vmgeomVst: Variance-Stabilizing Transformation for Geometric Visual...

In vismeteor: Analysis of Visual Meteor Data

Variance-Stabilizing Transformation for Geometric Visual Meteor Magnitudes

Description

Applies a variance-stabilizing transformation to visual meteor magnitudes under the geometric model.

Usage

vmgeomVstFromMagn(m, lm)

vmgeomVstToR(tm, log = FALSE, deriv.degree = 0L)

Arguments

m	integer; meteor magnitude.
lm	numeric; limiting magnitude.
tm	numeric; transformed magnitude.
log	logical; if TRUE, the logarithm of the population index r is returned.
deriv.degree	integer; the order of the derivative at tm to return instead of r or log(r). Must be 0, 1, or 2.

Details

Many linear models require the variance of visual meteor magnitudes to be homoscedastic. The function vmgeomVstFromMagn applies a transformation that produces homoscedastic distributions of visual meteor magnitudes if the underlying distribution follows a geometric model.

The geometric model of visual meteor magnitudes depends on the population index r and the limiting magnitude lm, resulting in a two-parameter distribution. Without detection probabilities, the magnitude distribution is purely geometric, and for integer limiting magnitudes the variance depends only on the population index r. 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 r using the function vmgeomVstToR.

A key advantage of this transformation is that the limiting magnitude lm is already incorporated into subsequent analyses. In this sense, the transformation acts as a normalization of meteor magnitudes and yields a variance close to 1.0.

This transformation is valid for 1.4 \le r \le 3.5. 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.

Value

• vmgeomVstFromMagn: numeric value, the transformed meteor magnitude.

• vmgeomVstToR: numeric value of the population index r, derived from the mean of tm.

The argument deriv.degree can be used to apply the delta method. If log = TRUE, the logarithm of r is returned.

See Also

vmgeom

Examples

N <- 100
r <- 2.0
limmag <- 6.3

# Simulate magnitudes
m <- rvmgeom(N, limmag, r)

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

# Estimator for r from the transformed mean
r.hat  <- vmgeomVstToR(tm.mean)

# Derivative dr/d(tm) at tm.mean (needed for the delta method)
dr_dtm <- vmgeomVstToR(tm.mean, deriv.degree = 1L)

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

# Delta method: variance and standard error of r.hat
var_r.hat <- (dr_dtm^2) * var_tm.mean
se_r.hat  <- sqrt(var_r.hat)

# Results
print(r.hat)
print(se_r.hat)

vismeteor documentation built on Sept. 9, 2025, 5:38 p.m.