Confidence ellipse of an Eigenvector of the orientation tensor

Usage

bingham_ellipse(lambda, nu, cross_moments, n, level = 0.95, which = 1L)

Source

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

Arguments

lambda: numeric. Eigenvalues of the orientation tensor, descending. See ot_eigen() for details
nu: Corresponding Eigenvectors. Object of class Spherical. See ot_eigen() for details
cross_moments: 3x3 matrix (see axial_cross_moments())
n: integer. Sample size
level: numeric. Confidence level ($1-\alpha$). Default is 0.95
which: integer. Index of the Eigenvector of the ellipse is centered on. One of 1 (the default), 2 or 3.

Value

list

Details

Large-sample (central limit theorem) approximation: the angle of the sample eigenvector $\nu_i$ toward $\nu_j$ is approximately normal with variance $m_{ij} / (n (\lambda_i - \lambda_j)^2)$, with $m_{ij} = \text{mean}((x \cdot \nu_i)^2 (x \cdot \nu_j)^2)$. The semi-angles of the region are $$\arcsin(\sqrt(\chi^2_2(\text{level}) \times m_{ij} / (n (\lambda_i - \lambda_j)^2)))$$, 90° for equal eigenvalues.

The moments make it valid for any distribution of axes; for rotationally symmetric data it is the cone of FLE eq. 6.11-6.13. Under the Bingham model $m_{ij} / (\lambda_i - \lambda_j)^2$ tends to $1 / (2 \Delta_{ij})$, the semi-axes $ \sqrt{\big(\chi^2_2(\text{level}) / (2 n \Delta_{ij})\big)}$ of Bingham (1974, eq. 7.24).

Examples

x <- example_planes
t <- ot_eigen(x)
m <- axial_cross_moments(x, ot_eigen(x)$vectors)
r4 <- bingham_ellipse(lambda = t$values, nu = t$vectors, cross_moments = m, n = nrow(x), which = 1)
print(r4)
#> $mu
#> Vector (Vec3) object (n = 1):
#>          x          y          z 
#> -0.9257945  0.1785999  0.3331765 
#> 
#> $level
#> [1] 0.95
#> 
#> $axes
#> Vector (Vec3) object (n = 2):
#>              x          y         z
#> [1,] 0.2407711 -0.4008846 0.8839235
#> [2,] 0.2914340  0.8985508 0.3281351
#> 
#> $gamma
#> [1] 9.028139 3.794581
#> 

stereoplot()
points(x, col = "grey", pch = 16)
stereo_confidence(bingham_confidence(r4), col = "red")