Directional (version 2.2)

Check visually whether matrix Fisher samples is correctly generated or not: Check visually whether matrix Fisher samples is correctly generated or not.

Description

It plots the log probability trace of matrix Fisher distribution which should close to the maximum value of the logarithm of matrix Fisher distribution, if samples are correctly generated.

Usage

visual.check(n, F)

Arguments

n

The sample size.

F

An arbitrary 3 x3 matrix represents the parameter matrix of this distribution.

Value

A plot which shows log probability trace of matrix Fisher distribution.

Details

For a given parameter matrix F, maximum value of the logarithm of matrix Fisher distribution is calculated via the form of singular value decomposition of $F = U \Lambda V^T$ which is $tr(\Lambda)$. Multiply the last column of $U$ by $-1$ and replace small eigenvalue, say, $\lambda_3$ by $-\lambda_3$ if $\left| UV^T\right| = -1$.

References

Habeck M. (2009). Generation of three-dimensional random rotations in fitting and matching problems. Computational Statistics, 24(4):719--731.

Examples

F <- 10^(-1) * matrix( c(85, 11, 41, 78, 39, 60, 43, 64, 48), ncol = 3)

# Here det(UV^{T}) is negative, so small eigenvalue is multiplied by -1.
# The maximum value in this case is (16.17 + 4.80 - 0.57) = 20.4

values <- visual.check(1000, F)

l1 <- values$l1
l2 <- values$l2

l1values <- l1[ seq(1, length(l1), by = 100) ]   # Habeck's method
l2values <- l2[ seq(1, length(l2), by = 100) ]   # Kent's method

plot( 1:length(l1values), l1values, type = "l", ylim = c( min(l1values, l2values ) -0.5,
max(l1values, l2values) + 0.5), col = "red" )
lines( 1:length(l2values), l2values, type = "l", col = "blue" )
abline(h = 20.4, lty = "solid")