vismeteor (version 2.0.1)

vmideal: Ideal Distribution of Visual Meteor Magnitudes

Description

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

Usage

dvmideal(m, lm, psi, log = FALSE, perception.fun = NULL)
pvmideal(m, lm, psi, lower.tail = TRUE, log = FALSE, perception.fun = NULL)
qvmideal(p, lm, psi, lower.tail = TRUE, perception.fun = NULL)
rvmideal(n, lm, psi, perception.fun = NULL)
cvmideal(lm, psi, log = FALSE, perception.fun = NULL)

Value

• dvmideal: density

• pvmideal: distribution function

• qvmideal: quantile function

• rvmideal: random generation

• cvmideal: partial convolution of the ideal distribution of meteor magnitudes with the perception probabilities.

The length of the result is determined by n for rvmideal, and is 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, qvmideal and rvmideal always return integer values. qvmideal may return NaN with a warning.

Arguments

m

integer; visual meteor magnitude.

lm

numeric; limiting magnitude.

psi

numeric; the location parameter of the probability distribution.

log

logical; if TRUE, probabilities are returned as log(p).

perception.fun

function; optional perception probability function. The 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

The density of the ideal distribution of meteor magnitudes is $$ {\displaystyle f(m) = \frac{\mathrm{d}p}{\mathrm{d}m} = \frac{3}{2} \, \log(r) \sqrt{\frac{r^{3 \, \psi + 2 \, m}}{(r^\psi + r^m)^5}}} $$ where \(m\) is the meteor magnitude, \(r = 10^{0.4} \approx 2.51189 \dots\) is a constant, and \(\psi\) is the only parameter of this magnitude distribution.

In visual meteor observations, magnitudes are usually estimated as integer values. Hence, this distribution is discrete and its probability mass function is given by $$ P[M = m] \sim g(m) \, \int_{m-0.5}^{m+0.5} f(m) \, \mathrm{d}m \, , $$ where \(g(m)\) denotes the perception probability. Thus, the distribution is the product of the perception probabilities and the underlying ideal distribution of meteor magnitudes.

If a perception probability function perception.fun is supplied, it must have the signature function(x) and 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 a perception probability of 1.0. The argument perception.fun is resolved using match.fun.

References

Richter, J. (2018) About the mass and magnitude distributions of meteor showers. WGN, Journal of the International Meteor Organization, vol. 46, no. 1, p. 34-38

See Also

mideal vmperception

Examples

N <- 100
psi <- 5.0
limmag <- 6.5
(m <- seq(6, -4))

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

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

# maximum likelihood estimation (MLE) of psi
est <- optim(2, lld, method='Brent', lower=0, upper=8, hessian=TRUE)

# estimations
est$par # mean of psi

# generate random meteor magnitudes
m <- rvmideal(N, limmag, psi)

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

# maximum likelihood estimation (MLE) of psi
est <- optim(2, llr, method='Brent', lower=0, upper=8, hessian=TRUE)

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

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

plot(
    function(lm) vismeteor::cvmideal(lm, psi, log = TRUE),
    -5, 10,
    main = paste0(
        'Partial convolution of the ideal meteor magnitude distribution\n',
        'with the perception probabilities (psi = ', psi, ')'
    ),
    col = "blue",
    xlab = 'lm',
    ylab = 'log(rate)'
)