rayleigh_test {tectonicr}

Rayleigh Test of Circular Uniformity

Description

Performs a Rayleigh test for uniformity of circular/directional data by assessing the significance of the mean resultant length.

Usage

rayleigh_test(x, mu = NULL, axial = TRUE)

Arguments

x

numeric vector. Values in degrees

mu

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

axial

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

Details

H_0:

angles are randomly distributed around the circle.

H_1:

angles are from unimodal distribution with unknown mean direction and mean resultant length (when mu is NULL. Alternatively (when mu is specified), angles are uniformly distributed around a specified direction.

If statistic > p.value, the null hypothesis is rejected, i.e. the length of the mean resultant differs significantly from zero, and the angles are not randomly distributed.

Value

a list with the components:

statistic

mean resultant length

p.value

significance level of the test statistic

p.value2

modified significance level (Cordeiro and Ferrari, 1991)

Note

Although the Rayleigh test is consistent against (non-uniform) von Mises alternatives, it is not consistent against alternatives with p = 0 (in particular, distributions with antipodal symmetry, i.e. axial data). Tests of non-uniformity which are consistent against all alternatives include Kuiper's test (kuiper_test()) and Watson's U^2 test (watson_test()).

References

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

Wilkie (1983): Rayleigh Test for Randomness of Circular Data. Appl. Statist. 32, No. 3, pp. 311-312

Jammalamadaka, S. Rao and Sengupta, A. (2001). Topics in Circular Statistics, Sections 3.3.3 and 3.4.1, World Scientific Press, Singapore.

See Also

mean_resultant_length(), circular_mean(), norm_chisq(), kuiper_test(), watson_test()

Examples

# Example data from Mardia and Jupp (2001), pp. 93
pidgeon_homing <- c(55, 60, 65, 95, 100, 110, 260, 275, 285, 295)
rayleigh_test(pidgeon_homing, axial = FALSE)

# Example data from Davis (1986), pp. 316
finland_stria <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
rayleigh_test(finland_stria, axial = FALSE)
rayleigh_test(finland_stria, mu = 105, axial = FALSE)

# Example data from Mardia and Jupp (2001), pp. 99
atomic_weight <- c(
  rep(0, 12), rep(3.6, 1), rep(36, 6), rep(72, 1),
  rep(108, 2), rep(169.2, 1), rep(324, 1)
)
rayleigh_test(atomic_weight, 0, axial = FALSE)

# San Andreas Fault Data:
data(san_andreas)
rayleigh_test(san_andreas$azi)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
rayleigh_test(sa.por$azi.PoR, mu = 135)