rayleigh-test: Rayleigh Test of Circular Uniformity

Description

A test to determine whether a sample of circular or directional data is evenly spread out or clustered around a single specific direction. The test assesses the significance of the mean resultant length. rayleigh_test_perm() estimates p-values by Monte Carlo simulation of uniform samples.

Usage

rayleigh_test(x, mu = NULL, axial = TRUE, alpha = 0.05, quiet = FALSE)
rayleigh_test_perm(x, mu = NULL, axial = TRUE, n_perm = 1000L)

Value

a list with the components:

R or C: mean resultant length or, if mu is specified, the mean cosine of the deviations from mu. Large values of R (large positive values of C) reject uniformity. Negative values of C indicate that the vectors point away from mu; the one-sided V-test does not reject them.
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. \(\pi\)-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.
alpha: Significance level of the test ranging between 0 and 1.
quiet: logical. Prints the test's decision.
n_perm: integer. Number of simulated uniform samples.

Details

Hypotheses

Null Hypothesis \(H_0\): The population is distributed uniformly (randomly) around the circle with no preferred direction.

Alternative Hypothesis \(H_1\): The population is not uniform and has a unimodal (single-peaked) concentration in a preferred direction. When mu is specified (V-test), the alternative is a unimodal distribution centred on the specified direction; the test is one-sided.

Mean Resultant Length (\(\bar{R}\) or R): A value between 0 and 1 that measures how concentrated the data points are.

\(\bar{R}\) = 0: The data is completely spread out around the circle
\(\bar{R}\) = 1: All data points point in the exact same direction.

p-value The probability of seeing data this clustered purely by chance under the assumption of uniformity.

Interpretation

Small p-value (p < 0.05): Reject the null hypothesis. The length of the mean resultant differs significantly from zero, and the angles are not randomly distributed. You have strong evidence that the data points share a significant preferred or mean direction (unimodal clustering).
Large p-value (p \(\ge\) 0.05): Fail to reject the null hypothesis. There is not enough evidence to claim a preferred direction, meaning the data looks random or uniform around the circle.

References

Fisher, N. I. (1993) Statistical Analysis of Circular Data, Cambridge University Press.

Examples

Run this code

# Example data from Mardia and Jupp (1999), pp. 93
rayleigh_test(homing, axial = FALSE) # Do not reject null hypothesis.
# R = 0.22; stat = 0.497, p = 0.62
rayleigh_test_perm(homing, axial = FALSE)

# Example data from Davis (1986), pp. 316
rayleigh_test(striae, axial = FALSE) # reject null hypothesis
rayleigh_test_perm(striae, axial = FALSE) # reject null hypothesis

rayleigh_test(striae, mu = 105, axial = FALSE) # reject null hypothesis
rayleigh_test_perm(striae, mu = 105, axial = FALSE) # reject null hypothesis

# Example data from Mardia and Jupp (1999), 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) # reject null hypothesis

# San Andreas Fault Data:
data(san_andreas)
rayleigh_test(san_andreas$azi) # reject null hypothesis
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
rayleigh_test(sa.por$azi.PoR, mu = 135) # reject null hypothesis
rayleigh_test_perm(sa.por$azi.PoR, mu = 135, n_perm = 1e3) # reject null hypothesis

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