weighted-rayleigh-test: Weighted Goodness-of-fit Test for Circular Data

Description

Weighted version of the Rayleigh test (or V0-test) for uniformity against a distribution with a priori expected von Mises concentration. weighted_rayleigh_perm() estimates p-values by Monte Carlo simulation of uniform samples with the same weights.

Usage

weighted_rayleigh(
  x,
  mu = NULL,
  w = NULL,
  axial = TRUE,
  alpha = 0.05,
  quiet = FALSE
)
weighted_rayleigh_perm(x, mu = NULL, w = NULL, axial = TRUE, n_perm = 1000L)
v0_test(x, mu, w = NULL, axial = TRUE, n_perm = 1000L)

Value

a list with the components:

R or C: weighted mean resultant length or, if mu is specified, the weighted mean cosine of the deviations from mu. Large values reject uniformity.
statistic: Test statistic
p.value: significance level of the test statistic
n_eff: effective sample size

weighted_rayleigh_perm() returns a list of class "htest".

Arguments

x: numeric vector. Values in degrees
mu: (optional) The specified or known mean direction (in degrees) in alternative hypothesis.
w: numeric vector weights of length length(x). If NULL, the non-weighted Rayleigh test is performed.
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

The Null hypothesis is uniformity (randomness). The alternative is a unimodal distribution with the specified mean direction mu (weighted V-test), or with an unknown mean direction if mu is NULL (weighted Rayleigh test). The null hypothesis is rejected if p.value < alpha.

Under the null hypothesis the weighted mean cosine has variance \(\sum w_i^2 / (2 (\sum w_i)^2)\), so the asymptotic tests use the effective sample size \(n_\text{eff} = (\sum w_i)^2 / \sum w_i^2\), which equals \(n\) for equal weights.

Examples

Run this code

# Load data
data("cpm_models")
data(san_andreas)
PoR <- equivalent_rotation(cpm_models[["NNR-MORVEL56"]], "na", "pa")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
data("iceland")
PoR.ice <- equivalent_rotation(cpm_models[["NNR-MORVEL56"]], "eu", "na")
ice.por <- PoR_shmax(iceland, PoR.ice, "out")
data("tibet")
PoR.tib <- equivalent_rotation(cpm_models[["NNR-MORVEL56"]], "eu", "in")
tibet.por <- PoR_shmax(tibet, PoR.tib, "in")

# GOF test:
weighted_rayleigh(tibet.por$azi.PoR, mu = 90, w = 1 / tibet$unc)
weighted_rayleigh(ice.por$azi.PoR, mu = 0, w = 1 / iceland$unc)
weighted_rayleigh(sa.por$azi.PoR, mu = 135, w = 1 / san_andreas$unc)

weighted_rayleigh_perm(tibet.por$azi.PoR, mu = 90, w = 1 / tibet$unc)
weighted_rayleigh_perm(ice.por$azi.PoR, mu = 0, w = 1 / iceland$unc)
weighted_rayleigh_perm(sa.por$azi.PoR, mu = 135, w = 1 / san_andreas$unc)