Watson's \(U^2\) Test of Circular Uniformity

Watson's test for circular random distribution.

Usage

watson_test(
  x,
  alpha = 0,
  dist = c("uniform", "vonmises"),
  axial = TRUE,
  mu = NULL
)

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.

dist

Distribution to test for. The default, "uniform", is the uniform distribution. "vonmises" tests the von Mises distribution.

axial

logical. Whether the data are axial, i.e. \(\pi\)-periodical (TRUE, the default) or circular, i.e. \(2 \pi\)-periodical (FALSE).

mu

(optional) The specified or known mean direction (in degrees) in alternative hypothesis

Value

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

Details

Watson's test statistic is a rotation-invariant Cramer - von Mises test statistic. If statistic > p.value, the null hypothesis is rejected. If not, randomness (uniform distribution) cannot be excluded.

References

Mardia and Jupp (2000). Directional Statistics. John Wiley and Sons.

Examples

# Example data from Mardia and Jupp (2001), pp. 93
pidgeon_homing <- c(55, 60, 65, 95, 100, 110, 260, 275, 285, 295)
watson_test(pidgeon_homing, alpha = .05)
#> Do Not Reject Null Hypothesis
#> $statistic
#> [1] 0.1153633
#> 
#> $p.value
#> [1] 0.187
#> 

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
watson_test(sa.por$azi.PoR, alpha = .05)
#> Reject Null Hypothesis
#> $statistic
#> [1] 50.38975
#> 
#> $p.value
#> [1] 0.187
#> 
watson_test(sa.por$azi.PoR, alpha = .05, dist = "vonmises")
#> Reject Null Hypothesis
#> $statistic
#> [1] 5.219359
#> 
#> $p.value
#> [1] 0.113
#>