vmideal {vismeteor}

Visual magnitude distribution of ideal distributed meteor magnitudes

Description

Density, distribution function, quantile function and random generation for the visual magnitude distribution of ideal distributed 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)

Arguments

m

integer; visual meteor magnitude.

lm

numeric; limiting magnitude.

psi

numeric; the location parameter of a probability distribution. 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

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

{\displaystyle P[M = m] \sim g(m) \, \int_{m-0.5}^{m+0.5} f(m) \, \, \mathrm{d}m} \, \mathrm{,}

where g(m) is the perception probability. This distribution is thus a product of the perception probabilities and the actual ideal distribution of the meteor magnitudes.

If the perception probabilities function perception.fun is given, it must have the signature ⁠function(M)⁠ and must return the perception probabilities of the difference M between the limiting magnitude and the meteor magnitude. If m >= 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

dvmideal gives the density, pvmideal gives the distribution function, qvmideal gives the quantile function, and rvmideal generates random deviates. cvmideal gives the partial convolution of the ideal meteor magnitude distribution with the perception probabilities.

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

Since the distribution is discrete, qvmideal and rvmideal always return integer values. qvmideal can return NaN value with a warning.

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)'
)