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    Re: That darned old cocked hat
    From: Antoine Couëtte
    Date: 2010 Dec 16, 09:49 -0800

    To the Attention of John Karl, George Huxtable, and George Brandeburg

    + others (among them Peter Fogg, Frank E. Reed, Gary Lapook and Herbert Prinz) who have been "building up" this thread.


    ??? WOULD THE FOLLOWING NOT BE "THE" PATH TOWARDS A FULL AND FINAL ANSWER ???

    ******* ******* ******* ******* ******* ******* *******

    I AM AGAIN REFERRING TO OUR (YET) UNSOLVED QUESTION :

    IS THE "ACTUAL FIX PROBABILITY TO LIE SOMEWHERE INSIDE THE COCKED HAT" (P) A "VARIABLE PROBABILITY" (WHICH SHRINKS TO ZERO WHEN THE COCKED HAT SIZES SHRINK TO ZERO, AS SHOWED BY JOHN) OR IS IT A "CONSTANT PROBABILITY" WHATEVER THE COCKED HAT SIZE (AND EQUAL TO 25%, A VALUE FIRMLY AND MOST BRAVELY DEFENDED BY GEORGE HUXTABLE ?)

    ******* ******* ******* ******* ******* ******* *******

    I have chosen NOT to list here all the specific references to the various earlier relevant posts which I am referring to because there are far too many of them.

    Still, and first of all, I want to thank you very much John for having taken your time for having computed and most recently displayed your probability ellipses tagged with their inner surface probabilities. MUCH EASIER now to get a better picture !

    Also, to you George B., it seems to me that you have earlier and recently addressed a view-point similar to the following one, which I just wish to cover without recourse to the chi-square concept.

    ******* ******* ******* ******* ******* ******* *******

    We will keep assuming that - for any and every single observation - we have ONLY random errors which are are GAUSSSIAN. We will also assume that the "LOP Probabilities" keep the same value for each one of the three LOP's depicting together one given Cocked Hat. For each and every LOP we then know that such probability can be uniquely defined by just one term : its SIGMA, with the "variable" here being the minimum distance from any point in the plan to such LOP. (I trust that this "fast" definition of a "LOP Probability" is sufficient).

    LOP's can be given/assigned DIFFERENT AND ARBITRARY SIGMA values.

    1 - For any given 3 LOP Cocked Hat, let us first assume that the SIGMA value of each LOP (again being constant for all 3 LOP's of one same Cocked Hat) is ARBITRARILY FORCED into being equal to the SD (Standard Deviation) observed for such specific Cocked Hat.

    Accordingly for LOP's forming a "big" Cocked Hat (which implies a "big" observed SD) we are to "tag" each of the LOP's with the same "big" SIGMA. On the other hand, for LOP's forming a "small" Cocked Hat (which implies a "small" observed SD) we will "tag" each of the LOP's with the same "small" SIGMA value.

    In such a case, there is absolutely no reason why (P) should not stay constant for each similarly shaped Cocked Hat - since it has now simply become a matter of scaling the Cocked Hat up or down - , and

    Even more !!!!... it is very likely that some math would soon show us that if we choose to arbitrarily assign to SIGMA a value equal to the "observed Cocked Hat SD", then not only will (P) remain constant for all similar shape Cocked Hats, but - Oh Miracle !!! - for ALL Cocked Hats and whatever their shapes, we will always and EVERY SINGLE TIME find for (P) the exact 25% value strongly supported by you George H., and by your Illustrious Predecessors.

    2 - On the other hand, and IF we give to such SIGMA one given constant and fixed value, totally independent of the Cocked Hat sizes, then all the conclusions brought up by you John are 100 % true.

    ******* ******* ******* ******* ******* ******* *******

    IN OTHER WORDS

    You both John and George H. are not starting from the same assumptions.

    George, one way or the other (still a "hidden" way to me) - and the same should also hold for you too Frank with your magic "Triangle Cocked Hat Computer" - for each Cocked Hat you are most probably directly linking the LOP SIGMA value (which by the way you are not using at all in your reasoning) to the actual value of each specific observed Cocked Hat SD value. Most likely, you are implicitly assuming that "INDIVIDUAL LOP SIGMA equals INDIVIDUAL COCKED HAT SD", which permits to always recover your (beloved) 25% probability for (P). Only remains here to unveil the "hidden link" which yourself and all your Predecessors have kept using through "unknowingly" assuming that "INDIVIDUAL LOP SIGMA equals COCKED HAT SD". (Any taker here ? Frank ? Herbert ???)

    John, whatever the quality of the observations, I think that you have kept assuming a constant value throughout for all your LOP SIGMA values, whatever the respective sizes of all the Cocked Hats.

    If such is your case John, then for any given fixed dimension Cocked Hat, if you were assigning a much smaller (and same for all) SIGMA to its LOP's, then the Cocked Hat probability would significantly increase. By the same token, for any given SIGMA, if you were to increase the Cocked Hat size, then (P) is to increase accordingly.

    Only remains here for us to know WHICH FIXED VALUE you have assigned to such LOP SIGMA value (is it 1 NM ?).

    John, in order to verify my intuition about George H.'s "hidden assumption", a simple way here would be for you to take an arbitrarily random Cocked Hat, and compute its inner surface probability through giving to all LOP's one (same) SIGMA value equal to the OBSERVED SD of such Cocked Hat. Let's go for it ...

    NOTE :

    John, if the first assumption " LOP SIGMA = COCKED HAT SD " does not work, would you mind trying with :

    LOP SIGMA = COCKED HAT SD * 0.5, (half value) or

    LOP SIGMA = COCKED HAT SD * 2 , (twice the value) ?

    The "trick" here is to find which same "hidden" relationship holds between LOP SIGMA and COCKED HAT SD under George H.'s "implicit" assumptions. I have initially thought that it should be some quite simple one, such as one the three possibilities listed here-above ...

    Sorry for the BIG computational work !!! :-((

    ******* ******* ******* ******* ******* ******* *******

    IN CONCLUSION, and if my intuition is true,

    As earlier indicated by you George B., the starting assumptions from you George H. and John are significantly different, and therefore your conclusions are different.

    But there is no contradiction whatsoever between your respectively stated results. You are simply not addressing the exact same topic.

    Thank you to all for giving this opportunity to think a bit in depth.

    Et maintenant ... And now ...

    "A vos Sextants et Chronomètres, la Méridienne n'attend point !!! "

    "Quickly grab your Sextants and Chronometers, Noon Time is not to be delayed !!! "

    Joyeux Noël 2010 et Bonne Année 2011 à vous tous

    Merry Christmas 2010 and Happy New Year 2011 to you all

    Antoine


    Antoine M. "Kermit" Couëtte

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