Kuiper Test of Circular Uniformity

A statistical test used to determine whether a set of angular or circular data points (such as times of day, compass directions, or degrees) are spread out evenly around a circle or if they cluster in some way.

Usage

kuiper_test(x, alpha = 0, axial = TRUE, quiet = FALSE)

Arguments

x

numeric vector. Values in degrees

alpha

Significance level of the test. Valid levels are 0.01, 0.05, and 0.1. This argument may be omitted (NULL, the default), in which case, a range for the p-value will be returned.

axial

logical. Whether the data are axial, i.e. \(\pi\)-periodical (TRUE, the default) or directional, i.e. \(2 \pi\)-periodical (FALSE). In case of axial data, the angles will be doubled for the test.

quiet

logical. Prints the test's decision.

Value

list containing the test statistic statistic and the significance level p.value.

Details

The Null Hypothesis (\(H_0\)): The data are distributed completely uniformly (randomly and evenly) around the circle.

The Alternative Hypothesis (\(H_1\)): The data are not uniform and show a preference, clustering, or pattern somewhere on the circle.

The Test Statistic (V or \(D^{+} + D^{-}\)): It measures the greatest positive and negative differences between your data's empirical cumulative distribution and a theoretical uniform distribution.

Interpreting the Results

• High Test Statistic / Low p-value (\(p < \alpha\), typically 0.05): You reject the null hypothesis. This means your data are not uniform; they have a significant preferred direction, grouping, or non-random pattern on the circle.

• Low Test Statistic / High p-value (\(p \ge 0.05\)): You fail to reject the null hypothesis. There is no strong evidence to say the data are different from a flat, uniform distribution. The points appear random across the circle.

Note

Kuiper's test statistic is a rotation-invariant Kolmogorov-type test statistic. The critical values of a modified Kuiper's test statistic are used according to the tabulation given in Stephens (1970).

See also

Other Tests: ar_test(), norm_chisq(), rayleigh-test, watson_test(), watson_two_sample, watson_wheeler_test_perm(), weighted-rayleigh-test

Examples

# Example data from Mardia and Jupp (1999), pp. 93
kuiper_test(homing, alpha = .05)
#> Reject Null Hypothesis
#> $statistic
#> [1] 2.262115
#> 
#> $p.value
#> [1] 1.747
#> 

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
kuiper_test(sa.por$azi.PoR, alpha = .05)
#> Reject Null Hypothesis
#> $statistic
#> [1] 16.60463
#> 
#> $p.value
#> [1] 1.747
#>