## Angular-Angular Correlation Tests

This example contains parametric and non-parametric tests with correlated and uncorrelated inputs.

### Angle-Angle Correlation

The following data sets contain angle measurements in radians:

data1 | data2 |

0 | 0.558505 |

0.174533 | 0.296706 |

0.261799 | 0.959931 |

0.349066 | 2.23402 |

0.436332 | 6.00393 |

The first test is the parametric angle-angle correlation. To run the test execute the command:

StatsCircularCorrelationTest/T=1/Q/PAA data1,data2

The results appear in the table "Circular Correlation Test":

r_{aa} | 0.0145224 |

avg | 0.00473055 |

variance | 0.00391019 |

L1 | -0.00112052 |

L2 | 0.1085 |

Here r_{aa} is the computed correlation coefficient, avg is the average correlation coefficient and variance
is the variance of the correlation coefficient computed for all N combinations when eliminating a single
pair of data. L1 and L2 provide the confidence interval at the specified significance (which in this
case is the default 0.05). If the confidence interval includes zero, as is the case above then H_{0}: there
is no relationship between the two waves can't be rejected.

### Nonparametric Test with uncorrelated inputs

To run the test execute the following command:

StatsCircularCorrelationTest/T=1/Q/NAA data1,data2

The results appear in the table "Circular Correlation Test":

N | 5 |

r_{p} | 0.523607 |

r_{pp} | 0.0763932 |

Statistic | 1.78885 |

alpha1 | 1.796 |

alpha2 | 4.004 |

Here r_{p} and r_{pp} are r' and r'' respectively of the Fisher and Lee formulation. The statistic is
(n-1)(r_{p}-r_{pp}) which is compared to one of the two critical values: alpha1 for one tail hypothesis
and alpha2 for a two tail hypothesis. In this case the test agrees with the results of the parametric
test above since the statistic is smaller than the critical value so the hypothesis of zero correlation
can't be rejected.

### Parameteric Test with correlated input

This example illustrates the result of a parametric test when there exists correlation between the input waves. To run the test execute the following commands:

Duplicate/O data2,data3

data3=data2+gnoise(0.2)

StatsCircularCorrelationTest/T=1/Q/PAA data2,data3

The results of the parametric test are:

r_{aa} | 0.887882 |

avg | 0.88599 |

variance | 0.00199459 |

L1 | 0.856304 |

L2 | 0.934596 |

It is clear from these values that there exists correlation between the two waves and since the values
of L1 and L2 are not on both sides of zero, H_{0} of zero correlation must be rejected.

### Nonparametric Test with Correlated Inputs

The fourth example consists of nonparametric test and correlated inputs (larger number of samples). To run the test execute the commands:

Make/O/D data4={0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55}

Duplicate/O data4,data5

data5+=enoise(0.05)

StatsCircularCorrelationTest/T=1/Q/NAA data4,data5

The results are:

N | 10 |

r_{p} | 0.925066 |

r_{pp} | 0.00145898 |

Statistic | 8.31246 |

alpha1 | 2.5 |

alpha2 | 3.19336 |

Here the test statistic is greater than the critical value (alpha2) so H_{0}: of zero correlation must be rejected.