Probability of being in a Cocked Hat

[weustace]

I would suggest that analysising noise in measurements is a well respected science when dealing with large numbers of samples.
A cocked hat is three measurements to determine two variables.
That is not a statistical process, it's a measurement with a vague check.

Statistics wise, it's like an opinion poll of 3 people.
In the same street.

Dealing with single measurements, a proper approach is to put absolute limits on each indicated value.

It depends what you want the answer for.
If you need to be absolutely sure you are avoiding a hazard, that is a different requirement from wanting an indication of progress.

How do you put an "absolute limit" on the measured values? The most common way in scientific measurement in my (admittedly rather limited) experience is to ascribe what is referred to as an "absolute uncertainty"; the manipulations for uncertainty that one is taught in school science lessons, and, I suspect, most less sophisticated experimental analysis, work (I think) on the principle that the measurement is subject to zero mean noise from a Gaussian distribution (if there were a mean error, then this would rest on calibrating your instrument to remove it). The "absolute uncertainty" is then the standard deviation of this noise, or some multiple thereof.

Of course, when drawing a lot of data points, the Central Limit Theorem will apply and push the result towards a Gaussian (hopefully), which is one of the other reasons this distribution is commonly used—but I think it still represents a reasonable model for the noise in single measurements, subject to the comment on tail probabilities in my previous post.

Your other point ("in the same street") is that there is likely to be some correlation between the errors in each measurement. Thinking about this in terms of a coastal three-point fix, this is certainly possible if, for example, the compass is held off-horizontal by some amount for each measurement. On the other hand, I usually take some care to set myself again for each bearing—so I don't think the errors need be strongly correlated.

I understood the whole point of Bayesian inference to be that one can take a noisy observation and make profitable inferences from it; in the case of a single observation, the confidence in that observation might be quite low (i.e. the variance of the noise quite high)—but so long as one proceeds without violating the axioms of probability, one should hopefully end up with something fairly sensible.

I'm in no position to lecture on this, by the way—it's an interesting debate, and I suspect you may well know more about it than I do, so please take above points in that spirit...

Regards
William

 

[Bi111ion]

You are right William, a Bayesian approach is quite relevant to navigation. You typically start with a prior distribution of your position (eg from DR) and update it with a fix. In this case you can work out the posterior distribution and draw contours which enclose a certain percentage probability of your position. With the DR case, or a running fix you can also think of this as just more lines of position with different variances, and it is a weighted least squares problem.

 

[Bi111ion]

For those mathematically inclined I just published a paper in the Journal of Navigation " The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position "

You should be able to see it for free on this link https://tinyurl.com/y5uaswsk

The point is given a three (or more) line fix there is a ruler and compasses construction of the major and minor axes of the probability ellipse, and a rectangular box touching the ellipse (which is almost as good as drawing the ellipse).

I don't think the procedure is really practical, although one could get quick at doing it with practice, but it may give some insight in to the relation between the shape of a cocked hat and the probability ellipse.

I made a you tube video of drawing the axes here https://www.youtube.com/watch?v=JS4ZgwZlAcc and the bounding rectangle here https://www.youtube.com/watch?v=6e-sXAdHNHg&t=5s .. sorry they are not very good... my technical drawing skills are not what they were when I was at school, but hopefully you will get the idea.

 

[johnalison]

Geometry wasn't like that at school. I have been equally baffled while watching Chinese opera, but with less pleasure.

 

[AntarcticPilot]

For those mathematically inclined I just published a paper in the Journal of Navigation " The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position "

You should be able to see it for free on this link https://tinyurl.com/y5uaswsk

The point is given a three (or more) line fix there is a ruler and compasses construction of the major and minor axes of the probability ellipse, and a rectangular box touching the ellipse (which is almost as good as drawing the ellipse).

I don't think the procedure is really practical, although one could get quick at doing it with practice, but it may give some insight in to the relation between the shape of a cocked hat and the probability ellipse.

I made a you tube video of drawing the axes here https://www.youtube.com/watch?v=JS4ZgwZlAcc and the bounding rectangle here https://www.youtube.com/watch?v=6e-sXAdHNHg&t=5s .. sorry they are not very good... my technical drawing skills are not what they were when I was at school, but hopefully you will get the idea.

Thank you! I had been considering the maths, but hadn't been able to get my head round the necessary visualisation of the problem. I must now read it in detail, not just skimming it!

 

[jdc]

It's a bit late, but it's not necessary to assume Gaussian distributions of uncertainties. Once there are several sources of error (say 6) even 'top-hat' distributions combine very well to give an extremely good approximation to the Gaussian (see Central limit Theorum).

By the way, it's a rare pleasure to see a discussion in PBO forum which talks of Bayesian analysis (I'm a Bayesian adherent rather than a frequentist I admit).

 

[mattonthesea]

For my first fix during my DS course the cocked hat indicated that I was in the eastern end of the Solent! Philosophic arguments aside the probability of being inside it was p=1.

What a good thread. I have bookmarked the pages to spend some leisure time dragging stats back from a distant past.

 

[ghostlymoron]

I've never really thought about this before and probability isn't my forté but now I consider it, there's no reason why the error on each line should be inside the triangle rather than outside. But, for practical purposes, does it matter?

 

[AntarcticPilot]

I've never really thought about this before and probability isn't my forté but now I consider it, there's no reason why the error on each line should be inside the triangle rather than outside. But, for practical purposes, does it matter?

Well, yes. At least one of the examples in the paper demonstrates a probability of being within the cocked hat of less than 50%, and with a strong bias in one direction. It is a case that I think we would all identify as a weak fix (two bearings at a shallow angle, the other approximately at right angles to them), but in that case the assumption you are within the cocked hat is more likely to be wrong than right.

 

[lw395]

I've never really thought about this before and probability isn't my forté but now I consider it, there's no reason why the error on each line should be inside the triangle rather than outside. But, for practical purposes, does it matter?

The tendency is to assume that the size of the triangle at least indicates the level on uncertainty in the position.
Say your plot produces a nice equilateral traingle of 1 cable on each side, you'd assume you were pretty surely within say 2 cables of its centre? But is that necessarily true?

Consider the case where all three bearings come from a handbearing compass, say the errors are quite large.
Any two reading from the first two objects will intersect at a point. If there is a finite chance that the third bearing goes right through that point, despite that point not being your true position, then a small cocked hat does not actually tell you anything.

Add in some sytematic error, perhaps deviation of the compass, incorrect conversion to true bearing or something, or operator bias such as favouring readings which are multiples of 5 and a tendency to draw a small triangle in the wrong place can develop.

Probability breaks down for small sample sizes. The third bearing has a sample size of 1. You can't deduce the probability of anything by tossing a coin once.

 

[[70521]]

I wonder if every navigation class has at some point exited to the pub still debating if the probability of being in a cocked hat from a three line position fix is really 1/4 - surely it depends on the size of the hat!

It's never been a debate I've been involved with!

I suspect it is about some peoples fear of not knowing where they are on the planet or those how have no natural feel for where they are.

We have a natural barrier as you come into Plymouth and I am content if I am three boat lenghts from with the breakwater or the shore, I do not need to know I am 53m west of the lighthouse at the end of the western end, some people do.

Several decades ago I was a member of a Mountain Rescue Team and somebody from the military joined us on a training weekend. He wanted to see if his navigation skills were up to scratch before visiting our friends in Hereford and going for long walks in the Brecon Beacons. I've never seen anybody cling onto a map and compass so tighty. We would randomly stop and ask where we were and he would take ages studying the map and doing three point fixes. He was totally gobsmacked where we would just point at the map and say we are here. He quicly found out his skills were nowhere near enough to find his way across the Brecons and returned on a few other weekends to hone his skills - It took him some time to read the landscape, count steps in his head and know what way he was walking.

Thanks for the link to NavList, I was unaware of it and am an associate member of the Royal Institution of Navigation.

 

[Skylark]

I don't mean to be uncharitable but is there a point to this?

I have a reasonable grasp of statistics having spent my career in manufacturing industry but this is well beyond my capability. So what am I missing?

What's the objective of navigating while at sea. Taking a 3-point fix includes any technique from forenoon sun-meridian passage-afternoon sun to using Calshot chimney plus two other identifiable marks in The Solent. With a resulting cocked hat, I was taught to assume a position nearest to any danger; steer away from it if required. Otherwise, the purpose of maintaining a log and plotting track is to avoid hitting something hard rising out of the seabed and, ultimately, finding landfall. How will the statistical analysis referred to above improve those primary objectives?

The post has motivated me to search my text books for any background info. I can't find any reference in Norrie's Nautical Tables or in Bowditch American Practical Navigator.

The post above by Sandy seems quite appropriate. Is this an example of the difference between pure and applied maths?

I say again, not being disrespectful, just don't get it and hoping someone can turn the light on for me :encouragement:

 

[Bi111ion]

I mentioned my paper "The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position " before and there is now a preprint version here if anyone is interested, as it is easier than on the JoN website if you dont have a subscription http://eprints.maths.manchester.ac.uk/2725/

Some versions of Bowditch describe the ellipse of uncertainty I think. If you want a good practical book on celestial navigation I recommend David Burch's books Complete Course https://www.starpath.com/catalog/books/1887.htm and Hawaii by Sextant. Good treatment of errors there.

If you use the plug in in OpenCPN you can put each sight in as a separate line and it
The main take home story about the least squares approach is that if the triangle is very non-equilateral you are likely to be closer to the short side.

It is maybe more interesting to think about non-equal variances. If you are forced to make a round of three sights with a bad triangle but you want to make the ellipse of uncertainty more circular, you can take more sights of teh lines of positions for the the longest to sides and average to reduce the error.

To can also think of a running fix, or using DR position as well as lines of position as a special case of non-equal variances, including cross track and distance run errors as though they were lines of position.

 

[lw395]

...

It is maybe more interesting to think about non-equal variances. If you are forced to make a round of three sights with a bad triangle but you want to make the ellipse of uncertainty more circular, you can take more sights of teh lines of positions for the the longest to sides and average to reduce the error.
.....

You can only average random error.

 

[lw395]

I don't mean to be uncharitable but is there a point to this?

I have a reasonable grasp of statistics having spent my career in manufacturing industry but this is well beyond my capability. So what am I missing?

What's the objective of navigating while at sea. Taking a 3-point fix includes any technique from forenoon sun-meridian passage-afternoon sun to using Calshot chimney plus two other identifiable marks in The Solent. With a resulting cocked hat, I was taught to assume a position nearest to any danger; steer away from it if required. Otherwise, the purpose of maintaining a log and plotting track is to avoid hitting something hard rising out of the seabed and, ultimately, finding landfall. How will the statistical analysis referred to above improve those primary objectives?

The post has motivated me to search my text books for any background info. I can't find any reference in Norrie's Nautical Tables or in Bowditch American Practical Navigator.

The post above by Sandy seems quite appropriate. Is this an example of the difference between pure and applied maths?

I say again, not being disrespectful, just don't get it and hoping someone can turn the light on for me :encouragement:

I too find it hard to see a point, particulary if you don't have a firm grasp of the underlying maths and the logic beneath that.
Statistics are all very well but what confidence level do you want?
Personally 99% sure I'm not going to get shipwrecked doesn't cut it.
OTOH, a best guess of how much progress I've made across safe water is fine.

The question is, is the fix or position line 'fit for purpose'?

 

[Bi111ion]

You can only average random error.

Yes exactly. You have to measure or calibrate out your index error. And systematic errors caused by your technique, like if you consistently miss estimate the horizon or use the wrong eight of eye, have to be reduced by practice when you know where you are.

After that the errors tend to be fairly random.

Interestingly a fixed systematic error, like an index error, can lead to a cocked hat which does NOT contain your true position, if the direction to the star's GP are not all inward or all outward relative to the triangle. In that case your position at one of the excentres, that is one of the circles tangent to the LOPs but outside the cocked hat.

 

[Skylark]

I mentioned my paper "The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position " before and there is now a preprint version here if anyone is interested, as it is easier than on the JoN website if you dont have a subscription http://eprints.maths.manchester.ac.uk/2725/

Some versions of Bowditch describe the ellipse of uncertainty I think. If you want a good practical book on celestial navigation I recommend David Burch's books Complete Course https://www.starpath.com/catalog/books/1887.htm and Hawaii by Sextant. Good treatment of errors there............

I can find no reference to an ellipse of uncertainty in my 2002 Bicentennial Edition of Bowditch . It does, of course, mention error circles.

I have YM Ocean and teach the shorebased course. My preferred text book is Adlard Coles Ocean Yachtmaster. Within Chapter 10 there are comments on Position Lines Errors and this concludes "Due to cumulative errors, position lines will rarely if ever exactly intersect and a cocked hat is formed. Each position line may be subject to different errors and it is not possible to determine the boat's position exactly. For safety the boat's position should be assumed to be at the corner of the cocked hat nearest to any danger"

Those words of wisdom have kept sailors safe for many centuries. I'm afaid that "the penny still hasn't dropped" for me to understand the message that you're trying to portray?

 

[Impaler]

Whilst an officer cadet in the Royal Navy I was required to carry out a number of Astro Navigation sun shots. My mentor, a very patient Lt. Cdr. in the training team thanked me after I presented my work. His comment was as far as I recall ‘ I know we are in the Mediterranean but perhaps you could be a little more precise than somewhere between Rome and Gibraltar.’ I didn’t really improve and became a Paratrooper. Never been lost since.

 

[ghostlymoron]

I can find no reference to an ellipse of uncertainty in my 2002 Bicentennial Edition of Bowditch . It does, of course, mention error circles.

I have YM Ocean and teach the shorebased course. My preferred text book is Adlard Coles Ocean Yachtmaster. Within Chapter 10 there are comments on Position Lines Errors and this concludes "Due to cumulative errors, position lines will rarely if ever exactly intersect and a cocked hat is formed. Each position line may be subject to different errors and it is not possible to determine the boat's position exactly. For safety the boat's position should be assumed to be at the corner of the cocked hat nearest to any danger"

Those words of wisdom have kept sailors safe for many centuries. I'm afaid that "the penny still hasn't dropped" for me to understand the message that you're trying to portray?

I think the point of all this is that you could be closer than the nearest corner. It's made me more thankful for GPS.

 

[Bi111ion]

. " For safety the boat's position should be assumed to be at the corner of the cocked hat nearest to any danger"

Those words of wisdom have kept sailors safe for many centuries. I'm afaid that "the penny still hasn't dropped" for me to understand the message that you're trying to portray?

David
First of all thanks for your comment and I value your experience. While I have many thousands of sea miles in small yachts (and a few big ones) and did my RYA shore based courses as a teenager in the 1970s I have very little practical experience taking celestial sights at sea on long passages, and I don't have an Ocean Yacht Master's ticket. I do know quite a bit about uncertainty quantification though as it is my job. If I can find the right way to help get the "penny to drop" for people such as yourself then I will be very pleased. I am still working how to explain it in a helpful way!

My paper starts with
"The (Admiralty Manual of Navigation 1938, p166) states
In the practice of navigation, when a cocked hat is obtained, it is customary to place the ship’s
position on a chart in the most dangerous position that can be derived from the observations because
the existence of a cocked hat is evidence that the observations are inaccurate, and by interpreting
them to his apparent disadvantage, the navigator gives himself a margin for safety which he might
not otherwise have."

Note the AMN does not tell you to choose a corner of the cocked hat. There are two parts. First that the lines don't meet at a point alerts you to the presence of an errors. Second that you should choose the most dangerous position you can infer from that.


And I think it dates way back to previous editions of AMN, I am just quoting the one I had to hand.

The first description of the least squares solution I could find (well actually Herbert Prinz on Navlist suggested it as the earliest) is Yvon-Villarceau, A. J. F. & Aved de Magnac, H. J. (1877), Nouvelle navigation astronomique, Gauthier-Villars, Paris. So this idea of a having more than two lines of position then finding the most probable position was "Nouvelle", ie new, in the late 1800s and as I understand it became more widespread n the early 1900s. The reference I give for ellipses of uncertainty is Stansfield, R. (1947) and it's motivation was RDF fixes.

The thing that isn't so useful about Coles' idea is that the "cocked hat" is not a useful guide to a probability contour, although it does at least have the idea that a long skinny triangle will result in less certainty in the long direction, the probability contours can stick out quite a long way outside the triangle. Remember the maximum likelihood point, or symmedian, is close to the short edge of the triangle. Have a look at Figure 6 d for example. If your danger was to the NE then Coles' approach would not be too bad an idea. If it was SW it would clearly be wrong, with the same assumed probability that would put you at NE vertex you would be almost as far the other side of the triangle rather than the vertices.

Please let me know if this makes sense to you..

Bill