shotGroups (version 0.7.1)

getHoytParam: Determine parameters q and omega of the Hoyt distribution

Description

Determines the Hoyt distribution's shape parameter q and scale parameter omega from the eigenvalues of a (2 x 2)-covariance matrix.

Usage

getHoytParam(x)
"getHoytParam"(x)
"getHoytParam"(x)
"getHoytParam"(x)
"getHoytParam"(x)

Arguments

x

one of the following: a (2 x 2)-covariance matrix, a list of (2 x 2)-covariance matrices, a data frame with either the variables X, Y or Point.X, Point.Y, a 2-vector with eigenvalues.

Value

A list with the following components:

q

A vector with values of the shape parameter q.

omega

A vector with values of the scale parameter omega.

Details

The parameters q and omega derive from the eigenvalues ev1, ev2 of the covariance matrix of the bivariate normal distribution as follows: q = 1 / sqrt(((ev1+ev2)/ev2) - 1) and omega = ev1 + ev2.

If x is a data frame, its sample covariance matrix is used to estimate the eigenvalues. Note that the Hoyt distribution is only approximately valid for large samples if estimated parameters are used.

References

Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26(2), 318-359.

http://reference.wolfram.com/mathematica/ref/HoytDistribution.html

See Also

Hoyt

Examples

## q and omega based on coordinates in a data frame
getHoytParam(DFscar17)

## q and omega based on a covariance matrix
cm1 <- cbind(c(8, 0), c(0, 2))
getHoytParam(cm1)

## q and omega based on a list of covariance matrices
cm2 <- cbind(c(6, 0), c(0, 4))
cmL <- list(cm1, cm2)
getHoytParam(cmL)

## q and omega based on eigenvalues
ev <- eigen(cm1)$values
getHoytParam(cm1)