The Fréchet (geodesic $L^2$) variance — geodesic-var • structr

Dispersion measured using the Fréchet variance, i.e the sum of the squared geodesic distances between all vectors and a specified vector.

Usage

geodesic_var(x, ...)

# S3 method for class 'Vec3'
geodesic_var(x, y = NULL, ...)

# S3 method for class 'Line'
geodesic_var(x, y = NULL, ...)

# S3 method for class 'Plane'
geodesic_var(x, y = NULL, ...)

# S3 method for class 'Ray'
geodesic_var(x, y = NULL, ...)

# S3 method for class 'Pair'
geodesic_var(x, y = NULL, group = NULL, ...)

Source

geologyGeometry (J.R. Davis): http://www.joshuadavis.us/software/

Arguments

x: object of class "Vec3", "Line", "Ray", "Plane", "Pair", or "Fault", where the rows are the observations and the columns are the coordinates.
...: parameters passed to geodesic_meanvariance_ray() (if x is a Ray), geodesic_meanvariance_line() (if x is a Vec3, Line or Plane) or geodesic_mean_pair() (if x is a Pair or a Fault).
y: Only for variance. object of class "Vec3", "Line", "Ray", "Plane", "Pair", or "Fault" about which the Fréchet variance should be calculated for. If NULL (the default), Fréchet variance about the Fréchet mean.
group: Symmetry group of x. See symmetry_group() for details If NULL, the group will be automatically picked based on the class of x.

Value

the Fréchet variance as a numeric number. Because distances in SO(3) never exceed $\pi$, the maximum possible variance is $\frac{\pi^2}{2} \approx 4.93$.

Details

The variance of a dataset ${x_1, \ldots, x_n}$ about a vector $y$ is defined as $$ \Psi(x) = \frac{1}{2n} \sum_{i=1}^n d_G(y, x_i)^2$$ where $d_G(x, y)$ is the geodesic distance between vectors $x$ and $y$ (see angle()).

References

Davis, J. R., & Titus, S. J. (2017). Modern methods of analysis for three-dimensional orientational data. Journal of Structural Geology, 96, 65–89. doi:10.1016/j.jsg.2017.01.002

Examples

set.seed(20250411)
geodesic_var(example_planes, example_planes[1, ])
#> [1] 0.559469
geodesic_var(example_planes)
#> [1] 0.2656372

The Fréchet (geodesic \(L^2\)) variance