Probability of being in a Cocked Hat

[Bi111ion]

Remember with celestial navigation you typically get a running fix from the sun at three different times or you can use several stars, planets and the moon at twilight. In between you use DR. So exactly, you have this charted reef that you know about (because you were wise enough to use a paper chart, not one that only shows the reef if you zoom in). You have a good idea of your uncertainty due to the errors in the fix and the DR and you set your course knowing that margin of error.

 

[[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!

Mine never did! More interested in the quality of the beer.

I would be more worried about the thickness of the pencil line in drawing the cocked hat, but I am old school and there or thereabouts at sea is usually accurate enough. In the mountains with a 100 meter cliff in front of you and a white out you are measuring paces and checking bearings several times before making the turn.

 

[weustace]

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?

Although your position is deterministic, your estimate of your position is made through measurements which inevitably have some noise (GPS timings, compass bearings, sextant altitudes)—hence the paper. Measurement noise is often considered Gaussian; this probably has something going for it. I would suggest that the errors are almost certainly random—but the next assumption one would usually make is that the error is zero mean, which may not be true if there is a systematic error in the measurement (sextant been bashed, compass next to a metal thing—steel glasses frame?—etc). I think it's probably a fair first order estimate though, for the sake of this discussion.

Regards
William

 

[johnalison]

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

I have to admit to finding probabilities difficult but it find his chart confusing. According to his lower diagram, if you take one position line, the probability given outside the line is given as 16.7 and twice 8.3, making about 33%, with 66% on the other side which includes the cocked hat. It looks as if there is twice the chance of being outside this one line than inside, which doesn't make sense to me, though all the probabilities add up to 100 as you'd expect.

 

[Bi111ion]

Although your position is deterministic, your estimate of your position is made through measurements which inevitably have some noise (GPS timings, compass bearings, sextant altitudes)—hence the paper. Measurement noise is often considered Gaussian; this probably has something going for it. I would suggest that the errors are almost certainly random—but the next assumption one would usually make is that the error is zero mean, which may not be true if there is a systematic error in the measurement (sextant been bashed, compass next to a metal thing—steel glasses frame?—etc). I think it's probably a fair first order estimate though, for the sake of this discussion.

Regards
William

Yes if you have three lines of position and only a fixed systematic error the true position is the incentre or excentre of the triangle.

here is a Geogebra animation to see the incircle and excircles, you see their centres are all the same distance from the LOPs but it depends which direction it is to the celestial body.

If the errors are zero mean and random with the same distribution then the best linear unbiased estimate is the symmedian (or Lemine) point. If the errors are zero mean and normally distributed with the same standard deviation this is the maximum likelihood estimate. If the variance are different (often one LOP is more accurate than others) you need the weighted symmedian point. If you have both systematic and random errors you need more than three LOPs. A back sight or a sight of another body with reciprocal zenith helps find systematic error.
https://www.geogebra.org/classic/ud8af3mr

 

[Joker]

"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)"

I remember an Open University programme on BBC2 [many years ago, obs], which asserted exactly that.

 

[lw395]

Although your position is deterministic, your estimate of your position is made through measurements which inevitably have some noise (GPS timings, compass bearings, sextant altitudes)—hence the paper. Measurement noise is often considered Gaussian; this probably has something going for it. I would suggest that the errors are almost certainly random—but the next assumption one would usually make is that the error is zero mean, which may not be true if there is a systematic error in the measurement (sextant been bashed, compass next to a metal thing—steel glasses frame?—etc). I think it's probably a fair first order estimate though, for the sake of this discussion.

Regards
William

Gaussian and Random are two big assumptions there.

A lot depends on the nature of the position lines.
If they are all from a HB compass, then a common deviation applied to the 3 bearings can have interesting effects.
If one of them is more dependable, like a transit, then the game is changed.

 

[Bi111ion]

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

Yes follow that link. It is really pretty confusing and it took me a while to understand. One of those problems that you can convince yourself of two equally plausible sounding answers for a while! But a diagram like the one in the link is what persuaded me in the end. When you have convinced your self of the 1/4, if you are still interested an know a bit of calculus you might find Robin Stuart's paper interesting. I expect the problem was bugging him for a while, and I think it has been discussed on NavList. I think it is instructive to see examples for different shaped cocked hats... but the message is maybe don't assume you are in the triangle, just fairly close to it!

 

[weustace]

Gaussian and Random are two big assumptions there.

A lot depends on the nature of the position lines.
If they are all from a HB compass, then a common deviation applied to the 3 bearings can have interesting effects.
If one of them is more dependable, like a transit, then the game is changed.

I still don't think that "random" is unreasonable—zero mean I must agree is a bigger jump to make.

I don't suggest that the errors are always Gaussian, but I still think it's a reasonable assumption provided you've identified the targets for the three sights correctly and have made no systematic errors. In practice there may be a heavier tailed distribution in play, as acknowledged in the paper's introduction—part of the reason for the Gaussian assumption might be that normalising the resulting multivariate distribution would otherwise be a challenge analytically.

Of course, at the end of the day most of us are interested in a point estimate of position, but, if one accepts the challenge of the paper to quantify the distribution on that position, how would you suggest modelling the measurement noise?

 

[Amulet]

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.

This is equivalent to what I meant when I said "once you've drawn the first line". No matter which side of the first line a point is, the remaining two lines can form a cocked hat. Once you've drawn the first line you have removed one of the possibilities.

I actually find that a three point fix as a single way of estimating position is rare. If more than three points are available you sight them. If only two you take them anyway. If you are worrying about your position and only one is available you take it as a credibility check. You look to see if the depth is plausible. You snatch any transit that turns up. You plot your position as best you can with all information available, and make a pessimistic assessment of its accuracy based on all factors. If you are near something unmarked and grisly you give it a helluva wide berth.

However, there is a certain satisfaction about the three distillery fix available off Islay.

 

[lw395]

I still don't think that "random" is unreasonable—zero mean I must agree is a bigger jump to make.

I don't suggest that the errors are always Gaussian, but I still think it's a reasonable assumption provided you've identified the targets for the three sights correctly and have made no systematic errors. In practice there may be a heavier tailed distribution in play, as acknowledged in the paper's introduction—part of the reason for the Gaussian assumption might be that normalising the resulting multivariate distribution would otherwise be a challenge analytically.

Of course, at the end of the day most of us are interested in a point estimate of position, but, if one accepts the challenge of the paper to quantify the distribution on that position, how would you suggest modelling the measurement noise?

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.

 

[Deleted member 36384]

Bowditch (sp?), the American manual of navigation has a section on this. IIRC it indicates the error lines as ellipses.

It’s easy to see the possible errors in the position lines by drawing a cocked hat then, re plotting by moving the position lines say + and - 5 degrees and observing how the cocked hat moves about. It will be noticed that it is possible to be outside the initial cocked hat.

As for probability, I guess one could over a period of time plot cocked hats and note GPS position, following some consistent method and calculate a probability. But why, it’s only ever an indication of a potential position anyway, which for all intent and purposes is good enough.

 

[AntarcticPilot]

Although your position is deterministic, your estimate of your position is made through measurements which inevitably have some noise (GPS timings, compass bearings, sextant altitudes)—hence the paper. Measurement noise is often considered Gaussian; this probably has something going for it. I would suggest that the errors are almost certainly random—but the next assumption one would usually make is that the error is zero mean, which may not be true if there is a systematic error in the measurement (sextant been bashed, compass next to a metal thing—steel glasses frame?—etc). I think it's probably a fair first order estimate though, for the sake of this discussion.

Regards
William

Well, theoretically you can't know your position and your speed simultaneously- Heisenbergs's Uncertainty principle. But I don't think it will matter very much at our speeds and mass!

 

[johnalison]

Well, theoretically you can't know your position and your speed simultaneously- Heisenbergs's Uncertainty principle. But I don't think it will matter very much at our speeds and mass!

I don't see what the difficulty is. If you just do a two-bearing fix you get a clear position. No cocked hat and no uncertainty!

 

[Bi111ion]

I don't see what the difficulty is. If you just do a two-bearing fix you get a clear position. No cocked hat and no uncertainty!

You still get elliptical contours for the uncertainty. The difference is that the measurements don't give you a way to estimate the uncertainty, but if you know that, a few degrees for hand bearings, a few minute or so of arc for celestial navigation fix, then you can draw say a 95% confidence ellipse. I think what you mean is that the Most Likely Position is obvious as it is where the lines cross, where as for three or more lines the least squares point is harder to guess/draw/ calculate.

(Its alright I know you were joking, but I have heard this said not in jest!)

 

[davidej]

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.

Can you explain why there are two combinations (or do I mean permutations) which put you in the hat.

My statistics classes were over 40 years ago, and now I can't fathom that one out.

 

[Amulet]

Can you explain why there are two combinations (or do I mean permutations) which put you in the hat.

My statistics classes were over 40 years ago, and now I can't fathom that one out.

I think an easier way to arrive at the same conclusion is to realise that it doesn't matter which side you are of the first line you are, it is still possible to end up in a cocked hat. You then need the two subsequent lines to err in the correct direction to get you in the cocked hat. !/2 x 1/2 = 1/4.

Or, if you like to think of it the other way round: once you have two intersecting lines and then plot the third bearing, it can form a cocked hat by going either to the left or to the right of the intersection point - two ways of getting a cocked hat.

 

[Elessar]

I think an easier way to arrive at the same conclusion is to realise that it doesn't matter which side you are of the first line you are, it is still possible to end up in a cocked hat. You then need the two subsequent lines to err in the correct direction to get you in the cocked hat. !/2 x 1/2 = 1/4.

Or, if you like to think of it the other way round: once you have two intersecting lines and then plot the third bearing, it can form a cocked hat by going either to the left or to the right of the intersection point - two ways of getting a cocked hat.

Plain English - thank you. I understood that! 1/4 it is.

 

[Bi111ion]

I think an easier way to arrive at the same conclusion is to realise that it doesn't matter which side you are of the first line you are, it is still possible to end up in a cocked hat. You then need the two subsequent lines to err in the correct direction to get you in the cocked hat. !/2 x 1/2 = 1/4.

Or, if you like to think of it the other way round: once you have two intersecting lines and then plot the third bearing, it can form a cocked hat by going either to the left or to the right of the intersection point - two ways of getting a cocked hat.

Well put. That is exactly the point.

 

[weustace]

Well, theoretically you can't know your position and your speed simultaneously- Heisenbergs's Uncertainty principle. But I don't think it will matter very much at our speeds and mass!

This is also true, though I don't think stemming from quite the same causes as the uncertainty I referred to...