One-sample inference about the mean of the Watson distribution.

Assumes large concentration — either kappa >> 0 or kappa << 0. From Mardia and Jupp (2000, Section 10.7.3).

Usage

watson_inference(x, alpha, shape)

# S3 method for class 'Vec3'
watson_inference(x, alpha = 0.05, shape = NULL)

# S3 method for class 'Line'
watson_inference(x, alpha = 0.05, shape = NULL)

# S3 method for class 'Plane'
watson_inference(x, alpha = 0.05, shape = NULL)

Arguments

x

object of class "Vec3", "Line", or "Plane", where the rows are the observations and the columns are the coordinates.

alpha

A real number, between 0 and 1. The significance level for the confidence region.

shape

NULL or character, either 'bipolar' or 'girdle'. If NULL, then this function chooses automatically.

Value

A list with members $shape, $tBar, $rhs, $pvalue.

shape

is either 'bipolar' or 'girdle'. If 'bipolar', then the confidence region consists of all lines u such that u^T %*% $tBar %*% u > $rhs. If 'girdle', then the confidence region consists of all lines u such that u^T %*% $tBar %*% u < $rhs.

tBar

orientation tesor

rhs

pvalue

is an R function that takes as input a line u0 and produces as output a real number in [0, 1] — the p-value for the null hypothesis that the Watson mean is u0.

See also

rwatson() for simulating a Watson distribution, and watson_MLE() to estimate distribution parameters.

Other distribution-inference: bingham-inference, fisher-inference

Examples

r <- watson_inference(example_lines)
print(r)
#> $shape
#> [1] "bipolar"
#> 
#> $tBar
#> Orientation tensor
#>            [,1]      [,2]       [,3]
#> [1,] 0.16204564 0.2552464 0.08096582
#> [2,] 0.25524643 0.7382838 0.19306540
#> [3,] 0.08096582 0.1930654 0.09967054
#> 
#> $rhs
#> [1] 0.8864225
#> 
#> $pvalue
#> function (u0) 
#> {
#>     u0 <- as.vector(Vec3(u0))
#>     f <- as.numeric((t1 - u0 %*% tBar %*% u0) * (n - 1)/(1 - 
#>         t1))
#>     1 - stats::pf(f, 2, 2 * n - 2)
#> }
#> <bytecode: 0x55c79da97150>
#> <environment: 0x55c79da8f8f8>
#> 

r$pvalue(Line(60, 10))
#> [1] 2.499618e-08