# Clustering the elliptic Fourier series coefficients

What do you think about the best way to perform a search for two discrete group membership in data that consist of four Fourier series coefficients like this:

$an [1] -1.099759e+02 -3.277411e+01 -2.069982e+01 1.294945e+00 -4.557681e+00 [6] 3.659611e+00 -2.266350e+00 2.290051e+00 -1.165922e+00 4.290447e-01 [11] -5.157878e-01 -4.373163e-02 -9.627400e-02 -2.160347e-01 -8.664855e-02 [16] -7.125669e-02 -1.661091e-01 5.034174e-02 -5.773743e-02 9.070195e-02 [21] -1.103436e-01 4.968477e-02 -3.843040e-02 -3.539704e-02 -2.001452e-03 [26] -6.103696e-02 5.080494e-03 -8.438730e-02 9.728283e-02 -1.201637e-01 [31] 5.214383e-03 -3.097534e-02$bn
[1] -9.950280480 21.060718135  1.749972609  3.125969744 -1.221718994
[6] -0.366008050  0.134939868 -1.433288279  1.286034494 -0.793100554
[11]  1.024338370 -0.645426109  0.752002518 -0.354023866  0.371995040
[16] -0.079819647  0.093705335 -0.085753179  0.062387444 -0.007351952
[21]  0.112815714 -0.071472761  0.180375050 -0.155871666  0.165280059
[26] -0.084238927  0.157971637 -0.115017123  0.058533444  0.021303459
[31] -0.009807206 -0.037191448

$cn [1] 186.42631186 -48.82963771 -13.05530543 -17.63800716 -8.78793762 [6] -5.13169720 -4.05998782 -1.53832276 -1.92963513 -1.04525653 [11] -0.80438591 -0.98701575 -0.24698734 -1.00053803 -0.21660353 [16] -0.85765069 -0.32822392 -0.54232592 -0.39971080 -0.22425141 [21] -0.46861898 -0.05909403 -0.45459051 -0.03218630 -0.39596045 [26] -0.06020157 -0.25675128 -0.08580023 -0.16002496 -0.14539536 [31] -0.13190927 -0.12939389$dn
[1] -60.193804743  17.089146986  -5.359459446   9.834089998  -1.841459861
[6]   2.894124403  -1.229890034  -0.060334681  -0.383461709  -0.414984724
[11]   0.161235406  -0.209664090   0.133169983   0.114066031  -0.113437218
[16]   0.326989075  -0.353357830   0.366442458  -0.379583013   0.327422310
[21]  -0.253858236   0.177601217  -0.120553856   0.010128884   0.009787827
[26]  -0.048527754   0.075992199  -0.090581996   0.043151283  -0.050768870
[31]   0.022838609   0.004311564


when I have 26 of such sets (and choosing the first 8 of each coefficient excluding the first one). These coefficients describe shape in mountain goat horns, and I need the best way to separate them according to sex which is accounting for horn curvature and other shape features. Now, PCA or LDA separate the extreme shapes well but what about the intermediate ones?

I tried numerous partitional clustering approaches but found no consistent grouping pattern with them. Also the majority of parametric statistic methods do not work because of high degree of correlation and non-normality of such data. Any advice?

-
 Two questions: 1. Why are you using Fourier series to represent your data? 2. You write that you "need the best way to separate them", which sounds like a discrimination problem (en.wikipedia.org/wiki/Linear_discriminant_analysis) rather than a clustering problem. Do you already know the sex of the goats from which the horns came? – MånsT Apr 3 '12 at 8:26 That is exactly the goal...to find mathematical way to separate two goat horns according to sex because both sexes have them and I don`t know in advance which one is it. Fourier series is commonly used for shape approximation of the complex outlines. LDA do sepatate them, but I can envision only extremes and not the intermediate shapes. – Ian Stuart Apr 3 '12 at 9:46