| vmgeomVst | R Documentation |
Applies a variance-stabilizing transformation to visual meteor magnitudes under the geometric model.
vmgeomVstFromMagn(m, lm)
vmgeomVstToR(tm, log = FALSE, deriv.degree = 0L)
m |
integer; meteor magnitude. |
lm |
numeric; limiting magnitude. |
tm |
numeric; transformed magnitude. |
log |
logical; if |
deriv.degree |
integer; the order of the derivative at |
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.
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.
vmgeom
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)
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