Parameter Estimation of the Ideal Distribution of Visual Meteor Magnitudes

Introduction

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 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 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$magn))
  subset(magnitudes, (magnitudes$lim.magn - magnitudes$magn) > -0.5)
})
head(magnitudes[magnitudes$Freq>0,], 5) # Example values

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

	magn.id	lim.magn	magn	Freq
9	225413	5.30	4	1.0
11	225413	5.30	1	2.0
14	225413	5.30	3	3.0
15	225432	5.95	4	2.0
17	225432	5.95	3	1.5

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.

# maximum likelihood estimation (MLE) of psi
result <- with(magnitudes, {
    # log likelihood function
    ll <- function(psi) -sum(Freq * dvmideal(magn, lim.magn, psi, log=TRUE))
    psi.start <- 6.0 # starting value
    psi.lower <- 4.0 # 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$par # mean of psi
print(psi.mean)
#> [1] 6.116217
psi.var <- 1/result$hessian[1][1] # variance of psi
print(psi.var)
#> [1] 0.3335565

We can additionally visualize the likelihood function here.

with(new.env(), {
  data.plot <- data.frame(psi = seq(4.0, 11, 0.1))
  data.plot$ll <- mapply(function(psi){
    with(magnitudes, {
      # log likelihood function
      sum(Freq * dvmideal(magn, lim.magn, psi, log=TRUE))
    })
  }, data.plot$psi)
  data.plot$l <- exp(data.plot$ll - max(data.plot$ll))
  data.plot$l <- data.plot$l / sum(data.plot$l)
  plot(data.plot$psi, data.plot$l,
    type = "l",
    col = "blue",
    xlab = "psi",
    ylab = "likelihood"
  )
  abline(v = result$par, col = "red", lwd = 1)
})

It is clearly visible that the likelihood function is not normally distributed. This distribution even belongs to the class of heavy-tailed distributions. While its maximum is indeed an unbiased estimator, this does not hold for the variance. This is important in this context because the variance of the estimated $\psi$-value is derived from the curvature (the second derivative at the maximum) of the log-likelihood function. Therefore, the estimator for the variance of $\psi$ is far too small.

Variance-Stabilizing Transformation as an Alternative Method

Estimation based on the maximum likelihood principle is computationally demanding. As an alternative, a variance-stabilizing transformation can be applied. This transformation maps meteor magnitudes onto a different scale, yielding a distribution whose variance no longer depends on the parameter $\psi$.

The variance-stabilizing transformation has the following additional advantages:

Since the variance does not depend on $\psi$, the accuracy of estimators (e.g., maximum likelihood estimators) is easier to assess,
The variance bound (Cramér–Rao lower bound) then often depends only on the sample size n, but not on the true value of $\psi$,
Estimators such as the sample mean are homoscedastic, i.e., their dispersion is constant across the parameter space. As a result, many classical results of linear estimation theory (BLUE properties, Gauss–Markov theorem) hold without additional transformations,
Because the variance is fixed, the calculation of standard errors is independent of the unknown value of $\psi$,
Confidence intervals can be more easily standardized and are equally well calibrated across the entire parameter space,
Test statistics (e.g., likelihood ratio or score tests) exhibit a more uniform distribution, since the variance does not need to be treated as an additional unknown. This improves test power and simplifies asymptotic approximations.

The resulting procedure is straightforward: it suffices to compute the mean of the transformed meteor magnitudes, from which an estimate of the parameter $\psi$ is obtained.

tm.mean <- with(magnitudes, {
  N <- sum(Freq)
  tm <- vmidealVstFromMagn(magn, lim.magn)
  tm.mean <- sum(Freq * tm)/N
  tm.var <- sum(Freq * (tm - tm.mean)^2)/(N-1)
  tm.mean.var <- tm.var / N
  list('val' = tm.mean, 'var' = tm.mean.var, 'sd' = sqrt(tm.mean.var))
})

Thus, one obtains the mean and the variance of the mean of tm.

print(paste('tm mean:', tm.mean$val))
#> [1] "tm mean: 0.0878019525445456"
print(paste('tm var:', tm.mean$var))
#> [1] "tm var: 0.00836015180816983"

Using the bootstrap method, it can be assessed whether the mean is normally distributed.

tm.means <- with(magnitudes, {
    N <- sum(Freq)
    tm <- vmidealVstFromMagn(magn, lim.magn)
    replicate(50000, {
      mean(sample(tm, size = N, replace = TRUE, prob = Freq))
    })
})

The graphical representation indicates that this is indeed approximately the case.

with(new.env(), {
  tm.min <- tm.mean$val - 3 * tm.mean$sd
  tm.max <- tm.mean$val + 3 * tm.mean$sd
  tm.means <- subset(tm.means, tm.means > tm.min & tm.means < tm.max)
  brks <- seq(min(tm.means) - 0.02, max(tm.means) + 0.02, by = 0.02)
  hist(tm.means,
    breaks = brks,
    col = "skyblue",
    border = "black",
    main = "Histogram of mean tm",
    xlab = "tm",
    ylab = "count",
    xaxt = "n"
  )
  axis(1, at = seq(round(min(brks), 1), round(max(brks), 1) + 0.1, by = 0.1))
  abline(v = 0, col = "red", lwd = 1)
})

A mean value of 0.0 implies that $\psi$ lies at infinity. Negative values can be interpreted as a kind of “beyond infinity”, which is not meaningful. There are two possible explanations:

The distribution is not ideal, i.e., the observations do not fit the model as described above.
Random variation led to this result.

Although the expected value is clearly above 0.0 and can therefore be used, it is more appropriate in this context to interpret the value as the median:

lim.magn.mean <- with(magnitudes, {
    N <- sum(Freq)
    sum(Freq * lim.magn)/N
})
print(paste('lim.magn.mean:', lim.magn.mean))
#> [1] "lim.magn.mean: 5.65752577319588"
print(paste('mean psi:', vmidealVstToPsi(tm.mean$val, lim.magn.mean)))
#> [1] "mean psi: 7.00387357931987"

The mean limiting magnitude lim.magn.mean is required to estimate the location parameter $\psi$. In visual meteor observations, different limiting magnitudes usually occur. In practice, the observed quantity is the difference between $\psi$ and the limiting magnitude for each observation. Since $\psi$ is unknown, one must assume that the mean tm is correlated with the limiting magnitude.

If this correlation can be described by a simple linear regression, the mean limiting magnitude serves as a reliable reference point for estimating $\psi$. This is typically the case when the limiting magnitudes of the individual observations differ only slightly, or $\psi$ is smaller than or equal to the limiting magnitudes.

By contrast, if $\psi$ is significantly larger than the limiting magnitude, estimation becomes problematic: $\psi$ effectively tends to infinity, and the observable magnitude distribution approaches the geometric model of visual meteor magnitudes with a population index of $r \approx 2.5$.

Nevertheless, if all observations share the same limiting magnitude, the procedure yields an exact estimate.

In practice, however, it is preferable to use a confidence interval estimate. For example, one can estimate that $\psi$ is, with 10 percent probability, not smaller than:

print(vmidealVstToPsi(qnorm(0.90, tm.mean$val, tm.mean$sd), lim.magn.mean))
#> [1] 5.827726

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:

psi.mean <- vmidealVstToPsi(tm.mean$val, lim.magn.mean)
magnitudes$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 
#>  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

Next, we check which magnitude classes need to be aggregated 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+ 
#> 11.64866 12.83943 19.87342 23.30279 29.33570

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] 0.5258994

If we set the level of significance at 5 percent, then it is clear that the p-value with 0.5258994 is greater than 0.05. Thus, under the assumption that the magnitude distribution follows the ideal meteor magnitude distribution and that the perception probabilities are correct (i.e., error-free or precisely known), these 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))

Parameter Estimation of the Ideal Distribution of Visual Meteor Magnitudes

2025-09-07

Introduction

Maximum Likelihood Method

Variance-Stabilizing Transformation as an Alternative Method

Residual Analysis