vismeteor (version 2.0.1)

vmidealVst: Variance-stabilizing Transformation for the Ideal Distribution of Visual Meteor Magnitudes

Description

Applies a variance-stabilizing transformation to meteor magnitudes under the assumption of the ideal magnitude distribution.

Usage

.vmidealVstFromMagn.params
vmidealVstFromMagn(m, lm)
vmidealVstToPsi(tm, lm, deriv.degree = 0L)

Value

• vmidealVstFromMagn: a numeric value, the transformed meteor magnitude.

• vmidealVstToPsi: a numeric value of the parameter psi, derived from the mean of tm. The argument deriv.degree can be used to apply the delta method.

Format

An object of class list of length 8.

Arguments

m

integer; the meteor magnitude.

lm

numeric; limiting magnitude.

tm

numeric; transformed magnitude.

deriv.degree

integer; the degree of the derivative at tm to return instead of psi. Must be 0, 1 or 2.

Details

Many linear models require the variance of visual meteor magnitudes to be homoscedastic. The function vmidealVstFromMagn applies a transformation that produces homoscedastic distributions of visual meteor magnitudes if the underlying magnitudes follow the ideal magnitude distribution. In this sense, the transformation acts as a normalization of meteor magnitudes and yields a variance close to 1.0.

The ideal distribution of visual meteor magnitudes depends on the parameter psi and the limiting magnitude lm, resulting in a two-parameter distribution. Without detection probabilities, the magnitude distribution reduces to a pure ideal magnitude distribution, which depends only on the parameter psi. Since the limiting magnitude lm is a fixed parameter and never estimated statistically, the magnitudes can be transformed such that, for example, the mean of the transformed magnitudes directly provides an estimate of psi using the function vmidealVstToPsi.

This transformation is valid for \(-10 \le \texttt{psi} \le 9\). The numerical form of the transformation is version-specific and may change substantially in future releases. Do not rely on equality of transformed values across package versions.

See Also

vmideal mideal vmperception

Examples

N <- 100
psi <- 5.0
limmag <- 6.3

# Simulate magnitudes
m <- rvmideal(N, limmag, psi)

# Variance-stabilizing transformation
tm <- vmidealVstFromMagn(m, limmag)
tm.mean <- mean(tm)
tm.var  <- var(tm)

# Estimator for psi from the transformed mean
psi.hat  <- vmidealVstToPsi(tm.mean, limmag)

# Derivative d(psi)/d(tm) at tm.mean (needed for the delta method)
dpsi_dtm <- vmidealVstToPsi(tm.mean, limmag, deriv.degree = 1L)

# Variance of the sample mean of tm
var_tm.mean <- tm.var / N

# Delta method: variance and standard error of psi.hat
var_psi.hat <- (dpsi_dtm^2) * var_tm.mean
se_psi.hat  <- sqrt(var_psi.hat)

# Results
print(psi.hat)
print(se_psi.hat)