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Round and round at sea with circular statistics

I was recently reminded of some work I had completed for the oil and gas industry many years ago and thought it would be useful to share with other analysts/statisticians.

For more general information see a case study I have put onto my personal blog (Offshore Storms Statistics and Oil Rigs).



It was important for the oil and gas engineers to gain a better understanding of the offshore environment during storm conditions, and in particular if there would be an impact upon the mooring of their semi-submersible drilling rigs. (See for example this paper: Bowers J, Morton I, Mould G 1997. Weathering the storm – how OR steered a course between extreme statistics & offshore design. OR Insight; 10 3: 16-21).


It was necessary for me to use circular statistics to analyse the wind and wave directions and come up with the mean directions and a 95% confidence interval.


Linear statistics aren’t appropriate because there is a crossover problem. If we consider three wind directions of 358, 0 and 2 degrees respectively then using linear statistics to find the mean we would add them together and divide by three, to arrive at a mean wind direction of 120 degrees (but it should have given an answer of 0 degrees).


I started to read about how to calculate basic circular statistics from a very useful book by NI Fisher Statistical analysis of circular data, and soon realised that I would have to revise my school maths on how to work with trigonometric functions. The mean direction is found from equation 1. It turns out that the calculation of a 95% confidence interval for the sample mean of circular data has mathematical equations that are intractable, and approximations which are unwieldy. Firstly, you could then assume that the data fit the “von Mises” distribution (It’s a wild assumption, other distributions might be more appropriate). Then, given this leap of faith, a reasonable approximation to determine the concentration  is as in equation 2 and following on from this, an estimate of the 95% confidence interval is given by equation 3. Maybe I have set the seeds for you to think about using circular statistics in your work, or alternatively you didn’t get this far because you were put off by the equations. Other practical examples are provided by Fisher and two examples are: the arrival times at an intensive care unit; and the vanishing directions of homing pigeons.


Please let me know if you are using circular statistics in your work, or you have any other comments.


Further reading

Bowers JA, Morton I, Mould GI 2000. Directional statistics of the wind and waves. Applied Ocean Research; 22: 13-30.

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