True Circular Mean

Routines to compute the true circular mean and standard deviation, as well as circular median.

Documentation and example usage for circular_mean.m and associated routines

https://www.mers.byu.edu/circularmean.html

David Long 08 Jul 2023, updated 31 Jul 2023

When computing the mean of angles the conventional approach is to take the arctangent of the mean sine and cosine of the angles. However, this is not true circular mean. Consider:

The mean M of a discrete set a={a_i} is formally defined as

M = argmin Sum distance_function(a_i,m) (1)

m a

where the distance_function(a_i,m) is some metric appropriate to the set.

For a linear mean, the one we are used to for linear values, the distance function is given by distance_function(a_i,m)=abs(a_i-m). However, for the conventional method commonly used for computing an average angle, i.e., the arctangent of the mean of the sine and cosine of the angles, it can be shown that the distance function is actually given by (conventional) distance_function(a_i,m)=1-cos(a_i-m), which is quite different than a linear mean.

To illustrate this, consider Fig. 1 (example.png), which shows three angles as stars on the unit circle. The average angle computed using the conventional method is given by the black line. The dashed red line shows the circular mean angle computed using the linear distance metric, which is visually much closer to the "middle" or mean angle than is the conventional angle average.

To solve for the mean using Eq. (1), we take the derivative and set it to zero. The conventional method has the advantage of being easily differentiable. The average angle metric (1-cos(a_i-m)) yields the conventional formula for an "average angle" Ma given by:

Ma = atan2(sum(sin(a*pi/180)), sum(cos(a*pi/180))) -equivalent to-

Ma = atan2(mean(sin(a*pi/180)), mean(cos(a*pi/180))) -or-

Ma = angle(exp(j*a*pi/180)) (in Matlab)

This works where the arctangent of its input arguments are defined. Note that it involves transcendential functions and that the distance function is related to the cosine of the angle difference. Further, when using circular sets such day of year, the value (which might be an integer) has to be converted to the range [0..2*pi), which involves the irrational number pi.

However, when considering the difference between two angles, the distance function can be better defined as what might called the minimum arc distance, i.e., the minimum of the two possible angles or arcs between the two angles. That is, the minimum angle moving clockwise or counterclockwise from one angle to the other. This minimum arc distance is between 0 and 180 deg. This is the linear distance as defined on a circular set. Computationally, this is the minimum of abs(a_i-m), abs(a_i-m-360), and abs(a_i-m+360), which can be written succintly as

distance_function(a_i,m)=180-abs(180-abs(mod(a_i-m,360))). (2)

where mod(x,y) is defined as in Matlab. Note: the signed (+CCW vs -CW in Matlab plots) distance_function for two angles a_i and m can be written as

distance_function(a_i,m)=mod(m-a_i+180,360)-180 (3)

Eq. (2) appears hard to differentiate. However, by dividing the modulo and absolute value functions into sets, the derivative of the distance function can be determined [1], which can be used to solve Eq. (1) for the true circular mean. This is computationally a little complicated since it requires sorting parts of the data, but it involves no transcendential functions. Further, Kogan's implementation [1] avoids the angle sampling limitations required by Matsuti's method [2]. Computation in degrees avoids irrational numbers such as pi and so is well-suited for other circular data sets. Note that Kogan's method is not a "single-pass" algorithm [2,3], i.e., it requires the entire list of input angles.

The weighted circular mean Mw is defined as

Mw = argmin Sum w_i*distance_function(a_i,m) (4)

m i

where {w_i} is the set weights associated with {a_i}.

The true circular standard deviation S is given by

S=sqrt( Sum distance_function(a_i,Mt)^2 / N) (5)

i

where Mt is the true circular mean and N is the number of angles in the set {a_i}. Note that this differs from common approximations for the angle standard deviation used by others [2],[3]. In computing both the conventional and circular standard deviation, this code uses the definition in Eq. (5) The weighted true circular standard deviation is given by

S=sqrt( Sum w_i*distance_function(a_i,Mt)^2 / Sum w_i). (6)

i i

Also of interest is the circular median, which has several potential definitions. Here the circular median is defined as the input angle which has the smallest summed absolute angular distance between it and all other input angles.

This package includes code for computing the circular mean, standard deviation, and median. Also provided is code to compute the conventional angle mean and standard deviation.

References:

[1] Lior Kogan, 2013, "Circular Values Math and Statistics with C++11", https://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus downloaded 1 Jul 2023.

[2] Mori, Y., 1986. Evaluation of Several Single-Pass Estimators of the Mean and the Standard Deviation of Wind Direction. J Climate Appl. Metro., Vol. 25, pp. 1387-1397.

[3] Yamartino, R.J., 1984. A Comparison of Several "Single-Pass" Estimators of the Standard Deviation of Wind Direction". Journal of Climate and Applied Meteorology. Vol. 23(9), pp. 1362-1366. doi:10.1175/1520-0450(1984)023<1362:ACOSPE>2.0.CO;2.

Routines:

circular_mean.m : function result = circular_mean(ang, sens) returns circular_mean

circular_std.m : function [mean_out std_out] = circular_std(ang) returns circular_mean and standard deviation

weighted_circular_mean.m : function result = weighted_circular_mean(ang, weights, sens)

cmediandeg.m : function out=cmediandeg(in) returns circular median (input angle with smallest total absolute angular difference)

diffangledeg.m : function out = diffangledeg(in1,in2) returns signed angular difference between two angles

meanangledeg.m : function out = meanangledeg(in,dim,sens) returns conventional angle mean computed using sin/cos/atan2

stdangledeg.m : function [mean_out mean_out2 std] = stdangledeg(in,dim,sens) returns conventional trigonometric angle mean estimate and the mean and standard deviation of the angular difference between the trig angle mean and the input angles

Example cases: (from example_usage.m)

Set of Angles: -10 90 5 (deg)

Circular mean and std: 28.33 44.03

Conventional mean and std: 24.76 44.03 std mean err: 3.58

Circular median & index: 5.00 3

Set of Angles: 30 90 200 (deg)

Circular mean and std: 106.67 70.40

Conventional mean and std: 93.64 70.40 std mean err: 13.03

Circular median & index: 90.00 2

Set of Angles: 45 48 42 (deg)

Circular mean and std: 45.00 2.45

Conventional mean and std: 45.00 2.45 std mean err: 0.00

Circular median & index: 45.00 1