## Wheeler Watson Test

The Wheeler-Watson tests the hypothesis H0: the samples came from the same population.

1. Testing two waves

The following waves represent angles in radians. Both are centered about pi/6.

 data1 data2 0.553951 0.572457 0.556496 0.516971 0.558007 0.619459 0.493574 0.580837 0.551672 0.585822 0.518077 0.534662 0.517558 0.563266 0.532961 0.49541 0.499501 0.489289 0.564656 0.628698 0.511045 0.555638 0.521595 0.46913 0.486378 0.523104 0.532568 0.492851 0.531593 0.425477 0.57336 0.489757 0.509024 0.544693

To test the hypothesis H0, execute:

`StatsWheelerWatsonTest/T=1/Q data1,data2`

The results are displayed in the Wheeler-Watson Test table:

 totalPoints 34 numSamples 2 W 7.10517 Critical 5.99146

Since the test statistic W is greater than the critical value, H0 is rejected.

2. Testing more than two waves (Mardia's method)

The additional sample is in the wave data3:

 data3 0.583937 0.492531 0.44756 0.528031 0.549251 0.609156 0.475164 0.502774 0.523131 0.635587 0.610486 0.489298 0.54757 0.506267 0.5838 0.419416 0.616708

Again, H0 corresponds to the samples being from the same distribution. To run the test execute:

`StatsWheelerWatsonTest/T=1/Q data1,data2,data3`

The results are displayed in the Wheeler-Watson Test table:

 totalPoints 51 numSamples 3 W 9.16895 Critical 9.48773

This time the test statistic is smaller than the critical value and H0 can't be rejected. This result may be explained, in part, by the fact that the wave data3 was constructed intentionally to have circular dispersion between that of data1 and data2. This can be seen by comparing the results of StatsCircularMoments for the three waves:

 number_of_points 15 19 17 number_of_NaNs 0 0 0 C 12.9487 16.3212 14.5841 S 7.56325 9.67855 8.67316 R 14.9957 18.9751 16.9682 cBar 0.863245 0.859011 0.857889 sBar 0.504217 0.509397 0.510186 rBar 0.999713 0.998692 0.99813 tBar 0.528642 0.53526 0.536513 V 0.000286899 0.00130836 0.00187004 v 0.0239558 0.0511707 0.0611848 median 0.531594 0.534664 0.528032 mean_deviation 0.0201722 0.042801 0.0517856 Circular_Dispersion 0.000573973 0.00261935 0.00374701 Skewness 0.404109 0.214704 0.22167 Kurtosis -2.24571 -1.05747 -1.95149 Confidence_d 0.0129445 0.0242404 0.0308417