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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.
>

Trevor.

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)
confidence.

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
centre.

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
sextant.

Geoffrey Kolbe.

   

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