Re: Average and Variance of a periodic angles
- From: junoexpress <MTBrenneman@xxxxxxxxx>
- Date: Fri, 7 Mar 2008 14:13:03 -0800 (PST)
You are on the right track: the answer to your problem lies in what is
commonly called "circular statistics".
To compute the avg of angles correctly you use the first trig moment.
Suppose you have N angles {q1,...,qN}, then compute:
C1 = Sum i=1 to N {cos(qk)}
S1 = Sum i=1 to N {sin(qk)}
Avg Angle = atan2(S1,C1) (use 4 quad arctan)
You'll notice that the avg angle is just the angle of the resultant
vector obtained by adding up all of the vectors {cos(qk),sin(qk)}.
The second-order stats are more difficult.
Here are some formulas you might find useful, but they are not
intuitively obvious
#1) Sample Circular Variance, V
V = 1 - R1/N
where R1 = length of resultant vector from data set
#2) Sample Circular Devtn, nu
nu = sqrt[ 1 - 2Log(1-V) ]
#3) Circular Dispersion, delta
Compute 2nd trig moment of cosines and sines
C2 = ( Sum i=1 to N {cos(2*qk)} )/N
S2 = ( Sum i=1 to N {sin(2*qk)} )/N
Now compute
R2 = C2^2 + S2^2
Then
delta = (1-R2)/(2R1^2)
These formulas comes from a very good reference, Nick Fisher's book
Statistical Analysis of Circular Data, Cambridge University Press,
1993. ISBN 0-521-35018-2
I
Also if you Google under
(variance AND statistics) AND (directional OR circular),
you'll find other refs and some codes too. In some cases, if the
distribution is well-known, the formulas will be worked out as well as
the code.
Maybe someone better in this area can give you more guidance.
HTH,
Matt
.
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- Average and Variance of a periodic angles
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