circular statistics with weight?

#1
Hi all,

I have a dataset that describes groups of 2-D linear features. Each group is a collection of angles (orientations) and lengths describing segments or linear features. The data is axial - meaning that 90 degrees and 270 degrees are the same for the purpose of the analysis.

I want to calculate a sort of mean that would reflect not only directions but also length, so that longer segments will have more weight. I can use regular circular statistics but I can't see any way to incorporate the length since the calculation of the mean involves sines and cosines (the length is in meters and can be several thousands of meters per segment).

Here is an example of my data:
orientation length
0.0000 787.5000
0.1041 567.7509
6.2346 183.5858
9.4623 307.9399
25.8636 1527.5020
39.8056 790.7878
45.0000 429.5674
56.6532 1592.2180
86.6252 535.1076
92.6115 1530.8080
97.9916 498.1029
99.0622 979.3255
104.036 695.7741

Any ideas?

p.s. I also need the standard deviation.

Thank you,
Neta
 
#2
As the data are now, you have a vector whose 2 elements are length and direction. Have you tried converting these vector to those whose 2 elements are lengths in orthogonal directions? For example, if one of your measurements is, say:

45 degrees, with length 2

this is equivalent to:

sqrt(2) units of length in the direction of 0 degrees and sqrt(2) units of length in the direction of 90 degrees.

This type of vector may be plotted in regular Cartesian (X, Y) space. It might also be modeled as a multivariate normal, allowing for statistical inferences. If you wanted to weight weight a mean based on the length of this vector, you could perform a weighted sum of the corresponding unit vectors, where the weight is the proportion of the vectors length to the total of all vector lengths.

~Matt