Statistical estimators of the distribution of a set of vectors

Usage

sph_mean(x, na.rm = TRUE, ...)

sph_sd(x, ...)

sph_var(x, ...)

sph_confidence_angle(x, w = NULL, alpha = 0.05, na.rm = TRUE)

rdegree(x, w = NULL, na.rm = FALSE)

sd_error(x, w = NULL, na.rm = FALSE)

delta(x, w = NULL, na.rm = TRUE)

estimate_k(x, w = NULL, na.rm = FALSE)

Arguments

x: object of class "Vec3", "Line", "Ray", or "Plane", where the rows are the observations and the columns are the coordinates.
na.rm: logical. Whether NA values should be removed before the computation proceeds.
...: arguments passed to function call
w: numeric. Optional weights for each observation.
alpha: numeric. Significance level for the confidence angle (default is 0.05 for a 95% confidence angle).

Details

These statistical estimators are based on the resultant vector of a set of $n$ vectors $x_1, \ldots, x_n$ (Mardia 1972). The resultant vector is given by $$\bar{\mathbf{x}} = \sum_{i=1}^{n} \mathbf{x}_i$$

The mean resultant is defined as $$\bar{\mathbf{R}} = ||\bar{\mathbf{x}}|| = \sqrt{x_x^2 + x_y^2 + x_z^2}$$

sph_mean returns the spherical mean of a set of vectors (object of class of x), that is the maximum likelihood estimate of the mean direction given by the arithmetic mean of the vector components normalized by the mean resultant vector: $$\mu = \frac{\bar{\mathbf{x}}}{\bar{\mathbf{R}}}$$

sph_var returns the spherical variance (numeric). $$S = 1 - \bar{\mathbf{R}}$$

sph_sd returns the spherical standard deviation (numeric) given as the half apical angle of a cone about the mean vector. In degrees if x is a "Plane" or "Line", or in radians if otherwise. $$s = \sqrt{\log(1 / \bar{\mathbf{R}}^2))}$$

delta returns the half apical angle of the cone containing ~63% of the data (in degrees if x is a "Plane" or "Line", or in radians if otherwise). For enough large sample it approaches the angular standard deviation ("csd") of the Fisher statistics. $$\delta = \arccos(\bar{\mathbf{R}})$$

rdegree returns the degree of preferred orientation of vectors, range: (0, 1). $$r = \frac{2 \bar{\mathbf{R}} - n}{n}$$

sd_error returns the spherical standard error (numeric). If the number of data is less than 25, if will print a additional message, that the output value might not be a good estimator. $$\text{SDE} = \sqrt{\frac{1 - \frac{1}{n} \sum_{i=1}^{n} (\mu \cdot x_i)^2}{n \bar{\mathbf{R}}^2}}$$

sph_confidence_angle returns the half-apical angle $q$ of a cone about the mean $\mu$ (in degrees if x is a "Plane" or "Line", or in radians if otherwise). The $100(1-\alpha)\%$ confidence interval is than given by $\mu \pm q$. $$q = \arcsin(\sqrt{-\log(\alpha)} \cdot \text{SDE})$$

estimate_k returns the estimated concentration of the von Mises-Fisher distribution $\hat{\kappa}$ (after Sra, 2011). $$\hat{\kappa} = \frac{\bar{R}(p - \bar{R}^2)}{1 - \bar{R}^2}$$ where $p$ is the dimension of the data (3 for spherical data).

References

Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. John Wiley & Sons Ltd. ISBN 978-0-471-95333-3.

Sra, S. A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I s (x). Comput Stat 27, 177–190 (2012). https://doi.org/10.1007/s00180-011-0232-x

Examples

set.seed(20250411)
x <- rvmf(100, mu = Line(120, 50), k = 5)
sph_mean(x)
#> Line object (n = 1):
#>   azimuth    plunge 
#> 115.21007  52.67744 
sph_sd(x)
#> [1] 40.82336
sph_var(x)
#> [1] 0.224176
delta(x)
#> [1] 39.1202
rdegree(x)
#> [1] -0.9844835
sd_error(x)
#> [1] 4.01451
sph_confidence_angle(x)
#> [1] 6.965534
estimate_k(x)
#> [1] 4.673486

#' weights:
x2 <- Line(c(0, 0), c(0, 90))
sph_mean(x2)
#> Line object (n = 1):
#> azimuth  plunge 
#>       0      45 
sph_mean(x2, w = c(1, 2))
#> Line object (n = 1):
#>  azimuth   plunge 
#>  0.00000 63.43495 
sph_var(x2)
#> [1] 0.2928932
sph_var(x2, w = c(1, 2))
#> [1] 0.254644