vmgeom: 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)

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. All arguments are vectorized; standard R recycling rules apply.

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

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. The argument perception.fun is resolved using match.fun.

Examples

Run this code

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))