# Parameter Estimation of the Ideal Magnitude Distribution

## Introduction

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 function. This distribution is thus a product of the perception probabilities and the actual ideal distribution of the meteor magnitudes.

Here we demonstrate a method for an unbiased estimation of $$\psi$$.

First, we obtain some magnitude observations from the example data set, which also includes the limiting magnitude.

observations <- with(PER_2015_magn$observations, { idx <- !is.na(lim.magn) & sl.start > 135.81 & sl.end < 135.87 data.frame( magn.id = magn.id[idx], lim.magn = lim.magn[idx] ) }) head(observations, 5) # Example values magn.id lim.magn 225413 5.30 225432 5.95 225438 6.01 225449 6.48 225496 5.50 Next, the observed meteor magnitudes are matched with the corresponding observations. This is necessary as we need the limiting magnitudes of the observations to determine the parameter. Using magnitudes <- with(new.env(), { magnitudes <- merge( observations, as.data.frame(PER_2015_magn$magnitudes),
by = 'magn.id'
)
magnitudes$magn <- as.integer(as.character(magnitudes$magn))
magnitudes
})
print(psi.mean)
#> [1] 6.118843
psi.var <- 1/result$hessian[1][1] # variance of r print(psi.var) #> [1] 0.3244365 ## Residual Analysis So far, we have operated under the assumption that the real distribution of meteor magnitudes is exponential and that the perception probabilities are accurate. We now use the Chi-Square goodness-of-fit test to check whether the observed frequencies match the expected frequencies. Then, using the estimated parameter, we retrieve the relative frequencies p for each observation and add them to the data frame magnitudes: magnitudes$p <- with(magnitudes, dvmideal(m = magn, lm = lim.magn, psi.mean))

We must also consider the probabilities for the magnitude class with the brightest meteors.

magn.min <- min(magnitudes$magn) The smallest magnitude class magn.min is -6. In calculating the probabilities, we assume that the magnitude class -6 contains meteors that are either brighter or equally bright as -6 and thus use the function pvmideal() to determine their probability. idx <- magnitudes$magn == magn.min
magnitudes$p[idx] <- with( magnitudes[idx,], pvmideal(m = magn + 1L, lm = lim.magn, psi.mean, lower.tail = TRUE) ) This ensures that the probability of observing a meteor of any given magnitude is 100%. This is known as the normalization condition. Accordingly, the Chi-Square goodness-of-fit test will fail if this condition is not met. We now create the contingency table magnitutes.observed for the observed meteor magnitudes and its margin table. magnitutes.observed <- xtabs(Freq ~ magn.id + magn, data = magnitudes) magnitutes.observed.mt <- margin.table(magnitutes.observed, margin = 2) print(magnitutes.observed.mt) #> magn #> -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 #> 0.0 0.0 0.0 0.0 3.0 4.0 7.0 10.0 23.0 26.5 20.0 3.0 0.5 0.0 Next, we check which magnitude classes need to be aggegated so that each contains at least 10 meteors, allowing us to perform a Chi-Square goodness-of-fit test. The last output shows that meteors of magnitude class 0 or brighter must be combined into a magnitude class 0-. Meteors with a brightness less than 4 are grouped here in the magnitude class 4+, and a new contingency table magnitudes.observed is created: magnitudes$magn[magnitudes$magn <= 0] <- '0-' magnitudes$magn[magnitudes$magn >= 4] <- '4+' magnitutes.observed <- xtabs(Freq ~ magn.id + magn, data = magnitudes) print(margin.table(magnitutes.observed, margin = 2)) #> magn #> 0- 1 2 3 4+ #> 14.0 10.0 23.0 26.5 23.5 We now need the corresponding expected relative frequencies magnitutes.expected <- xtabs(p ~ magn.id + magn, data = magnitudes) magnitutes.expected <- magnitutes.expected/nrow(magnitutes.expected) print(sum(magnitudes$Freq) * margin.table(magnitutes.expected, margin = 2))
#> magn
#>       0-        1        2        3       4+
#> 12.89177 14.34252 21.58284 23.59981 24.58307

and then carry out the Chi-Square goodness-of-fit test:

chisq.test.result <- chisq.test(
x = margin.table(magnitutes.observed, margin = 2),
p = margin.table(magnitutes.expected, margin = 2)
)

As a result, we obtain the p-value:

print(chisq.test.result\$p.value)
#> [1] 0.7528154

If we set the level of significance at 5 percent, then it is clear that the p-value with 0.7528154 is greater than 0.05. Thus, under the assumption that the magnitude distribution follows an ideal meteor magnitude distribution and assuming that the perception probabilities are correct (i.e., error-free or precisely known), the assumptions cannot be rejected. However, the converse is not true; the assumptions may not necessarily be correct. The total count of meteors here is too small for such a conclusion.

To verify the p-value, we also graphically represent the Pearson residuals:

chisq.test.residuals <- with(new.env(), {
chisq.test.residuals <- residuals(chisq.test.result)
v <- as.vector(chisq.test.residuals)
names(v) <- rownames(chisq.test.residuals)
v
})

plot(
chisq.test.residuals,
main="Residuals of the chi-square goodness-of-fit test",
xlab="m",
ylab="Residuals",
ylim=c(-3, 3),
xaxt = "n"
)
abline(h=0.0, lwd=2)
axis(1, at = seq_along(chisq.test.residuals), labels = names(chisq.test.residuals))