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    Re: That darned old cocked hat
    From: John Karl
    Date: 2010 Dec 12, 19:55 -0800

    Frank wrote:
    I don't think that it's appropriate to get the standard deviation from the current set of observations when they are so few in number …….
    ===================

    I agree. The whole topic should be given three LOPs with known standard deviation (from previous experience, e.g. with a sextant).

    Frank continues:
    If instead we treat the standard deviation of the observations as an INPUT to the problem, which I think is by far the better choice with a small number of LOPs, then the probability ellipses will have a FIXED size (different shapes and orientations, of course) with various different sizes of triangles. And in that case, the integrated probability over the triangle --the probability of being inside and not outside will vary from one triangle to another.
    ==================

    Yes, I agree, and this is the problem of a 3-body fix at sea. The given standard deviation is that of a single isolated LOP, not of the probability distribution around the combined three LOPs (around that darned cocked hat), that I’ve been calling P(x,y).

    However, P(x,y) does, in fact, have DIFFERENT SIZES (and shapes) around triangles of different sizes. Even so, you’re still right, as I’ve been saying, that the probability that the true fix is inside the hat depends on the size (and shape) of the hat, as in the examples I’ve computed and previously posted – a large hat has a large probability that the fix is inside, a small hat has a small probability that the fix is inside. (Here size means relative to the standard deviation of the individual LOP.) 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.

    Then Frank concludes:
    But in the long run, the probability of being inside ANY triangle formed from three LOPs will average out to 25%.
    =======================

    This is by no means obvious. And it’s clear to me that to make this definitive statistical statement, we need to know where the true fix is in every one of those large-number of 3-body fixes -- otherwise we can’t tell if the fix is in or out of the hat. This means doing the analysis (or real world experiment) using a known location. This is not what we do in navigation (but perhaps in evaluating some instrument or procedure). In spite of all the talk about his conclusion, I’ve yet to see a proof of it. Mover, even if this statement were correct, it obviously has no utility in estimating position from one given 3-body fix. (And now we all know to plot the symmedian – or eyeball it)

    JK
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