The StatsCircularMoments operation can be used as the WaveStats operation for circular data. You can also use the operation to perform uniformity tests or to convert periodic data of various formats to waves containing angles in radians.

### Data Conversion Examples

Consider the date data in the wave dateData1 can be converted into an angle wave (in radians) using the StatsCircularMoments operation with /MODE=5. To run the operation execute the command:

StatsCircularMoments/T=1/Q /SAW/Mode=5/KUPR/RAYL dateData1
 dateData1 W_AngleWave 12/25/04 20:58:45 5.06708 1/21/05 08:31:08 4.25635 1/2/05 07:53:44 0.405367 12/18/04 14:01:59 3.6483 1/28/05 01:50:50 5.67514 1/14/05 19:02:46 2.83757 1/18/05 21:32:49 3.6483 1/19/05 13:36:46 3.85098 1/8/05 00:05:24 1.62147 12/9/04 13:05:50 1.82415 12/26/04 15:18:29 5.26977 12/17/04 07:26:29 3.44562 12/4/04 00:37:08 0.810734 12/28/04 02:32:47 5.67514 1/13/05 08:02:51 2.63488 12/18/04 19:17:53 3.6483

Besides converting the date into circular data, the operation also performed the Kuiper and Rayleigh tests for uniformity. The results are stored in the wave W_CircularStats. The first part of the table represents various measures of circular statistics:

 number_of_points 16 number_of_NaNs 0 C -2.7804 S -2.36277 R 3.64874 cBar -0.173775 sBar -0.147673 rBar 0.228046 tBar 3.84597 V 0.771954 v 1.71942 median 3.6483 mean_deviation 1.22877 Circular_Dispersion 8.75803 Skewness -0.0902807 Kurtosis 0.10402

The second part of the table contains the results of the Kuiper and Rayleigh tests:

 Kuiper_V 1.13873 Kuiper_Critical 1.74726 Kuiper_P_Value 0.627286 Rayleigh_Z 0.832081 Rayleigh_CriticalZ 2.94819 Rayleigh_significance_prob 0.441939

Both tests indicate that H0 (uniform distribution) hypothesis can't be rejected.

### Axial data

Here the input is cyclical data as shown below:

To run the tests execute the following command:

StatsCircularMoments/mode=2/KUPR/RAYL/T=1/Q/AXD=2   axddata

The operation multiplies the data in axdData a factor of 2 (/AXD=2) and handles the input as if it contained angles in degrees (/mode=2).

The results in the "Circular Stats" table are:

 number_of_points 60 number_of_NaNs 0 C 10.9331 S 9.09466 R 14.2213 cBar 0.182218 sBar 0.151578 rBar 0.237022 tBar 0.693861 V 0.762978 v 1.69682 median 0.74489 mean_deviation 1.277 Circular_Dispersion 9.08257 Skewness 0.030089 Kurtosis 0.00192745 Confidence_d 0.848389 Kuiper_V 1.58639 Kuiper_Critical 1.74726 Kuiper_P_Value 0.118182 Rayleigh_Z 3.37076 Rayleigh_CriticalZ 2.98324 Rayleigh_significance_prob 0.0336911

These results are rather interesting in that the Kuiper test does not reject uniformity while the Rayleigh test does.

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