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.

data1data2
0.5539510.572457
0.5564960.516971
0.5580070.619459
0.4935740.580837
0.5516720.585822
0.5180770.534662
0.5175580.563266
0.5329610.49541
0.4995010.489289
0.5646560.628698
0.5110450.555638
0.5215950.46913
0.4863780.523104
0.5325680.492851
0.5315930.425477
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:

totalPoints34
numSamples2
W7.10517
Critical5.99146

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:

totalPoints51
numSamples3
W9.16895
Critical9.48773

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_points151917
number_of_NaNs000
C12.948716.321214.5841
S7.563259.678558.67316
R14.995718.975116.9682
cBar0.8632450.8590110.857889
sBar0.5042170.5093970.510186
rBar0.9997130.9986920.99813
tBar0.5286420.535260.536513
V0.0002868990.001308360.00187004
v0.02395580.05117070.0611848
median0.5315940.5346640.528032
mean_deviation0.02017220.0428010.0517856
Circular_Dispersion0.0005739730.002619350.00374701
Skewness0.4041090.2147040.22167
Kurtosis-2.24571-1.05747-1.95149
Confidence_d0.01294450.02424040.0308417