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    Re: On LOPs
    From: Trevor Kenchington
    Date: 2002 Apr 16, 11:50 -0300

    Steven Wepster wrote:
    >Let me add one cent. I don't understand what _the_ _true_ form of a
    >confidence region should mean.  IMO a confidence region is a region that
    >contains the MPP with a specified probability, often taken as 95%. It can
    >be circular, elliptical, square, heart-shaped or whatever you prefer, for
    >whatever theoretical, practical, or other reasons are important to you.
    My use of statistical terminology is probably not what it should be
    (comes of picking up stats as a biologist with little formal training!)
    and I have no idea of whatever conventions may have developed in the
    application of statistics to navigation. My lack of the right words
    should not, however, hide the existence of the thing I was trying to
    refer too.
    Given an MPP and sufficient information on the associated uncertainties,
    one could draw any number of boundaries around it, each defined so that
    there was a 95% probability (or any other probability one might select)
    that the true position lay within the boundary. Each boundary would
    encompass some areas where there was a high probability that the true
    position would lie and other areas where there would be a low
    probability. But there would only be one such boundary that not only met
    the criterion of a 95% probability of the true position lying within it
    but also had, at all points along the boundary, an equal probability of
    the true position lying at that point. That boundary (and only that
    boundary) would encompass the smallest area that had a 95% probability
    of containing the true position. It would also be the only boundary
    which did not exclude some areas where there was a relatively high
    probability of the true position lying -- all other boundaries excluding
    small high-probability areas but taking in larger low-probability ones
    to maintain the 95% encompassed.
    It is the area surrounded by that one boundary (for any chosen
    probability of enclosing the true position) which is also a "contour"
    (or isopleth, if you prefer) of constant probability which I meant by
    the "true" confidence region. I would contend that it, rather than the
    area with some other boundary, is the confidence region that we would
    like to have when using an MPP in practical navigation.
    The problem, of course, is that we do not have complete information on
    the shape of the boundary and thus must fall back on estimates from the
    available data. Moreover, we do not (at least, I do not) go to sea with
    the sort of computing power needed to make statistically-optimal
    estimates from such data as we do have. Hence, in practical
    applications, we fall back on assuming that our true position is
    somewhere inside a cocked hat, a circle around the MPP, an ellipse or
    some other simple figure that we can plot. My only reason in seeking to
    go beyond that, in a discussion of theory, is to explore the reliability
    or otherwise of such simple approximations to a complex topic. If, for
    example, George was right that the true position falls within the cocked
    hat only 25% of times (and I have yet to see his defence of that
    conclusion), navigators should know not to rely on that old rule of
    thumb. The same would be true of computed ellipses if they fail us too
    Trevor Kenchington

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