rayleigh_test: 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, quiet = FALSE)

Value

a list with the components:

R or C: mean resultant length or the dispersion (if mu is specified). Small values of R (large values of C) will reject uniformity. Negative values of C indicate that vectors point in opposite directions (also lead to rejection).
statistic: test statistic
p.value: significance level of the test statistic

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. $π$ -periodical (TRUE, the default) or directional, i.e. $2 π$ -periodical (FALSE).
quiet: logical. Prints the test's decision.

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.

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.

Examples

Run this code

# 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)

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