Probability of being in a Cocked Hat

[Bi111ion]

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!

The age old conundrum has been settled by Robin Stuart in the Journal of Navigation https://www.cambridge.org/core/jour...n-cocked-hat/E633B3F6347774A0E0E091FD51523959

In a nut shell, before you know where the lines are it is 1/4. Once you have drawn the lines it depends, and he works it out.

The admiralty manual of navigation from over 100 years ago suggest picking the point of the triangle closest to danger.

A better idea might be to consider the ellipse of uncertainty and pick the point of that closest to danger.

Anyway such debates abound on NavList, where we still discuss the best use of our sextants and chronometers, and maybe not practical enough for PBO forum!

 

[LittleSister]

I have heard numerous times people claiming that you must be somewhere inside the cocked hat. I could never understand how that could be true, and am pleased to hear that people much more knowledgeable than me can confirm that idea is mistaken.

I'll have peek at the article later, but I suspect the maths will be beyond me.

 

[AntarcticPilot]

There was a long thread on this some time ago, to which Tim Bartlett contributed - the spatial statistical problem is quite interesting. But you are quite correct - the chances of being within the cocked hat are not much better than 50%. I did figure out a methodology for working out the lines of equal likelihood more exactly, but never did it - it was one of those things that is easier to describe than to do!

 

[john_morris_uk]

Furthermore, the prudent navigator can place varying degrees of confidence in the position lines that have been drawn.. (eg a good transit of clearly identified objects might mean you have a high confidence whereas the ‘I think it’s about 22degrees on the hand gearing compass might be plus or minus a few...)

I’ve always taught students to use the point nearest danger as it’s easy to remember as a rule..

Prudence is key and assumption is extremely foolish in all things navigation...

 

[capnsensible]

If ever possible to revisit a most excellent PBO series of articles condensed into several booklets, try John Goodes slowly expanding 'circle of probability' and navigate that circle to your destination.

Or use an iphone......

(Navigation made simple).

 

[Amulet]

The 1/4 value is conceptually easy to justify. Having drawn the first line there is a 50% chance of each subsequent line erring in the direction which puts you outside of the cocked-hat. !/2 x 1/2 = 1/4. Simple.

More interesting is how far outside it to expect. The size of the cocked-hat could be regarded as evidence of how good you are at taking bearings. Certainly thinking that one corner of the cocked hat (nearest danger) is the worst possible, is hopelessly optimistic. I cannot do the statistics, but tend to assume that the distance outside of the cocked hat might easily equal the largest dimension of the cocked hat, and could be more.

 

[LittleSister]

The size of the cocked-hat could be regarded as evidence of how good you are at taking bearings.

I would have thought that a small cocked could be the result of being good at taking bearings, but could also be the result of chance. So it seems to me unreliable evidence of the former.

I could imagine that the size of the cocked hat would likely be inversely proportional to the proximity of a YM examiner.

 

[st599]

I would have thought that a small cocked could be the result of being good at taking bearings, but could also be the result of chance. So it seems to me unreliable evidence of the former.

I could imagine that the size of the cocked hat would likely be inversely proportional to the proximity of a YM examiner.

Surely the standard deviation in the line's position would be the measure of how accurate your bearing taking is.

 

[Richard10002]

Is there an argument for suggesting it doesn’t matter? other than as an academic exercise?

When proceeding along a course line, as long as the trend of the cocked hats is along the line, you are heading in roughly the right direction, and will continue to do so as long as the cocked hats continue with the same trend.

If there is a danger nearby, there is usually some other kind of navigational thing which helps you to avoid it - quadrant and red and green buoys, lighthouses, coloured light sectors, horns, similar, (its been a long time).

I always remember being a cadet taking a noon site in the middle of the Atlantic.

3 or 4 people would take a site, and it would always be the position of the 2nd Officer that would be “correct” and used for the Noon position.

I wondered firstly why it mattered and, secondly, how any of us could know whose position was correct.

At some point I decided that when I was a Second Officer, I would choose the Deck Cadets position as often as I would my own, (provided no glaring error in the calculation). I made it to 3rd Officer, but not 2nd, so didn’t get the chance to put my motivating management technique into practice

So I would suggest that it rarely matters where you are in relation to the cocked hat - you are in the region.

and, when it does matter, there is almost certainly something else around to help avoid danger.

 

[Bi111ion]

The 1/4 value is conceptually easy to justify. Having drawn the first line there is a 50% chance of each subsequent line erring in the direction which puts you outside of the cocked-hat. !/2 x 1/2 = 1/4. Simple.

More interesting is how far outside it to expect. The size of the cocked-hat could be regarded as evidence of how good you are at taking bearings. Certainly thinking that one corner of the cocked hat (nearest danger) is the worst possible, is hopelessly optimistic. I cannot do the statistics, but tend to assume that the distance outside of the cocked hat might easily equal the largest dimension of the cocked hat, and could be more.

The thing that confuses people is they thing 1/2 chance of being on each side of a LOP (1/2)x(1/2)x(1/2) is (1/8) . The problem with this is there still two ways of doing that and the position being inside the hat, so it is 1/4.

Remember that this is "a priori", before you know where the lines are. Once you have the lines the probability depends on the lines. That is what the JoN paper is about.

If you assume Gaussian (=normally distributed) errors, which is reasonable, then the contours of equal probability are ellipses. For a long skinny triangle you get a long skinny ellipse. For an equilateral triangle you get an ellipse. Even for more than three lines you still get an ellipse. It is the curve where the sum of squared distance to the lines is fixed. Some lines are more accurate than others this also changes the shape of the ellipse.

 

[Bi111ion]

So I would suggest that it rarely matters where you are in relation to the cocked hat - you are in the region.

and, when it does matter, there is almost certainly something else around to help avoid danger.

An exception would be unmarked reefs in the Pacific for example.

 

[Richard10002]

An exception would be unmarked reefs in the Pacific for example.

I agree, hence “rarely”

How have you navigated to avoid them? I’d have thought keep well clear, such that position line errors can’t possibly take you too close.

Having said that... where are you getting your 3 position lines from ?

 

[weustace]

Not that it impinges too much on the scope of the paper, but there is another point to consider if one is discussing the Bayesian approach to navigation (which is really just the intuitive way forwards)...

You presumably have an EP before you begin taking a fix. How confident are you in that EP? In reality, as in fig. 2 of the paper, there will be a spatial distribution of some nature around your EP. If you are infinitely confident in it, then of course you will not allow a fix placing you a long way away to affect your thinking... in a more typical way of things, the EP is very approximate and your estimate of position will be substantially updated by the fix.

The paper also makes a very good case against the sole focus of life being on minimising the area covered by your cocked hat: as the area tends to zero, the probability that you're in it also tends to zero. This is trivially obvious, but I had never really thought about it in that sense—probably because I tend to plot my GPS position immediately after taking a fix, so never worry too much about how accurate I have been except for training purposes!

Regards
William

 

[zoidberg]

An exception would be unmarked reefs in the Pacific for example.

Don't be fixated on a fix.

Think 'Limiting Danger Line' parallel to intended Course Made Good.

Seek/acquire multiple LOPs parallel to intended CMG, then plot/use Gaussian distribution to determine Across Track Error and correct accordingly. Seek/acquire multiple LOPs 'normal' to intended CMG, then plot/use Gaussian distribution to determine Distance To Go until abeam.
Then observe sounder plot and sea surface behaviour.

If in doubt, stop. You can always stop the boat....

If in serious doubt, Williamson Turn away from hazard and backtrack.

( Or pull 3G and Chandelle.....or Immelman out of there. )

Do not forget the Limitations of Survey.

 

[lw395]

Where you are is not actually a matter of probability.
You are where you are.
That is a fact and it won't change due to you reading some numbers off a compass.

You might or might not be able to think about the probability of reading the bearings as you did. Can you justify a particular distrubution of indicated values? Are all the errors random? Why would anyone expect them to be?

 

[Elessar]

The 1/4 value is conceptually easy to justify. Having drawn the first line there is a 50% chance of each subsequent line erring in the direction which puts you outside of the cocked-hat. !/2 x 1/2 = 1/4. Simple.

More interesting is how far outside it to expect. The size of the cocked-hat could be regarded as evidence of how good you are at taking bearings. Certainly thinking that one corner of the cocked hat (nearest danger) is the worst possible, is hopelessly optimistic. I cannot do the statistics, but tend to assume that the distance outside of the cocked hat might easily equal the largest dimension of the cocked hat, and could be more.

Except there are 3 lines. 1/2x1/2x1/2. 12.5% not 25 %.
And you are right the smaller the hat the more confidence in its accuracy.

 

[Elessar]

The thing that confuses people is they thing 1/2 chance of being on each side of a LOP (1/2)x(1/2)x(1/2) is (1/8) . The problem with this is there still two ways of doing that and the position being inside the hat, so it is 1/4.

Remember that this is "a priori", before you know where the lines are. Once you have the lines the probability depends on the lines. That is what the JoN paper is about.

If you assume Gaussian (=normally distributed) errors, which is reasonable, then the contours of equal probability are ellipses. For a long skinny triangle you get a long skinny ellipse. For an equilateral triangle you get an ellipse. Even for more than three lines you still get an ellipse. It is the curve where the sum of squared distance to the lines is fixed. Some lines are more accurate than others this also changes the shape of the ellipse.

Ok I should have read that before I posted. Don’t understand what you said but it sounded confident.

 

[superheat6k]

Except there are 3 lines. 1/2x1/2x1/2. 12.5% not 25 %.
And you are right the smaller the hat the more confidence in its accuracy.

Except in your case Mark, and indeed most folk on a fast planing boat it will invariably be where you were, and expect elongated cocked hats.

Do most modern fast planing boats even carry a hand bearing compass ?

With all the electronic wizardry do many skippers actually apply old fashioned methods of checking their position ?

I rarely do in sheltered waters, but always have a paper chart with ground track showing my intended route with a plot ~every 30 minutes, although I do cheat with the position source, I will also make rough checks with the compass to cross check providing an ongoing series of running fixes. After all it isn't so much where you are, but where you are not that matters (IMHO of course).

 

[dom]

Ok I should have read that before I posted. Don’t understand what you said but it sounded confident.

Conditional probability can be counterintuitive at first, but frame it right and it’s easy to visualise. Try this and you’ll see it in a jiffy:

http://onboardintelligence.com/CelestialNav/CelNav8.aspx

 

[Elessar]

Except in your case Mark, and indeed most folk on a fast planing boat it will invariably be where you were, and expect elongated cocked hats.

Do most modern fast planing boats even carry a hand bearing compass ?

With all the electronic wizardry do many skippers actually apply old fashioned methods of checking their position ?

I rarely do in sheltered waters, but always have a paper chart with ground track showing my intended route with a plot ~every 30 minutes, although I do cheat with the position source, I will also make rough checks with the compass to cross check providing an ongoing series of running fixes. After all it isn't so much where you are, but where you are not that matters (IMHO of course).

Yes i do carry a hand bearing compass and I make sure I go out a couple of times of year with no electronic aids other than depth - just so I don't forget how. This is more for my own satisfaction than necessity because we have so many independent GPSs these days.

If I need to do a 3 point fix I stop. By the time I took a fix on my flybridge at planing speed I'd be initiating a MOB for the chart.....