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    Re: Still on LOP's
    From: JC Sutherland
    Date: 2002 Apr 26, 18:22 +0100

    Quoting George Huxtable :
    > Response from George, to the message from my friend and near-neighbour
    > Clive Sutherland, who said-
    > >George�s presentation , for example, of three intersecting position
    > lines
    > >resulting in only 25% probability of the position being within the
    > cocked
    > >hat >has long been the argument in textbooks on this subject, but it
    > >doesn�t gel
    > >with me.
    > I would be most interested to learn in which textbooks it's to be
    > found
    > >To support this contention we are supposed to agree that for each one
    > >of these position lines the true position lies equally likely to the
    > left or to
    > >the right but not actually on the line! This is where I get off!
    > I think Clive and I agree better than he thinks we do.
    > We just need to define rather carefully exactly what we are talking
    > about.
    > Imagine dividing your chart up into thousands and thousands of (say)
    > one-foot square cells. Along the line of a bearing, it's the
    > "probability
    > density" that is at a maximum: the (small) probability of the vessel
    > being
    > in one of those one-foot cells. To find the probability of the vessel
    > being
    > inside a particular zone, one has to integrate the probability density
    > over
    > the area of that zone. That is, sum up all the individual probabilities
    > of
    > the squares within that zone
    > Though Clive would be right to claim that along the line of a bearing,
    > the
    > probability density is at a maximum, the probability of being exactly
    > on
    > that line is zero because the area of the line is zero, as it is with
    > all
    > lines (unless some finite line-width has been specified).
    > >
    > >Let�s imagine for a moment that the earth is still and many
    > observations of the
    > >same LOP can be taken. If these are analysed statistically ( assuming
    > that all
    > >errors are random) the most probable value will be represented by the
    > average
    > >and the further off this average  line you inspect the less likely you
    > are to
    > >find the true  position line.
    > >
    > > If we assume that the line is the actually the peak of a Gaussian
    > >distribution, yes, it has an equal probability either side, but this is
    >  a
    > >diminishing probability value the further you move away from it and the
    > MOST
    > I agree with all the above.
    > >
    > >If we take a three dimensional view, our observations could be better
    > >represented by a solid figure rather like a length of wood cut so that
    > it has
    > >Gaussian cross-section the same size and shape all along its length. We
    > could
    > >then lay this on the chart to represent our LOP.
    > >
    > >If you imagine two such position lines intersecting , merging and
    > adding. The
    > >probability curve at the point  of intersection would become a Bell
    > shape. It
    > >would have circular contours only if the error distributions of both
    > LOPs are
    > >the same , but would have elliptical contours if one LOP group is
    > tighter  than
    > >the other.
    > >
    > >Each contour will represent an uncertainty area at a particular
    > confidence
    > >level.(the smaller the ellipse the less confident you are of being
    > inside it).
    > >For simplicity it is usual to show only one contour for the confidence
    > level
    > >(usually 95%) the navigator prefers.
    > I agree with that too.
    > >
    > >Now it is possible to extend this theory to include three or more LOPs,
    > but my
    > >maths is not up to this, however I can convince myself that the true
    > position
    > >lies inside the cocked hat if I look at this plot in a more colouful
    > way.
    > >This is my way of imagination. On maps contours are often coloured. If
    > we
    > >wereto use dark transparent colour for high probability and  a light
    > colour for
    > >low,  this would reveal the shape of the probability terrain. The
    > darkest area
    > >would represent the highest confidence in your position. For example,
    > if we
    > >plot the Gaussian distribution of a single group of LOPs  as a ridge
    > line, the
    > >center would be a dark  line and the further off either side of this
    > line the
    > >lighter the shading  would be.
    > >
    > >For two position lines, at the point of intersection, the shading would
    > add and
    > >this would then produce a single darker peak with  the shading getting
    > lighter
    > >in all directions away from this point. This would be consistent with
    > the
    > >circular Bell shape,  and would reveal a ellipsoid bell, if this were
    > in fact
    > >the case as above.
    > >
    > >If you use this trick to imagine what three or more curves intersecting
    > as a
    > >cocked hat would look like, personally I am convinced that the
    > darkest
    > >cumulative shading would be a plateau inside the triangle. This would
    > show that
    > >the further you go from the �Center of gravity� of the triangle the
    > lighter the
    > >shading would be. Therefore by inference the highest probability of the
    > true
    > >position fix would be inside  the triangle.
    > >
    > >PS.  In surveying it is common to find �the most probable position� in
    > a
    > >triangle of error produced by three sight lines,  by drawing lines
    > inside this
    > >plotted  triangle, parallel to each side, at a distance from the side
    > >proportional to the length of the observed ray so as to produce a
    > smaller
    > >triangle inside the first. This is applied successively until the
    > smallest
    > >acceptable triangle is found and this then becomes the surveyed
    > point.
    > >
    > >Clive Sutherland.
    > I don't think I find any serious disagreement with anything Clive has
    > said.
    > I agree that the most probable position for a three-bearing fix is going
    > to
    > be somewhere within the triangle. I agree that when entering on the
    > chart
    > the result of a cocked hat plot, it is sensible to plot a point
    > somewhere
    > within the triangle as being the best estimate that can be made of the
    > vessel's position, though keeping aware of the deficiencies of that
    > process
    > when taking account of nearby dangers.
    > But that's not what we have been discussing. We have been asking
    > "What's
    > the probability of the position being within that triangle AT ALL?"
    > And
    > that's what has come out to be 25%, according to me and to JED
    > Williams.
    > That is quite compatible with the notion that the most probable
    > position
    > (the point where the probability density, or probability per square
    > foot,
    > is greatest) is within the triangle.
    > Does Clive maintain, I wonder, that the triangle of the cocked hat
    > MUST
    > contain the true position, the conventional wisdom that so many have
    > accepted in the past?
    > George Huxtable.
    > ------------------------------
    > george@huxtable.u-net.com
    > George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    > Tel. 01865 820222 or (int.) +44 1865 820222.
    > ------------------------------
    You are quite right to take me to task for answering the wrong question! But I
    had been trying to find the location of the True position by looking from this
    other direction. I now believe it is certain I will fail because we can never
    find the True position!  Indeed I postulate that the true position is an
    imaginary, zero dimensional point that can never  be found. Mind you I think
    Heisenberg  also had a similar theory .
    In the last week or so on this thread I have read a lot of interesting
    statistical analyses, some of which I have understood.  All of them have been
    about the decoration of George�s cocked hat. However while I have learnt
    something about millinery, I have learnt nothing about whether George�s head
    was in it or not.
    As every clever politician knows �Statistics is not about Truth� but about �our
    Perception of the truth�, two very different things.
    George asks me if �I believe the true position must be inside the cocked hat�,
    My question to George is �Does he think it must not?�
    Consider this. From a fixed point take many GPS fixes. Plot these on graph
    paper and it will show a scatter of points about a your (perceived) mean
    value.  Connect  three of these points to form a cocked hat.
    Which three points will you choose?. Is your perceived  average value closer to
    the truth than your  cocked hat? What does the size of the your chosen cocked
    hat tell you?
    The fact is, the answers you will give to these questions are all about Human
    Psychology and have nothing to do with Navigation!
    It is our psyche which makes us uncomfortable about uncertainty, Probability
    theory is merely a Teddy Bear.
    Some time ago  I saw an analysis as described by J.E.D.Williams below in a
    navigation text book  and I think it was the Admiralty Manual of Navigation but
    it could have been an RAF manual. I can�t  be sure as I don�t have either to
    hand. I have an area of uncertainty about it. :>) However I do have some other
    references  which might be useful.. I am sure you will have read most of them
    but the list may be of use to others. I also have a Journal Index which
    mentions a few others. If anyone would like a them I can send this section of
    it as a JPEG.
    I have not read many of these but it does show that the subject has been of
    interest to many people from the as far back as  1948.
    These references are all taken from the J. Inst of Nav unless otherwise stated
    Subject;        COCKED HAT
    The Cocked Hat                  J.E.D.Williams  Vol 44, 269  May 1991
    Refs                                    BBC Open Univ   � The Cocked Hat�
    The Cocked Hat                  P.D.Gething             Vol 45  143.    Jan 1992
                                            D.William Smith ditto
    J.E.D Williams replies
    Refs                                    �The Theory of position finding�
    Daniels,HE                                              J. R Stat. Soc. Series
    B.   Vol 13, 186.  1951
    Random Cocked Hats              Ian Cook                Vol 46, 132  Jan 1993
    J.E.D Williams replies
    Refs                                    Williams J.E.D. Vol 44 1991
    Subject;        ERROR PROBABILITY
    The Presentation of Fixing accuracy of navigation systems
                                            Jessel  and Trow  Vol 1. 313   OCT 1948
    On  Error Distributions in Navigation   O.D.Anderson    Vol 29 1969
    Refs                                    Several,  including �Is the gaussian
    distribution normal�
            Anderson E.W. Vol 18,  65
    The treatment of Navigational Errors    ,
                                            W/C  E.W. Anderson   Vol 50 362.    Sep
    Refs                                    Many,  including Tech Guide No 1,
    R.Inst.Nav.  1996
    Here�s to �Crystal clear thinking on this subject� to everyone.
    Clive Sutherland, 26 Apr 02
    Remember �Mis-understanding statistics is quite �Normal�,�

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