vmgeom {vismeteor}

Visual magnitude distribution of geometric distributed meteor magnitudes

Description

Density, distribution function, quantile function and random generation for the visual magnitude distribution of geometric distributed 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. It is the only parameter of the distribution.

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 observation, it is common to estimate meteor magnitudes in integer values. Hence, this distribution is discrete and has the density

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

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

The parameter lm indicate what the parameter m refers to. m must be an integer meteor magnitude. The length of the vector lm must then be equal to the length of the vector m or lm is a scalar value. In case of rvmgeom, the length of the vector lm must be n or lm is a scalar value.

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

Value

dvmgeom gives the density, pvmgeom gives the distribution function, qvmgeom gives the quantile function, and rvmgeom generates random deviates.

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

Since the distribution is discrete, qvmgeom and rvmgeom always return integer values. qvmgeom can return NaN value 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))