# Variance-stabilizing Transformation for the Ideal Distribution of Visual Meteor Magnitudes Applies a variance-stabilizing transformation to meteor magnitudes under the assumption of the ideal magnitude distribution. ```r vmidealVstFromMagn(m, lm) vmidealVstToPsi(tm, lm, deriv.degree = 0L) ``` ## 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`. ## Returns * `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. ## 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. ## Note The internal approximations used here are derived from the perception probabilities produced by vmperception . For details on the derivation, see the script `inst/derivation/vmideal_vst.R` in the package's source code. ## Examples ```r 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) ``` ## See Also vmideal mideal vmperception

vmidealVst function