 : [Navlist # 14823] dtd 201012 by George Brandeburg (http://www.fer3.com/arc/m2.aspx?i=114823&y=201012), and

    Re 2 : [NavList # 14828] dtd 201012 by John Karl (http://www.fer3.com/arc/m2.aspx?i=114828&y=201012)

    Dear John,


    In continuation of the most interesting debate on "That darned old cocked hat", thank you very much for having included 3 different size cocked hats, with the same shape but not the same dimensions in your post Re 2.

    Still, through studying again your error ellipses and especially the ones in Re 2, I keep having the following (big) difficulty. Maybe you can help me here, and if so, I am sure that you will also render your explanations much easier to follow.

    The big "issue" I have with your published probability ellipses is easily interpreting them. The ellipses you are publishing are tagged with "density probability" values. Visually starting from such density probability tagged values + ellipse shapes and dimensions into mentally estimating a PROBABILITY OF ACTUAL FIX PRESENCE INSIDE A COCKED HAT (isn't THAT exactly which we have been researching/debating for so long ???) is - at least for me - impracticable. I am totally unable to perform such a mental double integration on a surface for lack of immediately understandable and familiar scale. I can't prevent either from finding the values of the tagged probability density coefficients as being very (too) low, which certainly can NOT the case.

    COULD YOUR ERROR ELLIPSES BE TAGGED DIFFERENTLY WITH THE (CUMULATED)PROBABILITY OF THEIR (INNER) SURFACE ?

    If so, we would have an adequate way to "guesstimate" for each case the order of magnitude of the inner area of every cocked hat, and it would greatly help all of us (still) interested in the current debate.

    REMARK :

    In Re 2, you wrote that :

    QUOTE
    "Attached are three contour plots of P(x,y) formed from different size hats. They clearly show how the probability contours vary in size with the hat size.".
    QUOTE

    If I understand this last statement correctly, it means that the smaller the size of the cocked hat, the smaller the probability for the actual fix to be inside a shrinking cocked hat. As a consequence, a "zero surface" cocked hat (i.e. 3 LOP's crossing in one single point) has a zero probability to contain/to be the actual fix.


    *******

    Dear George,

    in your post Re 1, you wrote the following :

    QUOTE

    Now if we scale the triangle up in size keeping its shape the same we will get the same MPP AND the same error ellipse. In other words the error ellipse and the associated probability contours do not depend on the size of this triangle.

    UNQUOTE

    Should I interpret your statement as meaning that "for all similar cocked hat shapes, and whatever their actual dimensions/sizes, the probability for the fix to be inside the cocked hat remains constant" ?

    If my interpretation of your statement is correct, I should conclude that in the limiting case when the size of the cocked hat shrinks down to be only one single point (which becomes a similar case as the one addressed in the "REMARK" to John hereabove, i.e. 3 LOP's crossing in one single point) a "zero surface" cocked hat a NON zero probability to contain/to be the actual fix, such probability being equal to the constant probability of the (earlier shrinking) similar shape cocked hats. It also brings an interesting conclusion about different Crossing LOP's having different probabilities according to the various body azimuts.

    AS A CONCLUSION,

    John, if you could tag your error ellipses in Probabilities (vs. Probability Densities) that would be certainly quite helpful here.

    For George and John, I also pointed out what seems to me a possible/potential contradiction between your 2 statements. So, please be so kind as to give me additional explanations on this specific point.

    Thanking both of you for your Kind Attention and expected Help, I am extending you my

    Very Best Regards


    Antoine

    Antoine M. Couëtte


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