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Hello Antoine,

I composed a nice message for you last night, but due to the server problems it is sitting waiting for you on my home computer. I'll may send it later tonight for what it is worth, but since I have a few minutes today let me try to answer your questions.

I tried to give an explanation in terms of "quality of fit" a few days ago, but I fear my message was too long and filled with jargon. Let me repeat the important points. You advance two different scenarios below for calculating the SIGMA or the error ellipse associated with the most probable point (MPP).

I have always felt that your second one is the one that should be used, namely that the error ellipse size and shape should only depend on the measurement errors for the input LOPs. In other words you know what the final uncertainty will be based on your ability to do the measurements. You then measure three LOPs, determine the MPP from the triangle they form, and draw an error ellipse around this point. This error ellipse (which is the one standard deviation contour) will have the same shape and size regardless of how big the resulting triangle is. Its shape is determined by the angles that the LOPs make with each other, which in turn was determined by your choice of celestial fixes. The size or area of the ellipse depends on the measurement error associated with the LOPs, or their width if you like. (Actually if the LOPs have different errors then this will also influence the shape of the ellipse.)

This scenario is the one that John Karl has used when making the contour plots that he posted. And as you surmise it does mean that a very large intersection triangle then has a large probability that the location is enclosed, and when all 3 LOPs cross at one point there is zero enclosed area in the triangle and so the probability is zero that the location is enclosed.

To talk about your first scenario let me return to the concept of "quality of fit" as discussed in my previous posting. When you make a set of measurements like this you should first know how well you can do. Formally this means estimating you measurement error, but in practice you generally know how well you will do. You then make the measurements and fit them to an hypothesis. Formally that might mean doing a chi squared (chisq) fit to the hypothesis that the LOPs cross at one point, but in practice it may mean drawing your measured LOPs on a chart and taking the LOP to be near the center of the triangle formed by the LOPs. If you actually do the fit you will get two things: the coordinates of the MPP and the value of chisq. Chisq is nothing more than a measure of the quality of the fit (see my previous posting for more details about chisq, but it in this case it is defined as the sum over the three LOPs of their distance from the MPP squared divided by their measurement error squared). It turns out that chisq is directly proportional to the enclosed area of the LOP triangle. A very large chisq means that you were unlucky (or lacking skill) in your measurement. A very small chisq means you lucked out and by chance did a nearly perfect measurement (or you didn't know how well you could do). A value near one (since this problem has one constraint) means that you did an average job. If you got chisq less than or equal to one you can safely take a point near the center of the triangle as your fix and congratulate yourself on a job well done. If the chisq and triangle are large then you may want to add more LOPs or do the whole measurement over.

Some people like to mix their measurement errors together with the quality of the fit. I do not believe this is a good idea, but to do it you multiply your determined measurement errors (the size of the ellipse) by the square root of chisq (divided by the number of constraints which is one here). This means that if you do a poor measurement (large triangle) you inflate your errors to reflect how poorly you did. It also has the silly consequence that if you do a lucky measurement where the LOPs almost intersect and chisq is very small then you credit yourself with a measurement error that is much smaller than you are capable of doing! This is why I believe it is best to keep measurement error and quality of fit separate.

However, if you do make the kludge I have described above, then you get your first scenario, namely the size of the error ellipse scales with the size of the LOP triangle. This also means that the integrated probability enclosed within the triangle remains constant, even when the enclosed area goes to zero! If you are silly enough to think that your measurement error has gone to zero when your three LOPs just happen to cross at a point, then maybe you will like this scenario.

Finally your first scenario should not be confused with George H's (and Samuel Goudsmit's) very nice demonstration that if you make many 3 LOP measurements the average probability that the location is enclosed in the LOP triangle will approach 25%. The more such measurements you make the closer it will get to 25%.

I hope this is helpful!
George B

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From: antoine.m.couette---fr
Date: 16 Dec 2010 09:49

To the Attention of John Karl, George Huxtable, and George Brandeburg

+ others (among them Peter Fogg, Frank E. Reed, Gary Lapook and Herbert Prinz) who have been "building up" this thread.

??? WOULD THE FOLLOWING NOT BE "THE" PATH TOWARDS A FULL AND FINAL ANSWER ???

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I AM AGAIN REFERRING TO OUR (YET) UNSOLVED QUESTION :

IS THE "ACTUAL FIX PROBABILITY TO LIE SOMEWHERE INSIDE THE COCKED HAT" (P) A "VARIABLE PROBABILITY" (WHICH SHRINKS TO ZERO WHEN THE COCKED HAT SIZES SHRINK TO ZERO, AS SHOWED BY JOHN) OR IS IT A "CONSTANT PROBABILITY" WHATEVER THE COCKED HAT SIZE (AND EQUAL TO 25%, A VALUE FIRMLY AND MOST BRAVELY DEFENDED BY GEORGE HUXTABLE ?)

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I have chosen NOT to list here all the specific references to the various earlier relevant posts which I am referring to because there are far too many of them.

Still, and first of all, I want to thank you very much John for having taken your time for having computed and most recently displayed your probability ellipses tagged with their inner surface probabilities. MUCH EASIER now to get a better picture !

Also, to you George B., it seems to me that you have earlier and recently addressed a view-point similar to the following one, which I just wish to cover without recourse to the chi-square concept.

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We will keep assuming that - for any and every single observation - we have ONLY random errors which are are GAUSSSIAN. We will also assume that the "LOP Probabilities" keep the same value for each one of the three LOP's depicting together one given Cocked Hat. For each and every LOP we then know that such probability can be uniquely defined by just one term : its SIGMA, with the "variable" here being the minimum distance from any point in the plan to such LOP. (I trust that this "fast" definition of a "LOP Probability" is sufficient).

LOP's can be given/assigned DIFFERENT AND ARBITRARY SIGMA values.

1 - For any given 3 LOP Cocked Hat, let us first assume that the SIGMA value of each LOP (again being constant for all 3 LOP's of one same Cocked Hat) is ARBITRARILY FORCED into being equal to the SD (Standard Deviation) observed for such specific Cocked Hat.

Accordingly for LOP's forming a "big" Cocked Hat (which implies a "big" observed SD) we are to "tag" each of the LOP's with the same "big" SIGMA. On the other hand, for LOP's forming a "small" Cocked Hat (which implies a "small" observed SD) we will "tag" each of the LOP's with the same "small" SIGMA value.

In such a case, there is absolutely no reason why (P) should not stay constant for each similarly shaped Cocked Hat - since it has now simply become a matter of scaling the Cocked Hat up or down - , and

Even more !!!!... it is very likely that some math would soon show us that if we choose to arbitrarily assign to SIGMA a value equal to the "observed Cocked Hat SD", then not only will (P) remain constant for all similar shape Cocked Hats, but - Oh Miracle !!! - for ALL Cocked Hats and whatever their shapes, we will always and EVERY SINGLE TIME find for (P) the exact 25% value strongly supported by you George H., and by your Illustrious Predecessors.

2 - On the other hand, and IF we give to such SIGMA one given constant and fixed value, totally independent of the Cocked Hat sizes, then all the conclusions brought up by you John are 100 % true.

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IN OTHER WORDS

You both John and George H. are not starting from the same assumptions.

George, one way or the other (still a "hidden" way to me) - and the same should also hold for you too Frank with your magic "Triangle Cocked Hat Computer" - for each Cocked Hat you are most probably directly linking the LOP SIGMA value (which by the way you are not using at all in your reasoning) to the actual value of each specific observed Cocked Hat SD value. Most likely, you are implicitly assuming that "INDIVIDUAL LOP SIGMA equals INDIVIDUAL COCKED HAT SD", which permits to always recover your (beloved) 25% probability for (P). Only remains here to unveil the "hidden link" which yourself and all your Predecessors have kept using through "unknowingly" assuming that "INDIVIDUAL LOP SIGMA equals COCKED HAT SD". (Any taker here ? Frank ? Herbert ???)

John, whatever the quality of the observations, I think that you have kept assuming a constant value throughout for all your LOP SIGMA values, whatever the respective sizes of all the Cocked Hats.

If such is your case John, then for any given fixed dimension Cocked Hat, if you were assigning a much smaller (and same for all) SIGMA to its LOP's, then the Cocked Hat probability would significantly increase. By the same token, for any given SIGMA, if you were to increase the Cocked Hat size, then (P) is to increase accordingly.

Only remains here for us to know WHICH FIXED VALUE you have assigned to such LOP SIGMA value (is it 1 NM ?).

John, in order to verify my intuition about George H.'s "hidden assumption", a simple way here would be for you to take an arbitrarily random Cocked Hat, and compute its inner surface probability through giving to all LOP's one (same) SIGMA value equal to the OBSERVED SD of such Cocked Hat. Let's go for it ...

NOTE :

John, if the first assumption " LOP SIGMA = COCKED HAT SD " does not work, would you mind trying with :

LOP SIGMA = COCKED HAT SD * 0.5, (half value) or

LOP SIGMA = COCKED HAT SD * 2 , (twice the value) ?

The "trick" here is to find which same "hidden" relationship holds between LOP SIGMA and COCKED HAT SD under George H.'s "implicit" assumptions. I have initially thought that it should be some quite simple one, such as one the three possibilities listed here-above ...

Sorry for the BIG computational work !!! :-((

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IN CONCLUSION, and if my intuition is true,

As earlier indicated by you George B., the starting assumptions from you George H. and John are significantly different, and therefore your conclusions are different.

But there is no contradiction whatsoever between your respectively stated results. You are simply not addressing the exact same topic.

Thank you to all for giving this opportunity to think a bit in depth.

Et maintenant ... And now ...

"A vos Sextants et Chronomètres, la Méridienne n'attend point !!! "

"Quickly grab your Sextants and Chronometers, Noon Time is not to be delayed !!! "

Joyeux Noël 2010 et Bonne Année 2011 à vous tous

Merry Christmas 2010 and Happy New Year 2011 to you all

Antoine

Antoine M. "Kermit" Couëtte

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