CircStats.circ_axialFunction

Transforms p-axial data to a common scale.

1. α: sample of angles in radians
2. p: number of modes(default=1)

return: transformed data

CircStats.circ_clustMethod

Performs a simple agglomerative clustering of angular data.

1. α: sample of angles in radians
• k: number of clusters desired, default=2

return:

• cid: cluster id for each entry of α
• α: sorted angles, matched with cid
• μ: mean direction of angles in each cluster
CircStats.circ_confmeanMethod

Computes the confidence limits on the mean for circular data.

1. α: sample of angles in radians
• xi: (1-xi) confidence limits are computed, default=0.05
• w: number of incidences in case of binned angle data
• d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians
• dims: compute along this dimension(default=1)
return: mean ± d yields upper/lower (1-xi)% confidence limit
CircStats.circ_corrccMethod

Circular correlation coefficient for two circular random variables.

1. α₁: sample of angles in radians
2. α₂: sample of angles in radians

return:

• ρ: correlation coefficient
• p: p-value
CircStats.circ_corrclMethod

Correlation coefficient between one circular and one linear random variable.

1. α: sample of angles in radians
2. x: sample of linear random variable

return:

• ρ: correlation coefficient
• p: p-value
CircStats.circ_distMethod

Pairwise difference α-β around the circle computed efficiently.

1. α: sample of linear random variable
2. β: sample of linear random variable or one single angle

return: matrix with differences

CircStats.circ_dist2Method

All pairwise difference α-β around the circle computed efficiently.

1. α: sample of linear random variable
2. β: sample of linear random variable

return: matrix with pairwise differences

CircStats.circ_hktestMethod

Parametric two-way ANOVA for circular data with interations.

!!!note The test assumes underlying von-Mises distributrions. All groups are assumed to have a common concentration parameter k, between 0 and 2.

2. idp: indicates the level of factor 1 (1:p)
3. idq: indicates the level of factor 2 (1:q)
- inter: whether to include effect of interaction
- fn: string array containing names of the factors

return:

• p: pvalues for factors and interaction
• s: statistic of each p value
CircStats.circ_kappaMethod

Computes an approximation to the ML estimate of the concentration parameter kappa of the von Mises distribution.

1. α: angles in radians OR α is length resultant
- w: number of incidences in case of binned angle data

return: estimated value of kappa

CircStats.circ_ktestMethod

A parametric two-sample test to determine whether two concentration parameters are different.

H0: The two concentration parameters are equal. HA: The two concentration parameters are different.

Assumptions: both samples are drawn from von Mises type distributions and their joint resultant vector length should be > .7

1. α₁: sample of angles in radians
2. α₂: sample of angles in radians

return:

• p: p-value that samples have different concentrations
• f: f-statistic calculated
CircStats.circ_kuipertestMethod

The Kuiper two-sample test tests whether the two samples differ significantly. The difference can be in any property, such as mean location and dispersion. It is a circular analogue of the Kolmogorov-Smirnov test.

H0: The two distributions are identical. HA: The two distributions are different.

1. α₁: sample of angles in radians
2. α₂: sample of angles in radians
- n: resolution at which the cdf are evaluated

return:

• p: p-value; the smallest of .10, .05, .02, .01, .005, .002, .001, for which the test statistic is still higher than the respective critical value. this is due to the use of tabulated values. if p>.1, pval is set to 1.
• k: test statistic
• K: critical value
CircStats.circ_kurtosisMethod

Calculates a measure of angular kurtosis.

1. α: sample of angles in radians
- w: weightings in case of binned angle data
- dims: compute along this dimension(default=1)

return:

• k: kurtosis (from Pewsey)
• k0: kurtosis (from Fisher)
CircStats.circ_meanMethod

Computes the mean direction for circular data.

1. α: sample of angles in radians
- w: weightings in case of binned angle data
- dims: compute along this dimension(default=1)

return:
- μ: mean direction
- ul: upper 95% confidence limit
- ll: lower 95% confidence limit
CircStats.circ_medianMethod

Computes the median direction for circular data.

1. α: sample of angles in radians

return: median direction

CircStats.circ_medtestMethod

Tests for significance of the median.

H0: the population has median angle md HA: the population has not median angle md

1. α: sample of angles in radians
• md: median to test, default=0

return: p-value

CircStats.circ_momentMethod

Calculates the complex p-th centred or non-centred moment of the angular data in angle.

1. α: sample of angles in radians
• w: number of incidences in case of binned angle data
• p: p-th moment to be computed, default=1
• cent: if true, central moments are computed, default = false
• dims: compute along this dimension(default=1)

return:

• mp: complex p-th moment
• ρp: magnitude of the p-th moment
• μp: angle of th p-th moment
CircStats.circ_mtestMethod

One-Sample test for the mean angle. H0: the population has mean m. HA: the population has not mean m.

!!!note: This is the equvivalent to a one-sample t-test with specified mean direction.

1. α: sample of angles in radians
• m: assumed mean direction, default=0
• w: number of incidences in case of binned angle data
• d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians
• xi: alpha level of the test

return:

• h: false if H0 can not be rejected, true otherwise
• μ: mean
• ul: upper (1-xi) confidence level
• ll: lower (1-xi) confidence level
CircStats.circ_otestMethod

Computes Omnibus or Hodges-Ajne test for non-uniformity of circular data.

H0: the population is uniformly distributed around the circle
HA: the population is not distributed uniformly around the circle
Alternative to the Rayleigh and Rao's test. Works well for unimodal,
bimodal or multimodal data. If requirements of the Rayleigh test are met, the latter is more powerful.

1. α: sample of angles in radians
- sz: step size for evaluating distribution, default 1 degree
- w: number of incidences in case of binned angle data

return:
- p: p-value
- m: minimum number of samples falling in one half of the circle
CircStats.circ_rMethod

Computes mean resultant vector length for circular data.

1. α: sample of angles in radians
- w: number of incidences in case of binned angle data
- d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians
- dims: compute along this dimension(default=1)

return: mean resultant length
CircStats.circ_raotestMethod

Calculates Rao's spacing test by comparing distances between points on a circle to those expected from a uniform distribution. H0: Data is distributed uniformly around the circle. H1: Data is not uniformly distributed around the circle.

Alternative to the Rayleigh test and the Omnibus test. Less powerful than the Rayleigh test when the distribution is unimodal on a global scale but uniform locally.

Due to the complexity of the distributioin of the test statistic, we resort to the tables published by Russell, Gerald S. and Levitin, Daniel J.(1995) An expanded table of probability values for rao's spacing test, Communications in Statistics - Simulation and Computation Therefore the reported p-value is the smallest α level at which the test would still be significant. If the test is not significant at the α=0.1 level, we return the critical value for α = 0.05 and p = 0.5.

1. α: sample of angles in radians

return:

• p: smallest p-value at which test would be significant
• u: computed value of the test-statistic u
• UC: critical value of the test statistic at sig-level
CircStats.circ_rtestMethod

Computes Rayleigh test for non-uniformity of circular data.

H0: the population is uniformly distributed around the circle

HA: the populatoin is not distributed uniformly around the circle Assumption: the distribution has maximally one mode and the data is sampled from a von Mises distribution!

1. α: sample of angles in radians
- w: number of incidences in case of binned angle data
- d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians

return:
- p: p-value of Rayleigh's test
- z: value of the z-statistic
CircStats.circ_samplecdfMethod

Helper function for circ_kuipertest. Evaluates CDF of sample.

1. α: sample of angles in radians
- n: resolution at which the cdf are evaluated

return:

• ϕ: angles at which CDF are evaluated
• c: CDF values at ϕ
CircStats.circ_skewnessMethod

Calculates a measure of angular skewness.

1. α: sample of angles in radians
- w: weightings in case of binned angle data
- dims: compute along this dimension(default=1)

return:
- b: skewness (from Pewsey)
- b0: alternative skewness measure (from Fisher)
CircStats.circ_statsMethod

Computes descriptive statistics for circular data.

1. α: sample of angles in radians
- w: weightings in case of binned angle data
- d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians

return: descriptive statistics

CircStats.circ_stdMethod

Computes circular standard deviation for circular data (equ. 26.20, Zar).

1. α: sample of angles in radians
• w: weightings in case of binned angle data
• d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians
• dims: compute along this dimension(default=1)
return:
• s: angular deviation
• s0: circular standard deviation
CircStats.circ_symtestMethod

Tests for symmetry about the median. H0: the population is symmetrical around the median HA: the population is not symmetrical around the median

1. α: sample of angles in radians

return: p-value

CircStats.circ_varMethod

Computes circular variance for circular data (equ. 26.17/18, Zar).

1. α: sample of angles in radians
• w: number of incidences in case of binned angle data
• d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians
- dims: compute along this dimension(default=1)

return:
- S: circular variance 1-r
- s: angular variance 2(1-r)
CircStats.circ_vtestMethod

Computes V test for non-uniformity of circular data with a specified mean direction.

H0: the population is uniformly distributed around the circle HA: the population is not distributed uniformly around the circle but has a mean of m.

!!!note Not rejecting H0 may mean that the population is uniformly distributed around the circle OR that it has a mode but that this mode is not centered at m.

The V test has more power than the Rayleigh test and is preferred if there is reason to believe in a specific mean direction.

1. α: sample of angles in radians
• m: suspected mean direction, default=0
• w: number of incidences in case of binned angle data
• d: spacing of bin centers for binned data, used to correct for bias in estimation of r, in radians

return:

• p: p-value of V test
• v: value of the V statistic
CircStats.circ_wwtestMethod

Parametric Watson-Williams multi-sample test for equal means. Can be used as a one-way ANOVA test for circular data.

H0: the s populations have equal means HA: the s populations have unequal means

!!! note Use with binned data is only advisable if binning is finer than 10 deg. In this case, α is assumed to correspond to bin centers.

The Watson-Williams two-sample test assumes underlying von-Mises distributrions. All groups are assumed to have a common concentration parameter k.