# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 > >PROBABLE VALUE MUST BE THE LINE ITSELF. > > 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. > ------------------------------ > George; 1 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. 2 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. 3 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 1997 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�,�