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    Re: Still on LOP's
    From: Geoffrey Kolbe
    Date: 2002 Apr 22, 19:06 +0100

    Trevor Kenchington wrote:
    >Yet that 'hat can be very small or very large. Those variations in size
    >can be due to different levels of uncertainty in the measurement of each
    >bearing, in which case the probability of being in the 'hat need not
    >change. (Worse bearings give bigger area in which you might be located.)
    >But the varied sizes can also be due to chance effects: Three "bad"
    >bearings cam all pass through (very nearly) the same intersection point.
    >I could understand that the best estimate of the MPP was at the
    >intersection and the best estimate of a confidence circle around that
    >point would be a very small circle. But those are only estimates. Your
    >logic leads to the conclusion that, even when a cocked hat is very small
    >because of chance effects, there is exactly a 0.25 probability of the
    >true position lying inside it. And that doesn't make any sense to me at all.
    My "proof", as you were kind enough to call it, has nothing to say
    regarding the size of the 'hat. All that is demonstrated is that the 'hat
    will enclose the actual position 25% of the time.
    However, I think you are confusing yourself by then talking about
    "confidence circles" and Most Probable Positions in relation to 'hats.
    Getting from the statistics about the size and distribution of 'hats to the
    statistics of Most Probable Positions and circles of confidence is not
    trivial as they are quite different animals.
    I have some expertise with the statistics of groups and grouping where
    bullets are hitting a target. The statistics of MPP's is, I think, quite
    similar to finding the centre of a group.
    If you fire just one shot at a target and ask yourself where the centre of
    the group is, the best you can do is say it is in the centre of the bullet
    hole. You cannot do statistics on one datum point. But if you have two data
    points, you can do all the statistics in the world.
    If you now fire three or four more shots, you will have a distribution of
    holes in the target - what we call a group. You can now crank the
    statistical handle to find where the centre of the group is. The more shots
    you fire, the greater the level of certainty of your calculated centre of
    group. This level of certainly will increase (and your confidence circles
    decrease) as the square root of the number of shots in the group.
    Similarly, if you only have one 'hat, then placing the MPP in the centre of
    it is the best you can do. But if you perform multiple observations
    resulting in a number of 'hats, then the MPP can be placed at the centre of
    the distribution of 'hats. You can now also start doing statistics on the
    distance of the 'hats from the MPP and so establish circles (or ellipses)
    When thinking of MPP's and circles of confidence, I think the trick is to
    consider the centre of the 'hat as one datum point sitting somewhere on a
    probability distribution around the MPP. Until you do some more
    observations and establish some more 'hats, you will not know where the
    centre of that first 'hat is on the probability distribution of 'hats. When
    thinking of the relationship of 'hats to MPP's in this way, you can see
    that the actual size of the 'hat does not matter, only the position of its
    However, if you do perform multiple observations on each of the three
    landmarks, then producing a distribution of 'hats is not an efficient use
    of the data when determining the levels of confidence of your MPP. It is
    much more efficient to find the mean bearing and standard deviation from
    the mean for the measurements on each landmark. Plot the mean bearings and
    there will be a 'hat at the intersection. The centre of _this_ 'hat should
    now be a good MPP and from the standard deviations around your three mean
    bearings, you can establish ellipses of confidence.
    Once again though, this 'hat formed from the mean of multiple observations
    on each of three landmarks only has once chance in four of enclosing the
    actual position...!
    You will see why when you also plot bearings one standard deviation away
    from the mean bearing for each landmark. This will enable you to draw an
    ellipse which is in effect a 50% confidence ellipse The actual position has
    a 50% chance of being inside this ellipse Your 'hat should lay comfortably
    inside this ellipse with perhaps just the vertices outside it. If it
    evidently bigger than half the 50% confidence ellipse then there is some
    systematic error which is opening up your 'hat. Your compass needs
    calibrating or, when considering LOP's, you have an index error on your
    Geoffrey Kolbe.

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