Bingham test of uniformity of axes on the sphere

Bingham test of uniformity of axes on the sphere

Usage

bingham_test(D, n, level = 0.95)

Source

modified after the python package apsg (v.2.0.6) by O. Lexa

Arguments

D

3x3 orientation tensor (see ortensor())

n

integer. Number of vectors

level

numeric. Confidence level (\(1-\alpha\)). Default is 0.95

Value

object of class "Test"

Details

The statistic is $$S = 15 n / 2 (\text{tr}(T^2) - 1/3)$$ with the orientation tensor \(T\) (scaled to trace 1), asymptotically chi-square with 5 degrees of freedom (MJ eq. 10.7.1-10.7.2), which gives the p-value.

The modified statistic \(S^*\) of MJ eq. 10.7.3, a polynomial correction meant for \(S\) near the critical values, is given in "extra".

\(S\) equals Bingham's (1974, eq. 5.9) \(X_U^2\) and \(n I\) with Lisle's (1985) intensity $$I = 15/2 \sum((\lambda_i - 1/3)^2)$$.

Large \(S\) indicates a preferred orientation (cluster or girdle); a very small \(S\) (lower tail) indicates data more regular than random.

See also

Other tests: fisher_ftest(), fisher_goodness_of_fit(), fisher_kent_test(), rayleigh_test3(), watson_symmetry_test(), wellner()

Examples

t <- example_lines
bingham_test(ortensor(t), n = nrow(t))
#> 
#> 	Bingham Test
#> 
#> statistic = 293.51, df = 5, p-value < 2.2e-16
#> critical value (95% confidence level) = 11.07
#> null hypothesis rejected at the 95% confidence level: yes
#> further statistics:
#>  modified 
#> -703.6408 
#> source: Mardia & Jupp (2000), eq. 10.7.1-10.7.3 
#>