vmgeom: Geometric Model of Visual Meteor Magnitudes

In vismeteor: Analysis of Visual Meteor Data

Geometric Model of Visual Meteor Magnitudes

Description

Density, distribution function, quantile function, and random generation for the geometric model of visual meteor magnitudes.

Usage

dvmgeom(m, lm, r, log = FALSE, perception.fun = NULL)

pvmgeom(m, lm, r, lower.tail = TRUE, log = FALSE, perception.fun = NULL)

qvmgeom(p, lm, r, lower.tail = TRUE, perception.fun = NULL)

rvmgeom(n, lm, r, perception.fun = NULL)

Arguments

m	numeric; the meteor magnitude.
lm	numeric; limiting magnitude.
r	numeric; the population index.
log	logical; if TRUE, probabilities p are given as log(p).
perception.fun	function; perception probability function (optional). Default is vmperception.
lower.tail	logical; if TRUE (default) probabilities are P[M < m], otherwise, P[M \ge m].
p	numeric; probability.
n	numeric; count of meteor magnitudes.

Details

In visual meteor observations, magnitudes are usually estimated as integer values. Hence, this distribution is discrete and its probability mass function is

{\displaystyle P[X = x] \sim f(x) \, \mathrm r^{-x}} \,\mathrm{,}

where x \ge -0.5 denotes the difference between the limiting magnitude lm and the meteor magnitude m, and f(x) is the perception probability function. Thus, the distribution is the product of the perception probabilities and the underlying geometric distribution of meteor magnitudes. Therefore, the parameter p of the geometric distribution is given by p = 1 - 1/r.

The parameter lm specifies the reference for the meteor magnitude m. m must be an integer meteor magnitude. The length of the vector lm must either equal the length of the vector m, or lm must be a scalar value. In the case of rvmgeom, the length of the vector lm must equal n, or lm must be a scalar value.

If a perception probability function perception.fun is provided, it must have the signature ⁠function(x)⁠ and return the perception probability of the difference x between the limiting magnitude and the meteor magnitude. If x >= 15.0, the function perception.fun should return a perception probability of 1.0. If log = TRUE is specified, the logarithm of the perception probabilities must be returned. The argument perception.fun is resolved using match.fun.

Value

• dvmgeom: density

• pvmgeom: distribution function

• qvmgeom: quantile function

• rvmgeom: random generation

The length of the result is determined by n for rvmgeom, and by the maximum of the lengths of the numeric vector arguments for the other functions.

Since the distribution is discrete, qvmgeom and rvmgeom always return integer values. qvmgeom may return NaN with a warning.

See Also

vmperception stats::Geometric

Examples

N <- 100
r <- 2.0
limmag <- 6.5
(m <- seq(6, -7))

# discrete density of `N` meteor magnitudes
(freq <- round(N * dvmgeom(m, limmag, r)))

# log likelihood function
lld <- function(r) {
    -sum(freq * dvmgeom(m, limmag, r, log=TRUE))
}

# maximum likelihood estimation (MLE) of r
est <- optim(2, lld, method='Brent', lower=1.1, upper=4)

# estimations
est$par # mean of r

# generate random meteor magnitudes
m <- rvmgeom(N, r, lm=limmag)

# log likelihood function
llr <- function(r) {
    -sum(dvmgeom(m, limmag, r, log=TRUE))
}

# maximum likelihood estimation (MLE) of r
est <- optim(2, llr, method='Brent', lower=1.1, upper=4, hessian=TRUE)

# estimations
est$par # mean of r
sqrt(1/est$hessian[1][1]) # standard deviation of r

m <- seq(6, -4, -1)
p <- vismeteor::dvmgeom(m, limmag, r)
barplot(
    p,
    names.arg = m,
    main = paste0('Density (r = ', r, ', limmag = ', limmag, ')'),
    col = "blue",
    xlab = 'm',
    ylab = 'p',
    border = "blue",
    space = 0.5
)
axis(side = 2, at = pretty(p))

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