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    Cocked hat vs Confidence ellipse
    From: Andrew Nikitin
    Date: 2013 Mar 17, 15:48 -0700

    I noticed that the expression "Confidence ellipse" was not mentioned during this month's discussion on cocked hats, etc. I think it is such an important concept that it is worth reiterating.

    Here is in compressed form what i got from reading numerous cocked hat discussions in this forum.

    1. Given several LOPs, each of them obtained from altitude measurement with independent normal unbiased error with the same sigma, MPP should be chosen as a point that minimizes sum of squared distances to the LOPs.

    2. When chosen in such a way, the probability density of a true location will be elliptical 2d gaussian function (http://en.wikipedia.org/wiki/Gaussian_function). The level sets of this function are ellipses. These ellipses are centered on MPP, they all have same eccentricity and their major axis are parallel. Moreover, eccentricity and orientation of major axis depends only on the slopes of LOPs, and not on the sigma. (And the slopes of LOPs is determined by choice of target stars and known practically exactly)

    3. Such ellipse is called "N% confidence ellipse". For example, 50% confidence ellipse is an ellipse which has 50% chance of containing true location.
    Since navigator seldom knows his sigma, the size of the, say 50% or 90% confidence ellipse is unknowable. However, the shape of this ellipse, its center and orientation of its major axis is knowable and provides valuable information.

    4. Mathematically, there is no reason to choose as your fix any point other than the center of confidence ellipse, no matter what your sigma is. There may be operational reasons to choose other point, however. If the ellipse has high eccentricity, it should be understood that error in the direction of major axis of this ellipse is higher than in the direction of its minor axis.

    5. Cocked hat is a special case of 3 LOPs. Probability of true location being inside cocked hat is exactly 25%. This is mere curiousity and this is not a reason to construct a cocked hat.

    6. For 2 LOPs the center of confidence ellipse is point of intersection of these 2 LOPs

    7. For 3 LOPs the center of confidence ellipse is a symmedian point of a triangle formed by 3 LOPs (This is actual rason to construct cocked hat -- to obtain the center of confidence ellipse)

    8. For 4 or more LOPs there is no simple geometric procedure to find the center of a confidence ellipse. There is, however, a simple* arithmetic procedure. ([*] simple with a calculator, that is)

    9. The more LOPs you obtain, the closer the MPP will be to the true location on average. Each LOP slightly reduces the size of an N% confidence ellipse.

    Andres Ruiz has a paper on a subject of confidence ellipses.

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