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

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$magn))
  magnitudes
})
head(magnitudes[magnitudes$Freq>0,], 5) # Example values

we obtain a data frame with the absolute observed frequencies Freq for each observation of a magnitude class:

This data frame contains a total of 97 meteors. This is a sufficiently large number to estimate the parameter.

Maximum Likelihood Method

The maximum likelihood method can be used to estimate the parameter in an unbiased manner. For this, the function dvmideal() is needed, which returns the probability density of the observable meteor magnitudes when the parameter and the limiting magnitudes are known.

The following algorithm estimates the parameter by maximizing the likelihood with the optim() function. The function ll returns the negative log-likelihood, as optim() identifies a minimum. The expression subset(magnitudes, (magnitudes$lim.magn - magnitudes$magn) > -0.5 ensures that meteors fainter than the limiting magnitude are not used if they exist.

# maximum likelihood estimation (MLE) of psi
result <- with(subset(magnitudes, (magnitudes$lim.magn - magnitudes$magn) > -0.5), {
    # log likelihood function
    ll <- function(psi) -sum(Freq * dvmideal(magn, lim.magn, psi, log=TRUE))
    psi.start <- 5.0 # starting value
    psi.lower <- 0.0 # lowest expected value
    psi.upper <- 10.0 # highest expected value
    # find minimum
    optim(psi.start, ll, method='Brent', lower=psi.lower, upper=psi.upper, hessian=TRUE)
})

This gives the expected value and the variance of the parameter:

psi.mean <- result$par # mean of psi
print(psi.mean)
#> [1] 6.118843
psi.var <- 1/result$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))